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\title{Brief Article}
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\begin{document}

\section{First section}

\begin{eqnarray}
I_{system}&=& \frac{1}{2}\left(R^2_{in_{2}}-R^2_{in_{1}}\right)
\end{eqnarray}

The equation of motion of the system is:

\begin{eqnarray}
a\frac{d\omega}{dt}+b\omega+c&=&0 \label{EoM}\\
a&=& I_{system} \\
b&=& 2 \mu R^3_{in_2}\frac{\pi L}{t g}+\mu\pi\frac{R^4_{in_2}}{2 h}\\
c&=& R_{spool}mg - M_f
\end{eqnarray}

Rearranging the equation of motion (Eqn. \ref{EoM}) you get Eqn. \ref{EoM2}.

\begin{eqnarray}
a d \omega &=& \left(c-b \omega \right) dt \label{EoM2} \\
\int^{\omega_2}_{\omega_1} a d \omega &=& \int^{t_2}_{t_1}\left(c - b \omega\right) dt \\
a \left(\omega_2 - \omega_1\right) &=& \left(c-b \omega\right)\left(t_2-t_2\right) \\
t_1 &=& 0 \\
\omega_1 &=& 0
\end{eqnarray}

Using angular velