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Angular Momentum and Torque on a Rigid Body
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| <div class="stackedit__html"><h1 id="angular-momentum-and-torque-on-a-rigid-body">Angular Momentum and Torque on a Rigid Body</h1> | |
| <p>Angular momentum of a rigid body varies as the point of summation varies. Also, equipollent torque varies as the point of summation varies also. This posts examines the relationship between change in angular momentum and equipollent torque in the context of Newton’s 2nd law. It is <strike>expected</strike> required that at the center of mass equipollent torque and change in angular momentum at simply equal to each other. But what about when measured at an arbitrary point <strong>A</strong> not at the center of mass <strong>C</strong>.</p> | |
| <p><sub>Note that all quantities are expressed on the same basis vectors, and only reference point being <strong>C</strong> or <strong>A</strong> or <strong>O</strong> the origin describing the properties of whichever particles happens to be passing under this reference point.</sub></p> | |
| <h3 id="particle-forces--torques">Particle Forces & Torques</h3> | |
| <p>The combined loading on the rigid body can be reduced down to an equipollent system of forces <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="bold-italic">F</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.68611em; vertical-align: 0em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.15972em;">F</span></span></span></span></span></span></span> and torques <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">τ</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\tau}_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span> at the reference point <strong>A</strong>. In dynamics it is important to consider the combined torque about the center of mass which is evaluated with the transformation law(s)</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi mathvariant="bold-italic">τ</mi><mi>C</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi mathvariant="bold-italic">τ</mi><mi>A</mi></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">F</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi mathvariant="bold-italic">τ</mi><mi>A</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi mathvariant="bold-italic">τ</mi><mi>C</mi></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">F</mi></mrow></mstyle></mtd></mtr></mtable></mtd><mtd width="50%"></mtd><mtd><mtext>(1)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \begin{aligned} | |
| \boldsymbol{\tau}_C &= \boldsymbol{\tau}_A + (\boldsymbol{r}_A - \boldsymbol{r}_C) \times \boldsymbol{F} \\ \boldsymbol{\tau}_A& = \boldsymbol{\tau}_C + (\boldsymbol{r}_C - \boldsymbol{r}_A) \times \boldsymbol{F} | |
| \end{aligned} \tag{1} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 3em; vertical-align: -1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.75em;"><span class="" style="top: -3.91em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.25em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.75em;"><span class="" style="top: -3.91em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.15972em;">F</span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.15972em;">F</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.25em;"><span class=""></span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height: 3em; vertical-align: -1.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">1</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <table> | |
| <thead> | |
| <tr> | |
| <th align="right"><em>quantity</em></th> | |
| <th align="left"><em>description</em></th> | |
| </tr> | |
| </thead> | |
| <tbody> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="bold-italic">F</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.68611em; vertical-align: 0em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.15972em;">F</span></span></span></span></span></span></span></td> | |
| <td align="left">combined of forces applied on body.</td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">τ</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\tau}_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td align="left">combined torque applied on body about point <strong>A</strong>.</td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">τ</mi><mi>C</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\tau}_C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td align="left">combined torque applied on body about center of mass <strong>C</strong>.</td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{r}_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td align="left">location of point <strong>A</strong> from the origin.</td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{r}_C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td align="left">location of point <strong>C</strong> from the origin.</td> | |
| </tr> | |
| </tbody> | |
| </table><h3 id="center-of-mass">Center of mass</h3> | |
| <p>It is important to define the center of mass not only as the weighted sum of the particle locations</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>=</mo><mfrac><mn>1</mn><mi>m</mi></mfrac><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>m</mi><mi>i</mi></msub><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(3)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \boldsymbol{r}_C = \frac{1}{m} \sum_i m_i \boldsymbol{r}_i \tag{3} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 2.59911em; vertical-align: -1.27767em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.32144em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault">m</span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.686em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height: 2.59911em; vertical-align: -1.27767em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">3</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <p>but also as the relative location of each particle being zero</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>m</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="bold-italic">0</mn></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(4)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \sum_i m_i ( \boldsymbol{r}_i - \boldsymbol{r}_C) = \boldsymbol{0} \tag{4} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 2.32767em; vertical-align: -1.27767em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.64444em; vertical-align: 0em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">0</span></span></span></span><span class="tag"><span class="strut" style="height: 2.32767em; vertical-align: -1.27767em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">4</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <p>These two expressions are equivalent to each other. For simplicity you can define the relative position of each particle as <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">d</mi><mrow><mi>C</mi><mi>i</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{d}_{Ci}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.84444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">d</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>, but at this stage I want to keep things as explicit as possible.</p> | |
| <table> | |
| <thead> | |
| <tr> | |
| <th align="right">quantity</th> | |
| <th align="left">description</th> | |
| <th align="left">definition</th> | |
| </tr> | |
| </thead> | |
| <tbody> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathdefault">m</span></span></span></span></span></td> | |
| <td align="left">combined masss</td> | |
| <td align="left"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mo>∑</mo><mi>i</mi></msub><msub><mi>m</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\sum_i m_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.04971em; vertical-align: -0.29971em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position: relative; top: -5e-06em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.161954em;"><span class="" style="top: -2.40029em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.29971em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">d</mi><mrow><mi>C</mi><mi>i</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{d}_{Ci}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.84444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">d</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td align="left">relative position of particle</td> | |
| <td align="left"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">d</mi><mrow><mi>C</mi><mi>i</mi></mrow></msub><mo>=</mo><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{d}_{Ci} = \boldsymbol{r}_i - \boldsymbol{r}_C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.84444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">d</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.73333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| </tr> | |
| </tbody> | |
| </table><h3 id="momenta-definitions">Momenta Definitions</h3> | |
| <p>Momenta are <em>defined</em> by summing up the following contributions of all the particles that move together on a rigid body. This summation happens an at arbitrary location <strong>A</strong> in space <em>at every instance</em> in time.</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi mathvariant="bold-italic">p</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>m</mi><mi>i</mi></msub><msub><mi mathvariant="bold-italic">v</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi mathvariant="bold-italic">L</mi><mi>A</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo>∑</mo><mi>i</mi></munder><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><msub><mi>m</mi><mi>i</mi></msub><msub><mi mathvariant="bold-italic">v</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr></mtable></mtd><mtd width="50%"></mtd><mtd><mtext>(2)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \begin{aligned} | |
| \boldsymbol{p} & = \sum_i m_i \boldsymbol{v}_i \\ | |
| \boldsymbol{L}_A & = \sum_i (\boldsymbol{r}_i - \boldsymbol{r}_A) \times m_i \boldsymbol{v}_i | |
| \end{aligned} \tag{2} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 5.25535em; vertical-align: -2.37767em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.87767em;"><span class="" style="top: -4.87767em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span></span></span><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 2.37767em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.87767em;"><span class="" style="top: -4.87767em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 2.37767em;"><span class=""></span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height: 5.25535em; vertical-align: -2.37767em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">2</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <p>where:</p> | |
| <table> | |
| <thead> | |
| <tr> | |
| <th align="right"><em>quantity</em></th> | |
| <th><em>description</em></th> | |
| </tr> | |
| </thead> | |
| <tbody> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="bold-italic">p</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{p}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.63888em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span></span></span></span></span></td> | |
| <td><strike>linear</strike> translational momentum vector.</td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">L</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{L}_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83611em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td><strike>angular</strike> rotational momentum vector of the body measured at <strong>A</strong>.</td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>m</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">m_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td>infinitesimal mass of each particle such that the total mass is <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>m</mi><mo>=</mo><msub><mo>∑</mo><mi>i</mi></msub><msub><mi>m</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">m = \sum_i m_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathdefault">m</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1.04971em; vertical-align: -0.29971em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position: relative; top: -5e-06em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.161954em;"><span class="" style="top: -2.40029em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.29971em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>.</td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{r}_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td>position vector of the particle from the origin.</td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td>velocity vector of the particle.</td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{r}_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td>position vector of arbitrary point <strong>A</strong> from the origin.</td> | |
| </tr> | |
| </tbody> | |
| </table><p>Note that the point <strong>A</strong> might be fixed in space over time, or moving with constant velocity or riding on the body. At any time frame it might have non-zero velocity <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span> with respect to the origin.</p> | |
| <h3 id="kinematics">Kinematics</h3> | |
| <p>The motion of each particle can be decomposed into the motion of the center of mass and a rotation about the center of mass (Chasle’s Theorem).</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>i</mi></msub><mo>=</mo><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub><mo>+</mo><mi mathvariant="bold-italic">ω</mi><mo>×</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo stretchy="false">)</mo></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(3)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \boldsymbol{v}_i = \boldsymbol{v}_C + \boldsymbol{\omega} \times (\boldsymbol{r}_i - \boldsymbol{r}_C) \tag{3} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.73333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.66666em; vertical-align: -0.08333em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">ω</span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span><span class="tag"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">3</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <p>Additionally the combined motion of the center of mass point <strong>C</strong> is expressed on the reference point <strong>A</strong> using the following transformation law(s)</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi mathvariant="bold-italic">v</mi><mi>A</mi></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">ω</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi mathvariant="bold-italic">v</mi><mi>A</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">ω</mi></mrow></mstyle></mtd></mtr></mtable></mtd><mtd width="50%"></mtd><mtd><mtext>(4)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \begin{aligned} \boldsymbol{v}_C &= \boldsymbol{v}_A + (\boldsymbol{r}_A - \boldsymbol{r}_C) \times \boldsymbol{\omega} \\ \boldsymbol{v}_A &= \boldsymbol{v}_C + (\boldsymbol{r}_C - \boldsymbol{r}_A) \times \boldsymbol{\omega} | |
| \end{aligned} \tag{4} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 3em; vertical-align: -1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.75em;"><span class="" style="top: -3.91em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.25em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.75em;"><span class="" style="top: -3.91em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">ω</span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">ω</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.25em;"><span class=""></span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height: 3em; vertical-align: -1.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">4</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <p>Any resemblence of the above to (1) is not coincidence.</p> | |
| <h3 id="translational-momentum">Translational Momentum</h3> | |
| <p>Translational momentum is the easiest to evaluate from (2) and (3). Center of mass expression (4) is used for simplification below</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi mathvariant="bold-italic">p</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>m</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub><mo>+</mo><mi mathvariant="bold-italic">ω</mi><mo>×</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">(</mo><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>m</mi><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub><mo>+</mo><mi mathvariant="bold-italic">ω</mi><mo>×</mo><menclose notation="updiagonalstrike"><mrow><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>m</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo stretchy="false">)</mo></mrow></menclose></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex"> \begin{aligned} | |
| \boldsymbol{p} & = \sum_i m_i ( \boldsymbol{v}_C + \boldsymbol{\omega} \times ( \boldsymbol{r}_i - \boldsymbol{r}_C)) \\ | |
| & = (\sum_i m_i) \boldsymbol{v}_C + \boldsymbol{\omega} \times \cancel{ \sum_i m_i (\boldsymbol{r}_i - \boldsymbol{r}_C) } | |
| \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 5.25535em; vertical-align: -2.37767em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.87767em;"><span class="" style="top: -4.87767em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span></span></span><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 2.37767em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.87767em;"><span class="" style="top: -4.87767em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">ω</span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose">)</span></span></span><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mopen">(</span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">ω</span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord cancel-lap"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -4.32767em;"><span class="pstrut" style="height: 4.32767em;"></span><span class="mord cancel-pad"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span><span class="svg-align" style="top: -3.05001em;"><span class="pstrut" style="height: 4.32767em;"></span><span class="" style="height: 2.32767em;"><svg width="100%" height="2.327674em"><line x1="0" y1="100%" x2="100%" y2="0" stroke-width="0.046em"></line></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 2.37767em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></span></p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi mathvariant="bold-italic">p</mi><mo>=</mo><mi>m</mi> <msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(5)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \boldsymbol{p} = m\,\boldsymbol{v}_C \tag{5} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.63888em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord mathdefault">m</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">5</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <p>The is a well known result, undesputed as long as <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="bold-italic">p</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{p}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.63888em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span></span></span></span></span> and <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span> are measued form <em>the same coordinate frame</em>. Again, <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span> is the velocity vector of the center of mass, as in</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub><mo>=</mo><mstyle displaystyle="false" scriptlevel="0"><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></mstyle><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub></mrow><annotation encoding="application/x-tex"> \boldsymbol{v}_C = \tfrac{\rm d}{{\rm d}t} \boldsymbol{r}_C </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1.22511em; vertical-align: -0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></p> | |
| <h3 id="rotational-momentum-about-center-of-mass">Rotational Momentum about Center of Mass</h3> | |
| <p>Rotational momentum is also evaluated from (2) and (3) but we start with summing about the center of mass first</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi mathvariant="bold-italic">L</mi><mi>C</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo>∑</mo><mi>i</mi></munder><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo stretchy="false">)</mo><mo>×</mo><msub><mi>m</mi><mi>i</mi></msub><msub><mi>v</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>m</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub><mo>+</mo><mi mathvariant="bold-italic">ω</mi><mo>×</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><menclose notation="updiagonalstrike"><mrow><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>m</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo stretchy="false">)</mo></mrow></menclose><mo>×</mo><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub><mo>+</mo><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>m</mi><mi>i</mi></msub><msub><mi mathvariant="bold-italic">d</mi><mrow><mi>C</mi><mi>i</mi></mrow></msub><mo>×</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">ω</mi><mo>×</mo><msub><mi mathvariant="bold-italic">d</mi><mrow><mi>C</mi><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo>∑</mo><mi>i</mi></munder><mrow><mo fence="true">[</mo><mo>−</mo><msub><mi>m</mi><mi>i</mi></msub><msub><mi mathvariant="bold-italic">d</mi><mrow><mi>C</mi><mi>i</mi></mrow></msub><mo>×</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">d</mi><mrow><mi>C</mi><mi>i</mi></mrow></msub><mo>×</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo fence="true">]</mo></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex"> \begin{aligned} \boldsymbol{L}_C & = \sum_i (\boldsymbol{r}_i - \boldsymbol{r}_C) \times m_i v_i \\ | |
| & = \sum_i m_i ( \boldsymbol{r}_i - \boldsymbol{r}_C) \times ( \boldsymbol{v}_C + \boldsymbol{\omega} \times ( \boldsymbol{r}_i - \boldsymbol{r}_C)) \\ | |
| & = \cancel{ \sum_i m_i (\boldsymbol{r}_i - \boldsymbol{r}_C) } \times \boldsymbol{v}_C + \sum_i m_i \boldsymbol{d}_{Ci} \times ( \boldsymbol{\omega} \times \boldsymbol{d}_{Ci} ) \\ | |
| & = \sum_i \left[ - m_i \boldsymbol{d}_{Ci} \times ( \boldsymbol{d}_{Ci} \times \boldsymbol{\omega}) \right] \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 10.5107em; vertical-align: -5.00535em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 5.50535em;"><span class="" style="top: -7.50535em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -4.87767em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"></span></span><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"></span></span><span class="" style="top: 0.377674em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 5.00535em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 5.50535em;"><span class="" style="top: -7.50535em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -4.87767em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">ω</span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose">)</span></span></span><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord cancel-lap"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -4.32767em;"><span class="pstrut" style="height: 4.32767em;"></span><span class="mord cancel-pad"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span><span class="svg-align" style="top: -3.05001em;"><span class="pstrut" style="height: 4.32767em;"></span><span class="" style="height: 2.32767em;"><svg width="100%" height="2.327674em"><line x1="0" y1="100%" x2="100%" y2="0" stroke-width="0.046em"></line></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">d</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">ω</span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">d</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span><span class="" style="top: 0.377674em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="minner"><span class="mopen delimcenter" style="top: 0em;">[</span><span class="mord">−</span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">d</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">d</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">ω</span></span></span><span class="mclose">)</span><span class="mclose delimcenter" style="top: 0em;">]</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 5.00535em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></span></p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><msub><mi mathvariant="bold-italic">L</mi><mi>C</mi></msub><mo>=</mo><msub><mi mathvariant="bold">I</mi><mi>C</mi></msub><mi mathvariant="bold-italic">ω</mi></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(6)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \boldsymbol{L}_C = \mathbf{I}_C \boldsymbol{\omega} \tag{6} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83611em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.83611em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">ω</span></span></span></span><span class="tag"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">6</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <p>where <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold">I</mi><mi>C</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{I}_C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83611em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span> is the 3×3 mass moment of inertia tensor (inertia dyatic) derived from factoring <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="bold-italic">ω</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{\omega}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.44444em; vertical-align: 0em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">ω</span></span></span></span></span></span></span> from the rotational inertia expression above.</p> | |
| <p>Practically this is done by evaluating the following sum. For simplicity consider each relative position vector having components <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">d</mi><mrow><mi>C</mi><mi>i</mi></mrow></msub><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mi>i</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>y</mi><mi>i</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>z</mi><mi>i</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\boldsymbol{d}_{Ci} = \begin{pmatrix} x_i \\ y_i \\ z_i \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.84444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">d</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right: 0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.04398em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span> and evaluate</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><msub><mi mathvariant="bold">I</mi><mi>C</mi></msub><mo>=</mo><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>m</mi><mi>i</mi></msub><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msubsup><mi>y</mi><mi>i</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>z</mi><mi>i</mi><mn>2</mn></msubsup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>x</mi><mi>i</mi></msub><msub><mi>y</mi><mi>i</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>x</mi><mi>i</mi></msub><msub><mi>z</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>x</mi><mi>i</mi></msub><msub><mi>y</mi><mi>i</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>z</mi><mi>i</mi><mn>2</mn></msubsup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>y</mi><mi>i</mi></msub><msub><mi>z</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>x</mi><mi>i</mi></msub><msub><mi>z</mi><mi>i</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>y</mi><mi>i</mi></msub><msub><mi>z</mi><mi>i</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>y</mi><mi>i</mi><mn>2</mn></msubsup></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(7)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \mathbf{I}_C = \sum_i m_i \begin{bmatrix} y_i^2 + z_i^2 & -x_i y_i & -x_i z_i \\ -x_i y_i & x_i^2+z_i^2 & -y_i z_i \\ -x_i z_i & -y_i z_i & x_i^2+y_i^2 | |
| \end{bmatrix} \tag{7} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83611em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎣</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.814108em;"><span class="" style="top: -2.44134em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.258664em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.814108em;"><span class="" style="top: -2.44134em; margin-left: -0.04398em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.258664em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.04398em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span><span class="arraycolsep" style="width: 0.5em;"></span><span class="arraycolsep" style="width: 0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.814108em;"><span class="" style="top: -2.44134em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.258664em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.814108em;"><span class="" style="top: -2.44134em; margin-left: -0.04398em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.258664em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.04398em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span><span class="arraycolsep" style="width: 0.5em;"></span><span class="arraycolsep" style="width: 0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.04398em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.04398em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.814108em;"><span class="" style="top: -2.44134em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.258664em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.814108em;"><span class="" style="top: -2.44134em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.258664em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎦</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">7</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <p>Another common alterative to the above is to declare the 3×3 cross product matrix as <span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><msub><mi mathvariant="bold-italic">d</mi><mrow><mi>C</mi><mi>i</mi></mrow></msub><mo>×</mo><mo stretchy="false">]</mo><mo>≡</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>z</mi><mi>i</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>y</mi><mi>i</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>z</mi><mi>i</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>x</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>y</mi><mi>i</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mi>i</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex"> [\boldsymbol{d}_{Ci} \times] \equiv \begin{bmatrix} 0 & -z_i & y_i \\ z_i & 0 & -x_i \\ -y_i & x_i & 0 \end{bmatrix} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">d</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord">×</span><span class="mclose">]</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎣</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right: 0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.04398em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span><span class="arraycolsep" style="width: 0.5em;"></span><span class="arraycolsep" style="width: 0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.04398em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span><span class="arraycolsep" style="width: 0.5em;"></span><span class="arraycolsep" style="width: 0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎦</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></span> and evaluating</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><msub><mi mathvariant="bold">I</mi><mi>C</mi></msub><mo>=</mo><munder><mo>∑</mo><mi>i</mi></munder><mrow><mo fence="true">(</mo><mo>−</mo><msub><mi>m</mi><mi>i</mi></msub><mo stretchy="false">[</mo><msub><mi mathvariant="bold-italic">d</mi><mrow><mi>C</mi><mi>i</mi></mrow></msub><mo>×</mo><mo stretchy="false">]</mo><mo stretchy="false">[</mo><msub><mi mathvariant="bold-italic">d</mi><mrow><mi>C</mi><mi>i</mi></mrow></msub><mo>×</mo><mo stretchy="false">]</mo><mo fence="true">)</mo></mrow></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(8)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \mathbf{I}_C = \sum_i \left( -m_i [\boldsymbol{d}_{Ci} \times][\boldsymbol{d}_{Ci} \times] \right) \tag{8} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83611em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 2.32767em; vertical-align: -1.27767em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="minner"><span class="mopen delimcenter" style="top: 0em;">(</span><span class="mord">−</span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mopen">[</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">d</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord">×</span><span class="mclose">]</span><span class="mopen">[</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">d</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord">×</span><span class="mclose">]</span><span class="mclose delimcenter" style="top: 0em;">)</span></span></span><span class="tag"><span class="strut" style="height: 2.32767em; vertical-align: -1.27767em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">8</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <h3 id="rotational-momentum-about-arbitrary-point">Rotational Momentum about Arbitrary Point</h3> | |
| <p>Rotational momentum is also evaluated from (2) and (3) but we sum about the arbitrary point <strong>A</strong>.</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi mathvariant="bold-italic">L</mi><mi>A</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo>∑</mo><mi>i</mi></munder><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><msub><mi>m</mi><mi>i</mi></msub><msub><mi>v</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo>∑</mo><mi>i</mi></munder><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo stretchy="false">)</mo><mo>×</mo><msub><mi>m</mi><mi>i</mi></msub><msub><mi>v</mi><mi>i</mi></msub><mo>+</mo><munder><mo>∑</mo><mi>i</mi></munder><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><msub><mi>m</mi><mi>i</mi></msub><msub><mi>v</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi mathvariant="bold-italic">L</mi><mi>C</mi></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>m</mi><mi>i</mi></msub><msub><mi>v</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex"> \begin{aligned} \boldsymbol{L}_A & = \sum_i (\boldsymbol{r}_i - \boldsymbol{r}_A) \times m_i v_i \\ | |
| & = \sum_i (\boldsymbol{r}_i - \boldsymbol{r}_C) \times m_i v_i + \sum_i (\boldsymbol{r}_C - \boldsymbol{r}_A) \times m_i v_i \\ | |
| & = \boldsymbol{L}_C + (\boldsymbol{r}_C-\boldsymbol{r}_A) \times \sum_i m_i v_i | |
| \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 7.88302em; vertical-align: -3.69151em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 4.19151em;"><span class="" style="top: -6.19151em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.56384em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"></span></span><span class="" style="top: -0.936163em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 3.69151em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 4.19151em;"><span class="" style="top: -6.19151em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.56384em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -0.936163em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 3.69151em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></span></p> | |
| <p>Which leads the the following transformation law(s)</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi mathvariant="bold-italic">L</mi><mi>A</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi mathvariant="bold-italic">L</mi><mi>C</mi></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">p</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi mathvariant="bold-italic">L</mi><mi>C</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi mathvariant="bold-italic">L</mi><mi>A</mi></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">p</mi></mrow></mstyle></mtd></mtr></mtable></mtd><mtd width="50%"></mtd><mtd><mtext>(9)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \begin{aligned} \boldsymbol{L}_A & = \boldsymbol{L}_C + (\boldsymbol{r}_C-\boldsymbol{r}_A) \times \boldsymbol{p} \\ \boldsymbol{L}_C & = \boldsymbol{L}_A + (\boldsymbol{r}_A-\boldsymbol{r}_C) \times \boldsymbol{p} | |
| \end{aligned} \tag{9} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 3em; vertical-align: -1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.75em;"><span class="" style="top: -3.91em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.25em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.75em;"><span class="" style="top: -3.91em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.25em;"><span class=""></span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height: 3em; vertical-align: -1.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">9</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <p>Again, any resemblance to (1) and (4) is not a coincidence. This is because these are Plücker coordinates of different lines in space. The force line is called the <em>line of action</em>. The motion line is called the <em>rotation axis</em>. And the momentum line is called <em>axis of percussion</em>.</p> | |
| <h3 id="dynamics">Dynamics</h3> | |
| <p>We establish Newton’s second law about the center of mass <strong>C</strong> relating forces/torques to rate of change of momenta</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi mathvariant="bold-italic">F</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mstyle displaystyle="false" scriptlevel="0"><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></mstyle><mi mathvariant="bold-italic">p</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi mathvariant="bold-italic">τ</mi><mi>C</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mstyle displaystyle="false" scriptlevel="0"><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></mstyle><msub><mi mathvariant="bold-italic">L</mi><mi>C</mi></msub></mrow></mstyle></mtd></mtr></mtable></mtd><mtd width="50%"></mtd><mtd><mtext>(10)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \begin{aligned} | |
| \boldsymbol{F} & = \tfrac{\rm d}{{\rm d}t} \boldsymbol{p} \\ | |
| \boldsymbol{\tau}_C & = \tfrac{\rm d}{{\rm d}t} \boldsymbol{L}_C | |
| \end{aligned} \tag{10} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 3.08022em; vertical-align: -1.29011em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.79011em;"><span class="" style="top: -3.91em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.15972em;">F</span></span></span></span></span><span class="" style="top: -2.36989em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.29011em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.79011em;"><span class="" style="top: -3.91em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span></span></span><span class="" style="top: -2.36989em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.29011em;"><span class=""></span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height: 3.08022em; vertical-align: -1.29011em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">1</span><span class="mord">0</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <p>Using the definitions above from particle summation the expression of equations of motion are straigtforward when using rhe center of mass as a reference point.</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi mathvariant="bold-italic">F</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>m</mi><msub><mi mathvariant="bold-italic">a</mi><mi>C</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi mathvariant="bold-italic">τ</mi><mi>C</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi mathvariant="bold">I</mi><mi>C</mi></msub><mi mathvariant="bold-italic">α</mi><mo>+</mo><mi mathvariant="bold-italic">ω</mi><mo>×</mo><msub><mi mathvariant="bold-italic">L</mi><mi>C</mi></msub></mrow></mstyle></mtd></mtr></mtable></mtd><mtd width="50%"></mtd><mtd><mtext>(11)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \begin{aligned} | |
| \boldsymbol{F} & = m \boldsymbol{a}_C \\ | |
| \boldsymbol{\tau}_C & = \mathbf{I}_C \boldsymbol{\alpha} + \boldsymbol{\omega} \times \boldsymbol{L}_C | |
| \end{aligned} \tag{11} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 3em; vertical-align: -1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.75em;"><span class="" style="top: -3.91em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.15972em;">F</span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.25em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.75em;"><span class="" style="top: -3.91em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord mathdefault">m</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord boldsymbol">α</span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">ω</span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.25em;"><span class=""></span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height: 3em; vertical-align: -1.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">1</span><span class="mord">1</span></span><span class="mord">)</span></span></span></span></span></span></span><br> | |
| where</p> | |
| <table> | |
| <thead> | |
| <tr> | |
| <th align="right">quantity</th> | |
| <th align="left">description</th> | |
| <th align="left">definition</th> | |
| </tr> | |
| </thead> | |
| <tbody> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">a</mi><mi>C</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{a}_C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td align="left">translational acceleration of the center of mass</td> | |
| <td align="left"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub></mrow><annotation encoding="application/x-tex">\tfrac{\rm d}{{\rm d}t} \boldsymbol{v}_C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.22511em; vertical-align: -0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="bold-italic">α</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{\alpha}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.44444em; vertical-align: 0em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">α</span></span></span></span></span></span></span></td> | |
| <td align="left">rotational acceleration of the body</td> | |
| <td align="left"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mi mathvariant="bold-italic">ω</mi></mrow><annotation encoding="application/x-tex">\tfrac{\rm d}{{\rm d}t} \boldsymbol{\omega}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.22511em; vertical-align: -0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">ω</span></span></span></span></span></span></span></td> | |
| </tr> | |
| </tbody> | |
| </table><h3 id="arbitrary-point">Arbitrary Point</h3> | |
| <p>The question is can the equations of motion above can be derived from Newton’s 2nd law <em>at an arbitrary location</em>?</p> | |
| <p><strong>Is the following valid?</strong></p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><msub><mi mathvariant="bold-italic">τ</mi><mi>A</mi></msub><mo>=</mo><mstyle displaystyle="false" scriptlevel="0"><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></mstyle><msub><mi mathvariant="bold-italic">L</mi><mi>A</mi></msub></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(12)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \boldsymbol{\tau}_A = \tfrac{\rm d}{{\rm d}t} \boldsymbol{L}_A \tag{12} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1.22511em; vertical-align: -0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height: 1.22511em; vertical-align: -0.345em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">1</span><span class="mord">2</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <p>We can use the transformation laws (1) and (9) described above to see if (12) can lead to (11) which we know is correct.</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi mathvariant="bold-italic">τ</mi><mi>C</mi></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">F</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mstyle displaystyle="false" scriptlevel="0"><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></mstyle><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">L</mi><mi>C</mi></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">p</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mstyle displaystyle="false" scriptlevel="0"><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></mstyle><msub><mi mathvariant="bold-italic">L</mi><mi>C</mi></msub><mo>+</mo><mstyle displaystyle="false" scriptlevel="0"><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></mstyle><mrow><mo fence="true">(</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">p</mi><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><menclose notation="updiagonalstrike"><msub><mi mathvariant="bold-italic">τ</mi><mi>C</mi></msub></menclose><mo>+</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">F</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><menclose notation="updiagonalstrike"><msub><mi mathvariant="bold-italic">τ</mi><mi>C</mi></msub></menclose><mo>+</mo><mstyle displaystyle="false" scriptlevel="0"><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></mstyle><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">p</mi><mo>+</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mstyle displaystyle="false" scriptlevel="0"><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></mstyle><mi mathvariant="bold-italic">p</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><menclose notation="updiagonalstrike"><mrow><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">F</mi></mrow></menclose></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">v</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">p</mi><mo>+</mo><menclose notation="updiagonalstrike"><mrow><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">F</mi></mrow></menclose></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mn mathvariant="bold-italic">0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi mathvariant="bold-italic">p</mi><mo>×</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">v</mi><mi>A</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mn mathvariant="bold-italic">0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>m</mi> <msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub><mo>×</mo><msub><mi mathvariant="bold-italic">v</mi><mi>A</mi></msub></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex"> \begin{aligned} | |
| \boldsymbol{\tau}_C + (\boldsymbol{r}_C - \boldsymbol{r}_A) \times \boldsymbol{F} & = \tfrac{\rm d}{{\rm d}t} ( \boldsymbol{L}_C + (\boldsymbol{r}_C-\boldsymbol{r}_A) \times \boldsymbol{p} ) \\ | |
| & = \tfrac{\rm d}{{\rm d}t} \boldsymbol{L}_C + \tfrac{\rm d}{{\rm d}t} \left( (\boldsymbol{r}_C-\boldsymbol{r}_A) \times \boldsymbol{p} \right) \\ | |
| \cancel{\boldsymbol{\tau}_C} + (\boldsymbol{r}_C - \boldsymbol{r}_A) \times \boldsymbol{F} & = \cancel{\boldsymbol{\tau}_C} + \tfrac{\rm d}{{\rm d}t} (\boldsymbol{r}_C-\boldsymbol{r}_A) \times \boldsymbol{p} + (\boldsymbol{r}_C-\boldsymbol{r}_A) \times \tfrac{\rm d}{{\rm d}t} \boldsymbol{p} \\ | |
| \cancel{(\boldsymbol{r}_C - \boldsymbol{r}_A) \times \boldsymbol{F}} &= ( \boldsymbol{v}_C - \boldsymbol{v}_A)\times \boldsymbol{p} + \cancel{ (\boldsymbol{r}_C - \boldsymbol{r}_A) \times \boldsymbol{F}} \\ \boldsymbol{0} & = | |
| \boldsymbol{p} \times ( \boldsymbol{v}_A - \boldsymbol{v}_C) \\ | |
| \boldsymbol{0} & = m\,\boldsymbol{v}_C \times \boldsymbol{v}_A | |
| \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 9.12032em; vertical-align: -4.31016em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 4.81016em;"><span class="" style="top: -6.93005em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.15972em;">F</span></span></span></span></span><span class="" style="top: -5.38995em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"></span></span><span class="" style="top: -3.84984em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord cancel-lap"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.44444em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord cancel-pad"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="svg-align" style="top: -2.85em;"><span class="pstrut" style="height: 3em;"></span><span class="" style="height: 0.59444em;"><svg width="100%" height="0.59444em"><line x1="0" y1="100%" x2="100%" y2="0" stroke-width="0.046em"></line></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.15972em;">F</span></span></span></span></span><span class="" style="top: -2.34984em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord cancel-lap"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.75em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord cancel-pad"><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.15972em;">F</span></span></span></span></span><span class="svg-align" style="top: -2.75em;"><span class="pstrut" style="height: 3em;"></span><span class="" style="height: 1em;"><svg width="100%" height="1em"><line x1="0" y1="100%" x2="100%" y2="0" stroke-width="0.046em"></line></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.25em;"><span class=""></span></span></span></span></span></span></span><span class="" style="top: -0.849838em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathbf">0</span></span></span></span></span><span class="" style="top: 0.650162em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathbf">0</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 4.31016em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 4.81016em;"><span class="" style="top: -6.93005em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="mclose">)</span></span></span><span class="" style="top: -5.38995em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="minner"><span class="mopen delimcenter" style="top: 0em;">(</span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="mclose delimcenter" style="top: 0em;">)</span></span></span></span><span class="" style="top: -3.84984em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord cancel-lap"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.44444em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord cancel-pad"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="svg-align" style="top: -2.85em;"><span class="pstrut" style="height: 3em;"></span><span class="" style="height: 0.59444em;"><svg width="100%" height="0.59444em"><line x1="0" y1="100%" x2="100%" y2="0" stroke-width="0.046em"></line></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span></span></span><span class="" style="top: -2.34984em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord cancel-lap"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.75em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord cancel-pad"><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.15972em;">F</span></span></span></span></span><span class="svg-align" style="top: -2.75em;"><span class="pstrut" style="height: 3em;"></span><span class="" style="height: 1em;"><svg width="100%" height="1em"><line x1="0" y1="100%" x2="100%" y2="0" stroke-width="0.046em"></line></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.25em;"><span class=""></span></span></span></span></span></span></span><span class="" style="top: -0.849838em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span><span class="" style="top: 0.650162em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord mathdefault">m</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 4.31016em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></span></p> | |
| <p>The <em>necessary</em> condition(s) for (12) to be correct are as follows</p> | |
| <ul> | |
| <li>Reference point <strong>A</strong> is fixed in space.</li> | |
| <li>Reference point <strong>A</strong> is co-moving with the center of mass.</li> | |
| <li>Center of mass <strong>C</strong> is fixed in space.</li> | |
| </ul> | |
| <p>The general rule connecting equipollent torque to change in rotational momentum is thus</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><msub><mi mathvariant="bold-italic">τ</mi><mi>A</mi></msub><mo>=</mo><mstyle displaystyle="false" scriptlevel="0"><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></mstyle><msub><mi mathvariant="bold-italic">L</mi><mi>A</mi></msub><mo>+</mo><msub><mi mathvariant="bold-italic">v</mi><mi>A</mi></msub><mo>×</mo><mi mathvariant="bold-italic">p</mi></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(13)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \boldsymbol{\tau}_A = \tfrac{\rm d}{{\rm d}t} \boldsymbol{L}_A + \boldsymbol{v}_A \times \boldsymbol{p} \tag{13} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1.22511em; vertical-align: -0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.73333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.63888em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span></span><span class="tag"><span class="strut" style="height: 1.22511em; vertical-align: -0.345em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">1</span><span class="mord">3</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <p>where</p> | |
| <table> | |
| <thead> | |
| <tr> | |
| <th align="right">quantity</th> | |
| <th align="left">description</th> | |
| </tr> | |
| </thead> | |
| <tbody> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">τ</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\tau}_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td align="left">equipollent torque acting on the body (instanteneously) summed on the reference point <strong>A</strong>.</td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">L</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{L}_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83611em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td align="left">rotational momentum of rigid body summed on the reference point <strong>A</strong>.</td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="bold-italic">p</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{p}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.63888em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span></span></span></span></span></td> | |
| <td align="left">translational momentum of the rigid body.</td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td align="left">instanteneous velocity of reference point <strong>A</strong>.</td> | |
| </tr> | |
| </tbody> | |
| </table><h3 id="example">Example</h3> | |
| <p>Consider a rod sliding vertically along a rail with speed <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>v</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">v_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span> and pivoting with rotational speed <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span></span></span> about a point <strong>A</strong> away from the center of mass <strong>C</strong>. A force of magnitude <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>F</mi><mi>B</mi></msub></mrow><annotation encoding="application/x-tex">F_B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span> is applied horizontally at another point <strong>B</strong>.<br> | |
| <img src="https://i.imgur.com/JwNY2Oa.png" alt="sliding rod"></p> | |
| <p>The variable <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathdefault">c</span></span></span></span></span> is the distance between the pivot and the center of mass, and <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.69444em; vertical-align: 0em;"></span><span class="mord">ℓ</span></span></span></span></span> the distance between the pivot and the force application point.</p> | |
| <p><strong>Can (12) produce the correct equation of motion?</strong></p> | |
| <p>First we develop the equations of motion about the center of mass</p> | |
| <ol> | |
| <li><strong>Kinematics</strong> the center of mass <strong>C</strong> moves under two degrees of freedom, the sliding velocity <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>v</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">v_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span> and the rotation <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span></span></span> | |
| <ul> | |
| <li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub><mo>=</mo><msub><mi>v</mi><mi>A</mi></msub> <mover accent="true"><mi mathvariant="bold-italic">j</mi><mo>^</mo></mover><mo>+</mo><mi>ω</mi><mover accent="true"><mi mathvariant="bold-italic">k</mi><mo>^</mo></mover><mo>×</mo><mi>c</mi><mover accent="true"><mi mathvariant="bold-italic">j</mi><mo>^</mo></mover><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mi>c</mi> <mi>ω</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>v</mi><mi>A</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_C = v_A\, \boldsymbol{\hat{j}} + \omega \boldsymbol{\hat{k}} \times c \boldsymbol{\hat{j}} = \begin{pmatrix} -c\, \omega \\ v_A \\ 0\end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1.15114em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.9567em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.0622em;">j</span></span></span><span class="" style="top: -3.26226em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.25em;">^</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.19444em;"><span class=""></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1.04121em; vertical-align: -0.08333em;"></span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.95788em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.01852em;">k</span></span></span><span class="" style="top: -3.26344em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.25em;">^</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1.15114em; vertical-align: -0.19444em;"></span><span class="mord mathdefault">c</span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.9567em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.0622em;">j</span></span></span><span class="" style="top: -3.26226em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.25em;">^</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.19444em;"><span class=""></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></li> | |
| <li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">a</mi><mi>C</mi></msub><mo>=</mo><msub><mover accent="true"><mi>v</mi><mo>˙</mo></mover><mi>A</mi></msub><mover accent="true"><mi mathvariant="bold-italic">j</mi><mo>^</mo></mover><mo>+</mo><mover accent="true"><mi>ω</mi><mo>˙</mo></mover><mover accent="true"><mi mathvariant="bold-italic">k</mi><mo>^</mo></mover><mo>×</mo><mi>c</mi><mover accent="true"><mi mathvariant="bold-italic">j</mi><mo>^</mo></mover><mo>+</mo><mi>ω</mi><mover accent="true"><mi mathvariant="bold-italic">k</mi><mo>^</mo></mover><mo>×</mo><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mi>c</mi> <mover accent="true"><mi>ω</mi><mo>˙</mo></mover><mo>−</mo><mi>ω</mi> <msub><mi>v</mi><mi>A</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mover accent="true"><mi>v</mi><mo>˙</mo></mover><mi>A</mi></msub><mo>−</mo><mi>c</mi> <msup><mi>ω</mi><mn>2</mn></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\boldsymbol{a}_C =\dot{v}_A \boldsymbol{\hat{j}} + \dot{\omega}\boldsymbol{\hat{k}}\times c \boldsymbol{\hat{j}} + \omega \boldsymbol{\hat{k}} \times \boldsymbol{v}_C = \begin{pmatrix} -c\,\dot{\omega} - \omega\, v_A \\ \dot{v}_A - c\, \omega^2 \\ 0 \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1.15114em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.11111em;">˙</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.9567em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.0622em;">j</span></span></span><span class="" style="top: -3.26226em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.25em;">^</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.19444em;"><span class=""></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1.04121em; vertical-align: -0.08333em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.13889em;">˙</span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.95788em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.01852em;">k</span></span></span><span class="" style="top: -3.26344em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.25em;">^</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1.15114em; vertical-align: -0.19444em;"></span><span class="mord mathdefault">c</span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.9567em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.0622em;">j</span></span></span><span class="" style="top: -3.26226em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.25em;">^</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.19444em;"><span class=""></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1.04121em; vertical-align: -0.08333em;"></span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.95788em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.01852em;">k</span></span></span><span class="" style="top: -3.26344em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.25em;">^</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.13889em;">˙</span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.11111em;">˙</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.814108em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></li> | |
| </ul> | |
| </li> | |
| <li><strong>Momenta</strong> directly calculated from the motion of the center of mass | |
| <ul> | |
| <li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="bold-italic">p</mi><mo>=</mo><mi>m</mi><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mi>m</mi> <mi>c</mi> <mi>ω</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>m</mi> <msub><mi>v</mi><mi>A</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\boldsymbol{p} = m \boldsymbol{v}_C = \begin{pmatrix} -m \,c\,\omega \\ m\,v_A \\ 0 \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.63888em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord mathdefault">m</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord mathdefault">m</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></li> | |
| <li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">L</mi><mi>C</mi></msub><mo>=</mo><msub><mi mathvariant="bold">I</mi><mi>C</mi></msub><mi mathvariant="bold-italic">ω</mi><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>I</mi><mi>C</mi></msub><mi>ω</mi></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\boldsymbol{L}_C = \mathbf{I}_C \boldsymbol{\omega} = \begin{pmatrix} 0 \\ 0 \\ I_C \omega \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83611em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.83611em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">ω</span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right: 0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.07847em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></li> | |
| </ul> | |
| </li> | |
| <li><strong>Equipollent forces and moments</strong> at the center of mass <strong>C</strong> are a combination of the appled force <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>F</mi><mi>B</mi></msub></mrow><annotation encoding="application/x-tex">F_B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span> and the pin reaction force <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>A</mi><mi>x</mi></msub></mrow><annotation encoding="application/x-tex">A_x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathdefault">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>. | |
| <ul> | |
| <li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="bold-italic">F</mi><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>F</mi><mi>B</mi></msub><mo>+</mo><msub><mi>A</mi><mi>x</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\boldsymbol{F} = \begin{pmatrix} F_B+A_x \\ 0 \\ 0 \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.68611em; vertical-align: 0em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.15972em;">F</span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right: 0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord mathdefault">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></li> | |
| <li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">τ</mi><mi>C</mi></msub><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>c</mi><msub><mi>A</mi><mi>x</mi></msub><mo>−</mo><mo stretchy="false">(</mo><mi mathvariant="normal">ℓ</mi><mo>−</mo><mi>c</mi><mo stretchy="false">)</mo><msub><mi>F</mi><mi>B</mi></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\boldsymbol{\tau}_C = \begin{pmatrix} 0 \\ 0 \\ c A_x -(\ell-c) F_B \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault">c</span><span class="mord"><span class="mord mathdefault">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord">ℓ</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault">c</span><span class="mclose">)</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></li> | |
| </ul> | |
| </li> | |
| <li><strong>Equations of motion</strong> | |
| <ul> | |
| <li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>F</mi><mi>B</mi></msub><mo>+</mo><msub><mi>A</mi><mi>x</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><mo>=</mo><mi>m</mi><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mi>c</mi> <mover accent="true"><mi>ω</mi><mo>˙</mo></mover><mo>−</mo><mi>ω</mi> <msub><mi>v</mi><mi>A</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mover accent="true"><mi>v</mi><mo>˙</mo></mover><mi>A</mi></msub><mo>−</mo><mi>c</mi> <msup><mi>ω</mi><mn>2</mn></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{pmatrix} F_B+A_x \\ 0 \\ 0 \end{pmatrix} = m \begin{pmatrix} -c\,\dot{\omega} - \omega\, v_A \\ \dot{v}_A - c\, \omega^2 \\ 0 \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right: 0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord mathdefault">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="mord mathdefault">m</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.13889em;">˙</span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.11111em;">˙</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.814108em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></li> | |
| <li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>c</mi><msub><mi>A</mi><mi>x</mi></msub><mo>−</mo><mo stretchy="false">(</mo><mi mathvariant="normal">ℓ</mi><mo>−</mo><mi>c</mi><mo stretchy="false">)</mo><msub><mi>F</mi><mi>B</mi></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><mo>=</mo><msub><mi>I</mi><mi>C</mi></msub><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mover accent="true"><mi>ω</mi><mo>˙</mo></mover></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{pmatrix} 0 \\ 0 \\ c A_x -(\ell-c) F_B \end{pmatrix} = I_C \begin{pmatrix} 0 \\ 0 \\ \dot{\omega} \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault">c</span><span class="mord"><span class="mord mathdefault">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord">ℓ</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault">c</span><span class="mclose">)</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.07847em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.13889em;">˙</span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></li> | |
| </ul> | |
| </li> | |
| <li><strong>Solution</strong> | |
| <ul> | |
| <li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mover accent="true"><mi>ω</mi><mo>˙</mo></mover><mo>=</mo><mo>−</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi mathvariant="normal">ℓ</mi><msub><mi>F</mi><mi>B</mi></msub><mo>+</mo><mi>m</mi> <mi>c</mi> <mi>ω</mi> <msub><mi>v</mi><mi>A</mi></msub></mrow><mrow><msub><mi>I</mi><mi>C</mi></msub><mo>+</mo><mi>m</mi><msup><mi>c</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\dot{\omega} = - \dfrac{\ell F_B + m\,c\,\omega\,v_A}{I_C + m c^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.66786em; vertical-align: 0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.13889em;">˙</span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 2.20744em; vertical-align: -0.836em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.37144em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right: 0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.07847em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault">m</span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.740108em;"><span class="" style="top: -2.989em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">ℓ</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault">m</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.836em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></li> | |
| <li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mover accent="true"><mi>v</mi><mo>˙</mo></mover><mi>A</mi></msub><mo>=</mo><mi>c</mi> <msup><mi>ω</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\dot{v}_A = c\,\omega^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.81786em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.11111em;">˙</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.814108em; vertical-align: 0em;"></span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.814108em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></li> | |
| <li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>A</mi><mi>x</mi></msub><mo>=</mo><mo>−</mo><mrow><mo fence="true">(</mo><mn>1</mn><mo>−</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>m</mi> <mi>c</mi> <mi mathvariant="normal">ℓ</mi></mrow><mrow><msub><mi>I</mi><mi>C</mi></msub><mo>+</mo><mi>m</mi><msup><mi>c</mi><mn>2</mn></msup></mrow></mfrac></mstyle><mo fence="true">)</mo></mrow><msub><mi>F</mi><mi>B</mi></msub><mo>−</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>ω</mi> <msub><mi>v</mi><mi>A</mi></msub></mrow><mrow><mfrac><mn>1</mn><mi>m</mi></mfrac><mo>+</mo><mfrac><msup><mi>c</mi><mn>2</mn></msup><msub><mi>I</mi><mi>C</mi></msub></mfrac></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">A_x = -\left( 1-\dfrac{m\,c\,\ell}{I_C + m c^2} \right) F_B - \dfrac{\omega\,v_A}{ \tfrac{1}{m} + \tfrac{c^2}{I_C}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathdefault">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 2.40003em; vertical-align: -0.95003em;"></span><span class="mord">−</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="minner"><span class="mopen delimcenter" style="top: 0em;"><span class="delimsizing size3">(</span></span><span class="mord">1</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.37144em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right: 0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.07847em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault">m</span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.740108em;"><span class="" style="top: -2.989em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord">ℓ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.836em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top: 0em;"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 2.35928em; vertical-align: -1.25172em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.10756em;"><span class="" style="top: -2.19358em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.845108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.91642em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3448em;"><span class="" style="top: -2.35671em; margin-left: -0.07847em; margin-right: 0.0714286em;"><span class="pstrut" style="height: 2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.143293em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.746314em;"><span class="" style="top: -2.786em; margin-right: 0.0714286em;"><span class="pstrut" style="height: 2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.445305em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.25172em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></li> | |
| </ul> | |
| </li> | |
| </ol> | |
| <p>Now we apply equation (12) directly after the rotational momentum about <strong>A</strong> is calculated from (9).<br> | |
| - <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">L</mi><mi>A</mi></msub><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo stretchy="false">(</mo><msub><mi>I</mi><mi>C</mi></msub><mo>+</mo><mi>m</mi><msup><mi>c</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mi>ω</mi></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\boldsymbol{L}_A = \begin{pmatrix} 0 \\ 0 \\ (I_C + m c^2) \omega \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83611em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.07847em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault">m</span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.814108em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span><br> | |
| - <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">τ</mi><mi>A</mi></msub><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mi mathvariant="normal">ℓ</mi><msub><mi>F</mi><mi>B</mi></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\boldsymbol{\tau}_A = \begin{pmatrix} 0 \\ 0 \\ -\ell F_B \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">ℓ</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span><br> | |
| - <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><msub><mi mathvariant="bold-italic">L</mi><mi>A</mi></msub><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo stretchy="false">(</mo><msub><mi>I</mi><mi>C</mi></msub><mo>+</mo><mi>m</mi><msup><mi>c</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mover accent="true"><mi>ω</mi><mo>˙</mo></mover></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\tfrac{\rm d}{{\rm d}t}\boldsymbol{L}_A = \begin{pmatrix} 0 \\ 0 \\ (I_C + m c^2) \dot{\omega} \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.22511em; vertical-align: -0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.07847em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault">m</span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.814108em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.13889em;">˙</span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span><br> | |
| - <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>A</mi></msub><mo>×</mo><mi mathvariant="bold-italic">p</mi><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>m</mi> <mi>c</mi> <mi>ω</mi> <msub><mi>v</mi><mi>A</mi></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_A \times \boldsymbol{p} = \begin{pmatrix} 0 \\ 0 \\ m\,c\,\omega\,v_A \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.73333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.63888em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></p> | |
| <p>All together equation (12) is</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>−</mo><mi mathvariant="normal">ℓ</mi><msub><mi>F</mi><mi>B</mi></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>I</mi><mi>C</mi></msub><mo>+</mo><mi>m</mi><msup><mi>c</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mover accent="true"><mi>ω</mi><mo>˙</mo></mover><mo>+</mo><mi>m</mi><mi>c</mi><mi>ω</mi><msub><mi>v</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex"> -\ell F_B = (I_C + m c^2) \dot{\omega} + m c \omega v_A </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.84444em; vertical-align: -0.15em;"></span><span class="mord">−</span><span class="mord">ℓ</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.07847em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1.11411em; vertical-align: -0.25em;"></span><span class="mord mathdefault">m</span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.864108em;"><span class="" style="top: -3.113em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.13889em;">˙</span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord mathdefault">m</span><span class="mord mathdefault">c</span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></p> | |
| <p>which is solved for <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mover accent="true"><mi>ω</mi><mo>˙</mo></mover></mrow><annotation encoding="application/x-tex">\dot{\omega}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.66786em; vertical-align: 0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.13889em;">˙</span></span></span></span></span></span></span></span></span></span> to produce <strong>the exact same same solution as above</strong></p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mover accent="true"><mi>ω</mi><mo>˙</mo></mover><mo>=</mo><mo>−</mo><mfrac><mrow><mi mathvariant="normal">ℓ</mi><msub><mi>F</mi><mi>B</mi></msub><mo>+</mo><mi>m</mi> <mi>c</mi> <mi>ω</mi> <msub><mi>v</mi><mi>A</mi></msub></mrow><mrow><msub><mi>I</mi><mi>C</mi></msub><mo>+</mo><mi>m</mi><msup><mi>c</mi><mn>2</mn></msup></mrow></mfrac> <mi mathvariant="normal">✓</mi></mrow><annotation encoding="application/x-tex">\dot{\omega} = - \dfrac{\ell F_B + m\,c\,\omega\,v_A}{I_C + m c^2}\;\;\checkmark</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.66786em; vertical-align: 0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.13889em;">˙</span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 2.20744em; vertical-align: -0.836em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.37144em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right: 0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.07847em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault">m</span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.740108em;"><span class="" style="top: -2.989em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">ℓ</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault">m</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.836em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord amsrm">✓</span></span></span></span></span></span></p> | |
| <blockquote> | |
| <p>As in illustration we can see that <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">τ</mi><mi>A</mi></msub><mo>=</mo><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><msub><mi mathvariant="bold-italic">L</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\tau}_A = \tfrac{\rm d}{{\rm d}t} \boldsymbol{L}_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1.22511em; vertical-align: -0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span> would produce the incorrect result<br> | |
| <span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mover accent="true"><mi>ω</mi><mo>˙</mo></mover><mo>=</mo><mo>−</mo><mfrac><mrow><mi mathvariant="normal">ℓ</mi><msub><mi>F</mi><mi>B</mi></msub></mrow><mrow><msub><mi>I</mi><mi>C</mi></msub><mo>+</mo><mi>m</mi><msup><mi>c</mi><mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">\dot{\omega} = - \dfrac{\ell F_B}{I_C + m c^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.66786em; vertical-align: 0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.13889em;">˙</span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 2.20744em; vertical-align: -0.836em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.37144em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right: 0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.07847em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault">m</span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.740108em;"><span class="" style="top: -2.989em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">ℓ</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.836em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span></p> | |
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| <div class="stackedit__html"><h1 id="angular-momentum-and-torque-on-a-rigid-body">Angular Momentum and Torque on a Rigid Body</h1> | |
| <p>Angular momentum of a rigid body varies as the point of summation varies. Also, equipollent torque varies as the point of summation varies also. This posts examines the relationship between change in angular momentum and equipollent torque in the context of Newton’s 2nd law. It is <strike>expected</strike> required that at the center of mass equipollent torque and change in angular momentum at simply equal to each other. But what about when measured at an arbitrary point <strong>A</strong> not at the center of mass <strong>C</strong>.</p> | |
| <p><sub>Note that all quantities are expressed on the same basis vectors, and only reference point being <strong>C</strong> or <strong>A</strong> or <strong>O</strong> the origin describing the properties of whichever particles happens to be passing under this reference point.</sub></p> | |
| <h3 id="particle-forces--torques">Particle Forces & Torques</h3> | |
| <p>The combined loading on the rigid body can be reduced down to an equipollent system of forces <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="bold-italic">F</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.68611em; vertical-align: 0em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.15972em;">F</span></span></span></span></span></span></span> and torques <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">τ</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\tau}_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span> at the reference point <strong>A</strong>. In dynamics it is important to consider the combined torque about the center of mass which is evaluated with the transformation law(s)</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi mathvariant="bold-italic">τ</mi><mi>C</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi mathvariant="bold-italic">τ</mi><mi>A</mi></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">F</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi mathvariant="bold-italic">τ</mi><mi>A</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi mathvariant="bold-italic">τ</mi><mi>C</mi></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">F</mi></mrow></mstyle></mtd></mtr></mtable></mtd><mtd width="50%"></mtd><mtd><mtext>(1)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \begin{aligned} | |
| \boldsymbol{\tau}_C &= \boldsymbol{\tau}_A + (\boldsymbol{r}_A - \boldsymbol{r}_C) \times \boldsymbol{F} \\ \boldsymbol{\tau}_A& = \boldsymbol{\tau}_C + (\boldsymbol{r}_C - \boldsymbol{r}_A) \times \boldsymbol{F} | |
| \end{aligned} \tag{1} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 3em; vertical-align: -1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.75em;"><span class="" style="top: -3.91em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.25em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.75em;"><span class="" style="top: -3.91em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.15972em;">F</span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.15972em;">F</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.25em;"><span class=""></span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height: 3em; vertical-align: -1.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">1</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <table> | |
| <thead> | |
| <tr> | |
| <th align="right"><em>quantity</em></th> | |
| <th align="left"><em>description</em></th> | |
| </tr> | |
| </thead> | |
| <tbody> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="bold-italic">F</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.68611em; vertical-align: 0em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.15972em;">F</span></span></span></span></span></span></span></td> | |
| <td align="left">combined of forces applied on body.</td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">τ</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\tau}_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td align="left">combined torque applied on body about point <strong>A</strong>.</td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">τ</mi><mi>C</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\tau}_C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td align="left">combined torque applied on body about center of mass <strong>C</strong>.</td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{r}_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td align="left">location of point <strong>A</strong> from the origin.</td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{r}_C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td align="left">location of point <strong>C</strong> from the origin.</td> | |
| </tr> | |
| </tbody> | |
| </table><h3 id="center-of-mass">Center of mass</h3> | |
| <p>It is important to define the center of mass not only as the weighted sum of the particle locations</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>=</mo><mfrac><mn>1</mn><mi>m</mi></mfrac><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>m</mi><mi>i</mi></msub><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(3)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \boldsymbol{r}_C = \frac{1}{m} \sum_i m_i \boldsymbol{r}_i \tag{3} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 2.59911em; vertical-align: -1.27767em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.32144em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault">m</span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.686em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height: 2.59911em; vertical-align: -1.27767em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">3</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <p>but also as the relative location of each particle being zero</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>m</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="bold-italic">0</mn></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(4)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \sum_i m_i ( \boldsymbol{r}_i - \boldsymbol{r}_C) = \boldsymbol{0} \tag{4} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 2.32767em; vertical-align: -1.27767em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.64444em; vertical-align: 0em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">0</span></span></span></span><span class="tag"><span class="strut" style="height: 2.32767em; vertical-align: -1.27767em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">4</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <p>These two expressions are equivalent to each other. For simplicity you can define the relative position of each particle as <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">d</mi><mrow><mi>C</mi><mi>i</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{d}_{Ci}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.84444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">d</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>, but at this stage I want to keep things as explicit as possible.</p> | |
| <table> | |
| <thead> | |
| <tr> | |
| <th align="right">quantity</th> | |
| <th align="left">description</th> | |
| <th align="left">definition</th> | |
| </tr> | |
| </thead> | |
| <tbody> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathdefault">m</span></span></span></span></span></td> | |
| <td align="left">combined masss</td> | |
| <td align="left"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mo>∑</mo><mi>i</mi></msub><msub><mi>m</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\sum_i m_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.04971em; vertical-align: -0.29971em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position: relative; top: -5e-06em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.161954em;"><span class="" style="top: -2.40029em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.29971em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">d</mi><mrow><mi>C</mi><mi>i</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{d}_{Ci}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.84444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">d</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td align="left">relative position of particle</td> | |
| <td align="left"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">d</mi><mrow><mi>C</mi><mi>i</mi></mrow></msub><mo>=</mo><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{d}_{Ci} = \boldsymbol{r}_i - \boldsymbol{r}_C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.84444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">d</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.73333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| </tr> | |
| </tbody> | |
| </table><h3 id="momenta-definitions">Momenta Definitions</h3> | |
| <p>Momenta are <em>defined</em> by summing up the following contributions of all the particles that move together on a rigid body. This summation happens an at arbitrary location <strong>A</strong> in space <em>at every instance</em> in time.</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi mathvariant="bold-italic">p</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>m</mi><mi>i</mi></msub><msub><mi mathvariant="bold-italic">v</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi mathvariant="bold-italic">L</mi><mi>A</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo>∑</mo><mi>i</mi></munder><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><msub><mi>m</mi><mi>i</mi></msub><msub><mi mathvariant="bold-italic">v</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr></mtable></mtd><mtd width="50%"></mtd><mtd><mtext>(2)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \begin{aligned} | |
| \boldsymbol{p} & = \sum_i m_i \boldsymbol{v}_i \\ | |
| \boldsymbol{L}_A & = \sum_i (\boldsymbol{r}_i - \boldsymbol{r}_A) \times m_i \boldsymbol{v}_i | |
| \end{aligned} \tag{2} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 5.25535em; vertical-align: -2.37767em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.87767em;"><span class="" style="top: -4.87767em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span></span></span><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 2.37767em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.87767em;"><span class="" style="top: -4.87767em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 2.37767em;"><span class=""></span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height: 5.25535em; vertical-align: -2.37767em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">2</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <p>where:</p> | |
| <table> | |
| <thead> | |
| <tr> | |
| <th align="right"><em>quantity</em></th> | |
| <th><em>description</em></th> | |
| </tr> | |
| </thead> | |
| <tbody> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="bold-italic">p</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{p}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.63888em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span></span></span></span></span></td> | |
| <td><strike>linear</strike> translational momentum vector.</td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">L</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{L}_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83611em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td><strike>angular</strike> rotational momentum vector of the body measured at <strong>A</strong>.</td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>m</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">m_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td>infinitesimal mass of each particle such that the total mass is <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>m</mi><mo>=</mo><msub><mo>∑</mo><mi>i</mi></msub><msub><mi>m</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">m = \sum_i m_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathdefault">m</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1.04971em; vertical-align: -0.29971em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position: relative; top: -5e-06em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.161954em;"><span class="" style="top: -2.40029em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.29971em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>.</td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{r}_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td>position vector of the particle from the origin.</td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td>velocity vector of the particle.</td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{r}_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td>position vector of arbitrary point <strong>A</strong> from the origin.</td> | |
| </tr> | |
| </tbody> | |
| </table><p>Note that the point <strong>A</strong> might be fixed in space over time, or moving with constant velocity or riding on the body. At any time frame it might have non-zero velocity <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span> with respect to the origin.</p> | |
| <h3 id="kinematics">Kinematics</h3> | |
| <p>The motion of each particle can be decomposed into the motion of the center of mass and a rotation about the center of mass (Chasle’s Theorem).</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>i</mi></msub><mo>=</mo><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub><mo>+</mo><mi mathvariant="bold-italic">ω</mi><mo>×</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo stretchy="false">)</mo></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(3)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \boldsymbol{v}_i = \boldsymbol{v}_C + \boldsymbol{\omega} \times (\boldsymbol{r}_i - \boldsymbol{r}_C) \tag{3} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.73333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.66666em; vertical-align: -0.08333em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">ω</span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span><span class="tag"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">3</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <p>Additionally the combined motion of the center of mass point <strong>C</strong> is expressed on the reference point <strong>A</strong> using the following transformation law(s)</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi mathvariant="bold-italic">v</mi><mi>A</mi></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">ω</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi mathvariant="bold-italic">v</mi><mi>A</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">ω</mi></mrow></mstyle></mtd></mtr></mtable></mtd><mtd width="50%"></mtd><mtd><mtext>(4)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \begin{aligned} \boldsymbol{v}_C &= \boldsymbol{v}_A + (\boldsymbol{r}_A - \boldsymbol{r}_C) \times \boldsymbol{\omega} \\ \boldsymbol{v}_A &= \boldsymbol{v}_C + (\boldsymbol{r}_C - \boldsymbol{r}_A) \times \boldsymbol{\omega} | |
| \end{aligned} \tag{4} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 3em; vertical-align: -1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.75em;"><span class="" style="top: -3.91em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.25em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.75em;"><span class="" style="top: -3.91em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">ω</span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">ω</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.25em;"><span class=""></span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height: 3em; vertical-align: -1.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">4</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <p>Any resemblence of the above to (1) is not coincidence.</p> | |
| <h3 id="translational-momentum">Translational Momentum</h3> | |
| <p>Translational momentum is the easiest to evaluate from (2) and (3). Center of mass expression (4) is used for simplification below</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi mathvariant="bold-italic">p</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>m</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub><mo>+</mo><mi mathvariant="bold-italic">ω</mi><mo>×</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">(</mo><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>m</mi><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub><mo>+</mo><mi mathvariant="bold-italic">ω</mi><mo>×</mo><menclose notation="updiagonalstrike"><mrow><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>m</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo stretchy="false">)</mo></mrow></menclose></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex"> \begin{aligned} | |
| \boldsymbol{p} & = \sum_i m_i ( \boldsymbol{v}_C + \boldsymbol{\omega} \times ( \boldsymbol{r}_i - \boldsymbol{r}_C)) \\ | |
| & = (\sum_i m_i) \boldsymbol{v}_C + \boldsymbol{\omega} \times \cancel{ \sum_i m_i (\boldsymbol{r}_i - \boldsymbol{r}_C) } | |
| \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 5.25535em; vertical-align: -2.37767em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.87767em;"><span class="" style="top: -4.87767em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span></span></span><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 2.37767em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.87767em;"><span class="" style="top: -4.87767em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">ω</span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose">)</span></span></span><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mopen">(</span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">ω</span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord cancel-lap"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -4.32767em;"><span class="pstrut" style="height: 4.32767em;"></span><span class="mord cancel-pad"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span><span class="svg-align" style="top: -3.05001em;"><span class="pstrut" style="height: 4.32767em;"></span><span class="" style="height: 2.32767em;"><svg width="100%" height="2.327674em"><line x1="0" y1="100%" x2="100%" y2="0" stroke-width="0.046em"></line></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 2.37767em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></span></p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi mathvariant="bold-italic">p</mi><mo>=</mo><mi>m</mi> <msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(5)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \boldsymbol{p} = m\,\boldsymbol{v}_C \tag{5} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.63888em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord mathdefault">m</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">5</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <p>The is a well known result, undesputed as long as <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="bold-italic">p</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{p}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.63888em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span></span></span></span></span> and <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span> are measued form <em>the same coordinate frame</em>. Again, <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span> is the velocity vector of the center of mass, as in</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub><mo>=</mo><mstyle displaystyle="false" scriptlevel="0"><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></mstyle><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub></mrow><annotation encoding="application/x-tex"> \boldsymbol{v}_C = \tfrac{\rm d}{{\rm d}t} \boldsymbol{r}_C </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1.22511em; vertical-align: -0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></p> | |
| <h3 id="rotational-momentum-about-center-of-mass">Rotational Momentum about Center of Mass</h3> | |
| <p>Rotational momentum is also evaluated from (2) and (3) but we start with summing about the center of mass first</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi mathvariant="bold-italic">L</mi><mi>C</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo>∑</mo><mi>i</mi></munder><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo stretchy="false">)</mo><mo>×</mo><msub><mi>m</mi><mi>i</mi></msub><msub><mi>v</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>m</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub><mo>+</mo><mi mathvariant="bold-italic">ω</mi><mo>×</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><menclose notation="updiagonalstrike"><mrow><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>m</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo stretchy="false">)</mo></mrow></menclose><mo>×</mo><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub><mo>+</mo><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>m</mi><mi>i</mi></msub><msub><mi mathvariant="bold-italic">d</mi><mrow><mi>C</mi><mi>i</mi></mrow></msub><mo>×</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">ω</mi><mo>×</mo><msub><mi mathvariant="bold-italic">d</mi><mrow><mi>C</mi><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo>∑</mo><mi>i</mi></munder><mrow><mo fence="true">[</mo><mo>−</mo><msub><mi>m</mi><mi>i</mi></msub><msub><mi mathvariant="bold-italic">d</mi><mrow><mi>C</mi><mi>i</mi></mrow></msub><mo>×</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">d</mi><mrow><mi>C</mi><mi>i</mi></mrow></msub><mo>×</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo fence="true">]</mo></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex"> \begin{aligned} \boldsymbol{L}_C & = \sum_i (\boldsymbol{r}_i - \boldsymbol{r}_C) \times m_i v_i \\ | |
| & = \sum_i m_i ( \boldsymbol{r}_i - \boldsymbol{r}_C) \times ( \boldsymbol{v}_C + \boldsymbol{\omega} \times ( \boldsymbol{r}_i - \boldsymbol{r}_C)) \\ | |
| & = \cancel{ \sum_i m_i (\boldsymbol{r}_i - \boldsymbol{r}_C) } \times \boldsymbol{v}_C + \sum_i m_i \boldsymbol{d}_{Ci} \times ( \boldsymbol{\omega} \times \boldsymbol{d}_{Ci} ) \\ | |
| & = \sum_i \left[ - m_i \boldsymbol{d}_{Ci} \times ( \boldsymbol{d}_{Ci} \times \boldsymbol{\omega}) \right] \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 10.5107em; vertical-align: -5.00535em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 5.50535em;"><span class="" style="top: -7.50535em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -4.87767em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"></span></span><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"></span></span><span class="" style="top: 0.377674em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 5.00535em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 5.50535em;"><span class="" style="top: -7.50535em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; 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margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -4.87767em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">ω</span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose">)</span></span></span><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord cancel-lap"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -4.32767em;"><span class="pstrut" style="height: 4.32767em;"></span><span class="mord cancel-pad"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span><span class="svg-align" style="top: -3.05001em;"><span class="pstrut" style="height: 4.32767em;"></span><span class="" style="height: 2.32767em;"><svg width="100%" height="2.327674em"><line x1="0" y1="100%" x2="100%" y2="0" stroke-width="0.046em"></line></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">d</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">ω</span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">d</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span><span class="" style="top: 0.377674em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="minner"><span class="mopen delimcenter" style="top: 0em;">[</span><span class="mord">−</span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">d</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">d</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">ω</span></span></span><span class="mclose">)</span><span class="mclose delimcenter" style="top: 0em;">]</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 5.00535em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></span></p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><msub><mi mathvariant="bold-italic">L</mi><mi>C</mi></msub><mo>=</mo><msub><mi mathvariant="bold">I</mi><mi>C</mi></msub><mi mathvariant="bold-italic">ω</mi></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(6)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \boldsymbol{L}_C = \mathbf{I}_C \boldsymbol{\omega} \tag{6} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83611em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.83611em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">ω</span></span></span></span><span class="tag"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">6</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <p>where <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold">I</mi><mi>C</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{I}_C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83611em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span> is the 3×3 mass moment of inertia tensor (inertia dyatic) derived from factoring <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="bold-italic">ω</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{\omega}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.44444em; vertical-align: 0em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">ω</span></span></span></span></span></span></span> from the rotational inertia expression above.</p> | |
| <p>Practically this is done by evaluating the following sum. For simplicity consider each relative position vector having components <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">d</mi><mrow><mi>C</mi><mi>i</mi></mrow></msub><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mi>i</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>y</mi><mi>i</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>z</mi><mi>i</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\boldsymbol{d}_{Ci} = \begin{pmatrix} x_i \\ y_i \\ z_i \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.84444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">d</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right: 0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.04398em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span> and evaluate</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><msub><mi mathvariant="bold">I</mi><mi>C</mi></msub><mo>=</mo><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>m</mi><mi>i</mi></msub><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msubsup><mi>y</mi><mi>i</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>z</mi><mi>i</mi><mn>2</mn></msubsup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>x</mi><mi>i</mi></msub><msub><mi>y</mi><mi>i</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>x</mi><mi>i</mi></msub><msub><mi>z</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>x</mi><mi>i</mi></msub><msub><mi>y</mi><mi>i</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>z</mi><mi>i</mi><mn>2</mn></msubsup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>y</mi><mi>i</mi></msub><msub><mi>z</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>x</mi><mi>i</mi></msub><msub><mi>z</mi><mi>i</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>y</mi><mi>i</mi></msub><msub><mi>z</mi><mi>i</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>y</mi><mi>i</mi><mn>2</mn></msubsup></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(7)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \mathbf{I}_C = \sum_i m_i \begin{bmatrix} y_i^2 + z_i^2 & -x_i y_i & -x_i z_i \\ -x_i y_i & x_i^2+z_i^2 & -y_i z_i \\ -x_i z_i & -y_i z_i & x_i^2+y_i^2 | |
| \end{bmatrix} \tag{7} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83611em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎣</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.814108em;"><span class="" style="top: -2.44134em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.258664em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.814108em;"><span class="" style="top: -2.44134em; margin-left: -0.04398em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.258664em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.04398em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span><span class="arraycolsep" style="width: 0.5em;"></span><span class="arraycolsep" style="width: 0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.814108em;"><span class="" style="top: -2.44134em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.258664em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.814108em;"><span class="" style="top: -2.44134em; margin-left: -0.04398em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.258664em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.04398em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span><span class="arraycolsep" style="width: 0.5em;"></span><span class="arraycolsep" style="width: 0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.04398em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.04398em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.814108em;"><span class="" style="top: -2.44134em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.258664em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.814108em;"><span class="" style="top: -2.44134em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.258664em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎦</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">7</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <p>Another common alterative to the above is to declare the 3×3 cross product matrix as <span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><msub><mi mathvariant="bold-italic">d</mi><mrow><mi>C</mi><mi>i</mi></mrow></msub><mo>×</mo><mo stretchy="false">]</mo><mo>≡</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>z</mi><mi>i</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>y</mi><mi>i</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>z</mi><mi>i</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>x</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>y</mi><mi>i</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mi>i</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex"> [\boldsymbol{d}_{Ci} \times] \equiv \begin{bmatrix} 0 & -z_i & y_i \\ z_i & 0 & -x_i \\ -y_i & x_i & 0 \end{bmatrix} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">d</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord">×</span><span class="mclose">]</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎣</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right: 0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.04398em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span><span class="arraycolsep" style="width: 0.5em;"></span><span class="arraycolsep" style="width: 0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.04398em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span><span class="arraycolsep" style="width: 0.5em;"></span><span class="arraycolsep" style="width: 0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎦</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></span> and evaluating</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><msub><mi mathvariant="bold">I</mi><mi>C</mi></msub><mo>=</mo><munder><mo>∑</mo><mi>i</mi></munder><mrow><mo fence="true">(</mo><mo>−</mo><msub><mi>m</mi><mi>i</mi></msub><mo stretchy="false">[</mo><msub><mi mathvariant="bold-italic">d</mi><mrow><mi>C</mi><mi>i</mi></mrow></msub><mo>×</mo><mo stretchy="false">]</mo><mo stretchy="false">[</mo><msub><mi mathvariant="bold-italic">d</mi><mrow><mi>C</mi><mi>i</mi></mrow></msub><mo>×</mo><mo stretchy="false">]</mo><mo fence="true">)</mo></mrow></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(8)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \mathbf{I}_C = \sum_i \left( -m_i [\boldsymbol{d}_{Ci} \times][\boldsymbol{d}_{Ci} \times] \right) \tag{8} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83611em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 2.32767em; vertical-align: -1.27767em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="minner"><span class="mopen delimcenter" style="top: 0em;">(</span><span class="mord">−</span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mopen">[</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">d</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord">×</span><span class="mclose">]</span><span class="mopen">[</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">d</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord">×</span><span class="mclose">]</span><span class="mclose delimcenter" style="top: 0em;">)</span></span></span><span class="tag"><span class="strut" style="height: 2.32767em; vertical-align: -1.27767em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">8</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <h3 id="rotational-momentum-about-arbitrary-point">Rotational Momentum about Arbitrary Point</h3> | |
| <p>Rotational momentum is also evaluated from (2) and (3) but we sum about the arbitrary point <strong>A</strong>.</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi mathvariant="bold-italic">L</mi><mi>A</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo>∑</mo><mi>i</mi></munder><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><msub><mi>m</mi><mi>i</mi></msub><msub><mi>v</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo>∑</mo><mi>i</mi></munder><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo stretchy="false">)</mo><mo>×</mo><msub><mi>m</mi><mi>i</mi></msub><msub><mi>v</mi><mi>i</mi></msub><mo>+</mo><munder><mo>∑</mo><mi>i</mi></munder><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><msub><mi>m</mi><mi>i</mi></msub><msub><mi>v</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi mathvariant="bold-italic">L</mi><mi>C</mi></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>m</mi><mi>i</mi></msub><msub><mi>v</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex"> \begin{aligned} \boldsymbol{L}_A & = \sum_i (\boldsymbol{r}_i - \boldsymbol{r}_A) \times m_i v_i \\ | |
| & = \sum_i (\boldsymbol{r}_i - \boldsymbol{r}_C) \times m_i v_i + \sum_i (\boldsymbol{r}_C - \boldsymbol{r}_A) \times m_i v_i \\ | |
| & = \boldsymbol{L}_C + (\boldsymbol{r}_C-\boldsymbol{r}_A) \times \sum_i m_i v_i | |
| \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 7.88302em; vertical-align: -3.69151em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 4.19151em;"><span class="" style="top: -6.19151em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.56384em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"></span></span><span class="" style="top: -0.936163em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 3.69151em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 4.19151em;"><span class="" style="top: -6.19151em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.56384em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -0.936163em;"><span class="pstrut" style="height: 3.05001em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.05001em;"><span class="" style="top: -1.87233em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span class="" style="top: -3.05001em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.27767em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.311664em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 3.69151em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></span></p> | |
| <p>Which leads the the following transformation law(s)</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi mathvariant="bold-italic">L</mi><mi>A</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi mathvariant="bold-italic">L</mi><mi>C</mi></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">p</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi mathvariant="bold-italic">L</mi><mi>C</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi mathvariant="bold-italic">L</mi><mi>A</mi></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">p</mi></mrow></mstyle></mtd></mtr></mtable></mtd><mtd width="50%"></mtd><mtd><mtext>(9)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \begin{aligned} \boldsymbol{L}_A & = \boldsymbol{L}_C + (\boldsymbol{r}_C-\boldsymbol{r}_A) \times \boldsymbol{p} \\ \boldsymbol{L}_C & = \boldsymbol{L}_A + (\boldsymbol{r}_A-\boldsymbol{r}_C) \times \boldsymbol{p} | |
| \end{aligned} \tag{9} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 3em; vertical-align: -1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.75em;"><span class="" style="top: -3.91em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.25em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.75em;"><span class="" style="top: -3.91em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.25em;"><span class=""></span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height: 3em; vertical-align: -1.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">9</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <p>Again, any resemblance to (1) and (4) is not a coincidence. This is because these are Plücker coordinates of different lines in space. The force line is called the <em>line of action</em>. The motion line is called the <em>rotation axis</em>. And the momentum line is called <em>axis of percussion</em>.</p> | |
| <h3 id="dynamics">Dynamics</h3> | |
| <p>We establish Newton’s second law about the center of mass <strong>C</strong> relating forces/torques to rate of change of momenta</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi mathvariant="bold-italic">F</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mstyle displaystyle="false" scriptlevel="0"><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></mstyle><mi mathvariant="bold-italic">p</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi mathvariant="bold-italic">τ</mi><mi>C</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mstyle displaystyle="false" scriptlevel="0"><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></mstyle><msub><mi mathvariant="bold-italic">L</mi><mi>C</mi></msub></mrow></mstyle></mtd></mtr></mtable></mtd><mtd width="50%"></mtd><mtd><mtext>(10)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \begin{aligned} | |
| \boldsymbol{F} & = \tfrac{\rm d}{{\rm d}t} \boldsymbol{p} \\ | |
| \boldsymbol{\tau}_C & = \tfrac{\rm d}{{\rm d}t} \boldsymbol{L}_C | |
| \end{aligned} \tag{10} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 3.08022em; vertical-align: -1.29011em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.79011em;"><span class="" style="top: -3.91em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.15972em;">F</span></span></span></span></span><span class="" style="top: -2.36989em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.29011em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.79011em;"><span class="" style="top: -3.91em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span></span></span><span class="" style="top: -2.36989em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.29011em;"><span class=""></span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height: 3.08022em; vertical-align: -1.29011em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">1</span><span class="mord">0</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <p>Using the definitions above from particle summation the expression of equations of motion are straigtforward when using rhe center of mass as a reference point.</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi mathvariant="bold-italic">F</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>m</mi><msub><mi mathvariant="bold-italic">a</mi><mi>C</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi mathvariant="bold-italic">τ</mi><mi>C</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi mathvariant="bold">I</mi><mi>C</mi></msub><mi mathvariant="bold-italic">α</mi><mo>+</mo><mi mathvariant="bold-italic">ω</mi><mo>×</mo><msub><mi mathvariant="bold-italic">L</mi><mi>C</mi></msub></mrow></mstyle></mtd></mtr></mtable></mtd><mtd width="50%"></mtd><mtd><mtext>(11)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \begin{aligned} | |
| \boldsymbol{F} & = m \boldsymbol{a}_C \\ | |
| \boldsymbol{\tau}_C & = \mathbf{I}_C \boldsymbol{\alpha} + \boldsymbol{\omega} \times \boldsymbol{L}_C | |
| \end{aligned} \tag{11} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 3em; vertical-align: -1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.75em;"><span class="" style="top: -3.91em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.15972em;">F</span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.25em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.75em;"><span class="" style="top: -3.91em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord mathdefault">m</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord boldsymbol">α</span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">ω</span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.25em;"><span class=""></span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height: 3em; vertical-align: -1.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">1</span><span class="mord">1</span></span><span class="mord">)</span></span></span></span></span></span></span><br> | |
| where</p> | |
| <table> | |
| <thead> | |
| <tr> | |
| <th align="right">quantity</th> | |
| <th align="left">description</th> | |
| <th align="left">definition</th> | |
| </tr> | |
| </thead> | |
| <tbody> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">a</mi><mi>C</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{a}_C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td align="left">translational acceleration of the center of mass</td> | |
| <td align="left"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub></mrow><annotation encoding="application/x-tex">\tfrac{\rm d}{{\rm d}t} \boldsymbol{v}_C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.22511em; vertical-align: -0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="bold-italic">α</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{\alpha}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.44444em; vertical-align: 0em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">α</span></span></span></span></span></span></span></td> | |
| <td align="left">rotational acceleration of the body</td> | |
| <td align="left"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mi mathvariant="bold-italic">ω</mi></mrow><annotation encoding="application/x-tex">\tfrac{\rm d}{{\rm d}t} \boldsymbol{\omega}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.22511em; vertical-align: -0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">ω</span></span></span></span></span></span></span></td> | |
| </tr> | |
| </tbody> | |
| </table><h3 id="arbitrary-point">Arbitrary Point</h3> | |
| <p>The question is can the equations of motion above can be derived from Newton’s 2nd law <em>at an arbitrary location</em>?</p> | |
| <p><strong>Is the following valid?</strong></p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><msub><mi mathvariant="bold-italic">τ</mi><mi>A</mi></msub><mo>=</mo><mstyle displaystyle="false" scriptlevel="0"><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></mstyle><msub><mi mathvariant="bold-italic">L</mi><mi>A</mi></msub></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(12)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \boldsymbol{\tau}_A = \tfrac{\rm d}{{\rm d}t} \boldsymbol{L}_A \tag{12} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1.22511em; vertical-align: -0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height: 1.22511em; vertical-align: -0.345em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">1</span><span class="mord">2</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <p>We can use the transformation laws (1) and (9) described above to see if (12) can lead to (11) which we know is correct.</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi mathvariant="bold-italic">τ</mi><mi>C</mi></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">F</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mstyle displaystyle="false" scriptlevel="0"><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></mstyle><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">L</mi><mi>C</mi></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">p</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mstyle displaystyle="false" scriptlevel="0"><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></mstyle><msub><mi mathvariant="bold-italic">L</mi><mi>C</mi></msub><mo>+</mo><mstyle displaystyle="false" scriptlevel="0"><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></mstyle><mrow><mo fence="true">(</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">p</mi><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><menclose notation="updiagonalstrike"><msub><mi mathvariant="bold-italic">τ</mi><mi>C</mi></msub></menclose><mo>+</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">F</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><menclose notation="updiagonalstrike"><msub><mi mathvariant="bold-italic">τ</mi><mi>C</mi></msub></menclose><mo>+</mo><mstyle displaystyle="false" scriptlevel="0"><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></mstyle><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">p</mi><mo>+</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mstyle displaystyle="false" scriptlevel="0"><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></mstyle><mi mathvariant="bold-italic">p</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><menclose notation="updiagonalstrike"><mrow><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">F</mi></mrow></menclose></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">v</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">p</mi><mo>+</mo><menclose notation="updiagonalstrike"><mrow><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">r</mi><mi>C</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">r</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">F</mi></mrow></menclose></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mn mathvariant="bold-italic">0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi mathvariant="bold-italic">p</mi><mo>×</mo><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">v</mi><mi>A</mi></msub><mo>−</mo><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mn mathvariant="bold-italic">0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>m</mi> <msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub><mo>×</mo><msub><mi mathvariant="bold-italic">v</mi><mi>A</mi></msub></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex"> \begin{aligned} | |
| \boldsymbol{\tau}_C + (\boldsymbol{r}_C - \boldsymbol{r}_A) \times \boldsymbol{F} & = \tfrac{\rm d}{{\rm d}t} ( \boldsymbol{L}_C + (\boldsymbol{r}_C-\boldsymbol{r}_A) \times \boldsymbol{p} ) \\ | |
| & = \tfrac{\rm d}{{\rm d}t} \boldsymbol{L}_C + \tfrac{\rm d}{{\rm d}t} \left( (\boldsymbol{r}_C-\boldsymbol{r}_A) \times \boldsymbol{p} \right) \\ | |
| \cancel{\boldsymbol{\tau}_C} + (\boldsymbol{r}_C - \boldsymbol{r}_A) \times \boldsymbol{F} & = \cancel{\boldsymbol{\tau}_C} + \tfrac{\rm d}{{\rm d}t} (\boldsymbol{r}_C-\boldsymbol{r}_A) \times \boldsymbol{p} + (\boldsymbol{r}_C-\boldsymbol{r}_A) \times \tfrac{\rm d}{{\rm d}t} \boldsymbol{p} \\ | |
| \cancel{(\boldsymbol{r}_C - \boldsymbol{r}_A) \times \boldsymbol{F}} &= ( \boldsymbol{v}_C - \boldsymbol{v}_A)\times \boldsymbol{p} + \cancel{ (\boldsymbol{r}_C - \boldsymbol{r}_A) \times \boldsymbol{F}} \\ \boldsymbol{0} & = | |
| \boldsymbol{p} \times ( \boldsymbol{v}_A - \boldsymbol{v}_C) \\ | |
| \boldsymbol{0} & = m\,\boldsymbol{v}_C \times \boldsymbol{v}_A | |
| \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 9.12032em; vertical-align: -4.31016em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 4.81016em;"><span class="" style="top: -6.93005em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.15972em;">F</span></span></span></span></span><span class="" style="top: -5.38995em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"></span></span><span class="" style="top: -3.84984em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord cancel-lap"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.44444em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord cancel-pad"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="svg-align" style="top: -2.85em;"><span class="pstrut" style="height: 3em;"></span><span class="" style="height: 0.59444em;"><svg width="100%" height="0.59444em"><line x1="0" y1="100%" x2="100%" y2="0" stroke-width="0.046em"></line></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.15972em;">F</span></span></span></span></span><span class="" style="top: -2.34984em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord cancel-lap"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.75em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord cancel-pad"><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.15972em;">F</span></span></span></span></span><span class="svg-align" style="top: -2.75em;"><span class="pstrut" style="height: 3em;"></span><span class="" style="height: 1em;"><svg width="100%" height="1em"><line x1="0" y1="100%" x2="100%" y2="0" stroke-width="0.046em"></line></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.25em;"><span class=""></span></span></span></span></span></span></span><span class="" style="top: -0.849838em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathbf">0</span></span></span></span></span><span class="" style="top: 0.650162em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathbf">0</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 4.31016em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 4.81016em;"><span class="" style="top: -6.93005em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="mclose">)</span></span></span><span class="" style="top: -5.38995em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="minner"><span class="mopen delimcenter" style="top: 0em;">(</span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="mclose delimcenter" style="top: 0em;">)</span></span></span></span><span class="" style="top: -3.84984em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord cancel-lap"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.44444em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord cancel-pad"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="svg-align" style="top: -2.85em;"><span class="pstrut" style="height: 3em;"></span><span class="" style="height: 0.59444em;"><svg width="100%" height="0.59444em"><line x1="0" y1="100%" x2="100%" y2="0" stroke-width="0.046em"></line></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span></span></span><span class="" style="top: -2.34984em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord cancel-lap"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.75em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord cancel-pad"><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.15972em;">F</span></span></span></span></span><span class="svg-align" style="top: -2.75em;"><span class="pstrut" style="height: 3em;"></span><span class="" style="height: 1em;"><svg width="100%" height="1em"><line x1="0" y1="100%" x2="100%" y2="0" stroke-width="0.046em"></line></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.25em;"><span class=""></span></span></span></span></span></span></span><span class="" style="top: -0.849838em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span><span class="" style="top: 0.650162em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord mathdefault">m</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 4.31016em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></span></p> | |
| <p>The <em>necessary</em> condition(s) for (12) to be correct are as follows</p> | |
| <ul> | |
| <li>Reference point <strong>A</strong> is fixed in space.</li> | |
| <li>Reference point <strong>A</strong> is co-moving with the center of mass.</li> | |
| <li>Center of mass <strong>C</strong> is fixed in space.</li> | |
| </ul> | |
| <p>The general rule connecting equipollent torque to change in rotational momentum is thus</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><msub><mi mathvariant="bold-italic">τ</mi><mi>A</mi></msub><mo>=</mo><mstyle displaystyle="false" scriptlevel="0"><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></mstyle><msub><mi mathvariant="bold-italic">L</mi><mi>A</mi></msub><mo>+</mo><msub><mi mathvariant="bold-italic">v</mi><mi>A</mi></msub><mo>×</mo><mi mathvariant="bold-italic">p</mi></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(13)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex"> \boldsymbol{\tau}_A = \tfrac{\rm d}{{\rm d}t} \boldsymbol{L}_A + \boldsymbol{v}_A \times \boldsymbol{p} \tag{13} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1.22511em; vertical-align: -0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.73333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.63888em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span></span><span class="tag"><span class="strut" style="height: 1.22511em; vertical-align: -0.345em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">1</span><span class="mord">3</span></span><span class="mord">)</span></span></span></span></span></span></span></p> | |
| <p>where</p> | |
| <table> | |
| <thead> | |
| <tr> | |
| <th align="right">quantity</th> | |
| <th align="left">description</th> | |
| </tr> | |
| </thead> | |
| <tbody> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">τ</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\tau}_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td align="left">equipollent torque acting on the body (instanteneously) summed on the reference point <strong>A</strong>.</td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">L</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{L}_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83611em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td align="left">rotational momentum of rigid body summed on the reference point <strong>A</strong>.</td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="bold-italic">p</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{p}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.63888em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span></span></span></span></span></td> | |
| <td align="left">translational momentum of the rigid body.</td> | |
| </tr> | |
| <tr> | |
| <td align="right"><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></td> | |
| <td align="left">instanteneous velocity of reference point <strong>A</strong>.</td> | |
| </tr> | |
| </tbody> | |
| </table><h3 id="example">Example</h3> | |
| <p>Consider a rod sliding vertically along a rail with speed <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>v</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">v_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span> and pivoting with rotational speed <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span></span></span> about a point <strong>A</strong> away from the center of mass <strong>C</strong>. A force of magnitude <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>F</mi><mi>B</mi></msub></mrow><annotation encoding="application/x-tex">F_B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span> is applied horizontally at another point <strong>B</strong>.<br> | |
| <img src="https://i.imgur.com/JwNY2Oa.png" alt="sliding rod"></p> | |
| <p>The variable <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathdefault">c</span></span></span></span></span> is the distance between the pivot and the center of mass, and <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.69444em; vertical-align: 0em;"></span><span class="mord">ℓ</span></span></span></span></span> the distance between the pivot and the force application point.</p> | |
| <p><strong>Can (12) produce the correct equation of motion?</strong></p> | |
| <p>First we develop the equations of motion about the center of mass</p> | |
| <ol> | |
| <li><strong>Kinematics</strong> the center of mass <strong>C</strong> moves under two degrees of freedom, the sliding velocity <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>v</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">v_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span> and the rotation <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span></span></span> | |
| <ul> | |
| <li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub><mo>=</mo><msub><mi>v</mi><mi>A</mi></msub> <mover accent="true"><mi mathvariant="bold-italic">j</mi><mo>^</mo></mover><mo>+</mo><mi>ω</mi><mover accent="true"><mi mathvariant="bold-italic">k</mi><mo>^</mo></mover><mo>×</mo><mi>c</mi><mover accent="true"><mi mathvariant="bold-italic">j</mi><mo>^</mo></mover><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mi>c</mi> <mi>ω</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>v</mi><mi>A</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_C = v_A\, \boldsymbol{\hat{j}} + \omega \boldsymbol{\hat{k}} \times c \boldsymbol{\hat{j}} = \begin{pmatrix} -c\, \omega \\ v_A \\ 0\end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1.15114em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.9567em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.0622em;">j</span></span></span><span class="" style="top: -3.26226em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.25em;">^</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.19444em;"><span class=""></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1.04121em; vertical-align: -0.08333em;"></span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.95788em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.01852em;">k</span></span></span><span class="" style="top: -3.26344em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.25em;">^</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1.15114em; vertical-align: -0.19444em;"></span><span class="mord mathdefault">c</span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.9567em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.0622em;">j</span></span></span><span class="" style="top: -3.26226em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.25em;">^</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.19444em;"><span class=""></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></li> | |
| <li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">a</mi><mi>C</mi></msub><mo>=</mo><msub><mover accent="true"><mi>v</mi><mo>˙</mo></mover><mi>A</mi></msub><mover accent="true"><mi mathvariant="bold-italic">j</mi><mo>^</mo></mover><mo>+</mo><mover accent="true"><mi>ω</mi><mo>˙</mo></mover><mover accent="true"><mi mathvariant="bold-italic">k</mi><mo>^</mo></mover><mo>×</mo><mi>c</mi><mover accent="true"><mi mathvariant="bold-italic">j</mi><mo>^</mo></mover><mo>+</mo><mi>ω</mi><mover accent="true"><mi mathvariant="bold-italic">k</mi><mo>^</mo></mover><mo>×</mo><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mi>c</mi> <mover accent="true"><mi>ω</mi><mo>˙</mo></mover><mo>−</mo><mi>ω</mi> <msub><mi>v</mi><mi>A</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mover accent="true"><mi>v</mi><mo>˙</mo></mover><mi>A</mi></msub><mo>−</mo><mi>c</mi> <msup><mi>ω</mi><mn>2</mn></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\boldsymbol{a}_C =\dot{v}_A \boldsymbol{\hat{j}} + \dot{\omega}\boldsymbol{\hat{k}}\times c \boldsymbol{\hat{j}} + \omega \boldsymbol{\hat{k}} \times \boldsymbol{v}_C = \begin{pmatrix} -c\,\dot{\omega} - \omega\, v_A \\ \dot{v}_A - c\, \omega^2 \\ 0 \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1.15114em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.11111em;">˙</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.9567em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.0622em;">j</span></span></span><span class="" style="top: -3.26226em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.25em;">^</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.19444em;"><span class=""></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1.04121em; vertical-align: -0.08333em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.13889em;">˙</span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.95788em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.01852em;">k</span></span></span><span class="" style="top: -3.26344em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.25em;">^</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1.15114em; vertical-align: -0.19444em;"></span><span class="mord mathdefault">c</span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.9567em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.0622em;">j</span></span></span><span class="" style="top: -3.26226em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.25em;">^</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.19444em;"><span class=""></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1.04121em; vertical-align: -0.08333em;"></span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.95788em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.01852em;">k</span></span></span><span class="" style="top: -3.26344em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.25em;">^</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.13889em;">˙</span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.11111em;">˙</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.814108em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></li> | |
| </ul> | |
| </li> | |
| <li><strong>Momenta</strong> directly calculated from the motion of the center of mass | |
| <ul> | |
| <li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="bold-italic">p</mi><mo>=</mo><mi>m</mi><msub><mi mathvariant="bold-italic">v</mi><mi>C</mi></msub><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mi>m</mi> <mi>c</mi> <mi>ω</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>m</mi> <msub><mi>v</mi><mi>A</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\boldsymbol{p} = m \boldsymbol{v}_C = \begin{pmatrix} -m \,c\,\omega \\ m\,v_A \\ 0 \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.63888em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord mathdefault">m</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord mathdefault">m</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></li> | |
| <li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">L</mi><mi>C</mi></msub><mo>=</mo><msub><mi mathvariant="bold">I</mi><mi>C</mi></msub><mi mathvariant="bold-italic">ω</mi><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>I</mi><mi>C</mi></msub><mi>ω</mi></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\boldsymbol{L}_C = \mathbf{I}_C \boldsymbol{\omega} = \begin{pmatrix} 0 \\ 0 \\ I_C \omega \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83611em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.83611em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">ω</span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right: 0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.07847em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></li> | |
| </ul> | |
| </li> | |
| <li><strong>Equipollent forces and moments</strong> at the center of mass <strong>C</strong> are a combination of the appled force <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>F</mi><mi>B</mi></msub></mrow><annotation encoding="application/x-tex">F_B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span> and the pin reaction force <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>A</mi><mi>x</mi></msub></mrow><annotation encoding="application/x-tex">A_x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathdefault">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>. | |
| <ul> | |
| <li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="bold-italic">F</mi><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>F</mi><mi>B</mi></msub><mo>+</mo><msub><mi>A</mi><mi>x</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\boldsymbol{F} = \begin{pmatrix} F_B+A_x \\ 0 \\ 0 \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.68611em; vertical-align: 0em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.15972em;">F</span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right: 0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord mathdefault">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></li> | |
| <li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">τ</mi><mi>C</mi></msub><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>c</mi><msub><mi>A</mi><mi>x</mi></msub><mo>−</mo><mo stretchy="false">(</mo><mi mathvariant="normal">ℓ</mi><mo>−</mo><mi>c</mi><mo stretchy="false">)</mo><msub><mi>F</mi><mi>B</mi></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\boldsymbol{\tau}_C = \begin{pmatrix} 0 \\ 0 \\ c A_x -(\ell-c) F_B \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault">c</span><span class="mord"><span class="mord mathdefault">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord">ℓ</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault">c</span><span class="mclose">)</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></li> | |
| </ul> | |
| </li> | |
| <li><strong>Equations of motion</strong> | |
| <ul> | |
| <li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>F</mi><mi>B</mi></msub><mo>+</mo><msub><mi>A</mi><mi>x</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><mo>=</mo><mi>m</mi><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mi>c</mi> <mover accent="true"><mi>ω</mi><mo>˙</mo></mover><mo>−</mo><mi>ω</mi> <msub><mi>v</mi><mi>A</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mover accent="true"><mi>v</mi><mo>˙</mo></mover><mi>A</mi></msub><mo>−</mo><mi>c</mi> <msup><mi>ω</mi><mn>2</mn></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{pmatrix} F_B+A_x \\ 0 \\ 0 \end{pmatrix} = m \begin{pmatrix} -c\,\dot{\omega} - \omega\, v_A \\ \dot{v}_A - c\, \omega^2 \\ 0 \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right: 0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mord mathdefault">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="mord mathdefault">m</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.13889em;">˙</span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.11111em;">˙</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.814108em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></li> | |
| <li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>c</mi><msub><mi>A</mi><mi>x</mi></msub><mo>−</mo><mo stretchy="false">(</mo><mi mathvariant="normal">ℓ</mi><mo>−</mo><mi>c</mi><mo stretchy="false">)</mo><msub><mi>F</mi><mi>B</mi></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><mo>=</mo><msub><mi>I</mi><mi>C</mi></msub><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mover accent="true"><mi>ω</mi><mo>˙</mo></mover></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{pmatrix} 0 \\ 0 \\ c A_x -(\ell-c) F_B \end{pmatrix} = I_C \begin{pmatrix} 0 \\ 0 \\ \dot{\omega} \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault">c</span><span class="mord"><span class="mord mathdefault">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mopen">(</span><span class="mord">ℓ</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault">c</span><span class="mclose">)</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.07847em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.13889em;">˙</span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></li> | |
| </ul> | |
| </li> | |
| <li><strong>Solution</strong> | |
| <ul> | |
| <li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mover accent="true"><mi>ω</mi><mo>˙</mo></mover><mo>=</mo><mo>−</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi mathvariant="normal">ℓ</mi><msub><mi>F</mi><mi>B</mi></msub><mo>+</mo><mi>m</mi> <mi>c</mi> <mi>ω</mi> <msub><mi>v</mi><mi>A</mi></msub></mrow><mrow><msub><mi>I</mi><mi>C</mi></msub><mo>+</mo><mi>m</mi><msup><mi>c</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\dot{\omega} = - \dfrac{\ell F_B + m\,c\,\omega\,v_A}{I_C + m c^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.66786em; vertical-align: 0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.13889em;">˙</span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 2.20744em; vertical-align: -0.836em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.37144em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right: 0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.07847em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault">m</span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.740108em;"><span class="" style="top: -2.989em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">ℓ</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault">m</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.836em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></li> | |
| <li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mover accent="true"><mi>v</mi><mo>˙</mo></mover><mi>A</mi></msub><mo>=</mo><mi>c</mi> <msup><mi>ω</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\dot{v}_A = c\,\omega^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.81786em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.11111em;">˙</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.814108em; vertical-align: 0em;"></span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.814108em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></li> | |
| <li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>A</mi><mi>x</mi></msub><mo>=</mo><mo>−</mo><mrow><mo fence="true">(</mo><mn>1</mn><mo>−</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>m</mi> <mi>c</mi> <mi mathvariant="normal">ℓ</mi></mrow><mrow><msub><mi>I</mi><mi>C</mi></msub><mo>+</mo><mi>m</mi><msup><mi>c</mi><mn>2</mn></msup></mrow></mfrac></mstyle><mo fence="true">)</mo></mrow><msub><mi>F</mi><mi>B</mi></msub><mo>−</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>ω</mi> <msub><mi>v</mi><mi>A</mi></msub></mrow><mrow><mfrac><mn>1</mn><mi>m</mi></mfrac><mo>+</mo><mfrac><msup><mi>c</mi><mn>2</mn></msup><msub><mi>I</mi><mi>C</mi></msub></mfrac></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">A_x = -\left( 1-\dfrac{m\,c\,\ell}{I_C + m c^2} \right) F_B - \dfrac{\omega\,v_A}{ \tfrac{1}{m} + \tfrac{c^2}{I_C}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathdefault">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 2.40003em; vertical-align: -0.95003em;"></span><span class="mord">−</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="minner"><span class="mopen delimcenter" style="top: 0em;"><span class="delimsizing size3">(</span></span><span class="mord">1</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.37144em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right: 0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.07847em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault">m</span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.740108em;"><span class="" style="top: -2.989em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord">ℓ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.836em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top: 0em;"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 2.35928em; vertical-align: -1.25172em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.10756em;"><span class="" style="top: -2.19358em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.845108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.91642em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3448em;"><span class="" style="top: -2.35671em; margin-left: -0.07847em; margin-right: 0.0714286em;"><span class="pstrut" style="height: 2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.143293em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.746314em;"><span class="" style="top: -2.786em; margin-right: 0.0714286em;"><span class="pstrut" style="height: 2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.445305em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.25172em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></li> | |
| </ul> | |
| </li> | |
| </ol> | |
| <p>Now we apply equation (12) directly after the rotational momentum about <strong>A</strong> is calculated from (9).<br> | |
| - <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">L</mi><mi>A</mi></msub><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo stretchy="false">(</mo><msub><mi>I</mi><mi>C</mi></msub><mo>+</mo><mi>m</mi><msup><mi>c</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mi>ω</mi></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\boldsymbol{L}_A = \begin{pmatrix} 0 \\ 0 \\ (I_C + m c^2) \omega \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83611em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.07847em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault">m</span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.814108em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span><br> | |
| - <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">τ</mi><mi>A</mi></msub><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mi mathvariant="normal">ℓ</mi><msub><mi>F</mi><mi>B</mi></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\boldsymbol{\tau}_A = \begin{pmatrix} 0 \\ 0 \\ -\ell F_B \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">ℓ</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span><br> | |
| - <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><msub><mi mathvariant="bold-italic">L</mi><mi>A</mi></msub><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo stretchy="false">(</mo><msub><mi>I</mi><mi>C</mi></msub><mo>+</mo><mi>m</mi><msup><mi>c</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mover accent="true"><mi>ω</mi><mo>˙</mo></mover></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\tfrac{\rm d}{{\rm d}t}\boldsymbol{L}_A = \begin{pmatrix} 0 \\ 0 \\ (I_C + m c^2) \dot{\omega} \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.22511em; vertical-align: -0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.07847em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault">m</span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.814108em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.13889em;">˙</span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span><br> | |
| - <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>A</mi></msub><mo>×</mo><mi mathvariant="bold-italic">p</mi><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>m</mi> <mi>c</mi> <mi>ω</mi> <msub><mi>v</mi><mi>A</mi></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_A \times \boldsymbol{p} = \begin{pmatrix} 0 \\ 0 \\ m\,c\,\omega\,v_A \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.73333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.63888em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 3.60004em; vertical-align: -1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎝</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎛</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.21em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -3.01em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -1.81em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55em;"><span class=""></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05002em;"><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎠</span></span></span><span class="" style="top: -4.05002em;"><span class="pstrut" style="height: 3.155em;"></span><span class="delimsizinginner delim-size4"><span class="">⎞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.55002em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></p> | |
| <p>All together equation (12) is</p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>−</mo><mi mathvariant="normal">ℓ</mi><msub><mi>F</mi><mi>B</mi></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>I</mi><mi>C</mi></msub><mo>+</mo><mi>m</mi><msup><mi>c</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mover accent="true"><mi>ω</mi><mo>˙</mo></mover><mo>+</mo><mi>m</mi><mi>c</mi><mi>ω</mi><msub><mi>v</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex"> -\ell F_B = (I_C + m c^2) \dot{\omega} + m c \omega v_A </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.84444em; vertical-align: -0.15em;"></span><span class="mord">−</span><span class="mord">ℓ</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.07847em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1.11411em; vertical-align: -0.25em;"></span><span class="mord mathdefault">m</span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.864108em;"><span class="" style="top: -3.113em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.13889em;">˙</span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord mathdefault">m</span><span class="mord mathdefault">c</span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></p> | |
| <p>which is solved for <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mover accent="true"><mi>ω</mi><mo>˙</mo></mover></mrow><annotation encoding="application/x-tex">\dot{\omega}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.66786em; vertical-align: 0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.13889em;">˙</span></span></span></span></span></span></span></span></span></span> to produce <strong>the exact same same solution as above</strong></p> | |
| <p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mover accent="true"><mi>ω</mi><mo>˙</mo></mover><mo>=</mo><mo>−</mo><mfrac><mrow><mi mathvariant="normal">ℓ</mi><msub><mi>F</mi><mi>B</mi></msub><mo>+</mo><mi>m</mi> <mi>c</mi> <mi>ω</mi> <msub><mi>v</mi><mi>A</mi></msub></mrow><mrow><msub><mi>I</mi><mi>C</mi></msub><mo>+</mo><mi>m</mi><msup><mi>c</mi><mn>2</mn></msup></mrow></mfrac> <mi mathvariant="normal">✓</mi></mrow><annotation encoding="application/x-tex">\dot{\omega} = - \dfrac{\ell F_B + m\,c\,\omega\,v_A}{I_C + m c^2}\;\;\checkmark</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.66786em; vertical-align: 0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.13889em;">˙</span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 2.20744em; vertical-align: -0.836em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.37144em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right: 0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.07847em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault">m</span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.740108em;"><span class="" style="top: -2.989em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">ℓ</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault">m</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.836em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord amsrm">✓</span></span></span></span></span></span></p> | |
| <blockquote> | |
| <p>As in illustration we can see that <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="bold-italic">τ</mi><mi>A</mi></msub><mo>=</mo><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><msub><mi mathvariant="bold-italic">L</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\tau}_A = \tfrac{\rm d}{{\rm d}t} \boldsymbol{L}_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.59444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right: 0.13472em;">τ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1.22511em; vertical-align: -0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.880108em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span><span class="mord mathdefault mtight">t</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span> would produce the incorrect result<br> | |
| <span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mover accent="true"><mi>ω</mi><mo>˙</mo></mover><mo>=</mo><mo>−</mo><mfrac><mrow><mi mathvariant="normal">ℓ</mi><msub><mi>F</mi><mi>B</mi></msub></mrow><mrow><msub><mi>I</mi><mi>C</mi></msub><mo>+</mo><mi>m</mi><msup><mi>c</mi><mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">\dot{\omega} = - \dfrac{\ell F_B}{I_C + m c^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.66786em; vertical-align: 0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.66786em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">ω</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.13889em;">˙</span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 2.20744em; vertical-align: -0.836em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.37144em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right: 0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.07847em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord mathdefault">m</span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.740108em;"><span class="" style="top: -2.989em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">ℓ</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right: 0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.836em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span></p> | |
| </blockquote> | |
| </div> | |
| </body> | |
| </html> |
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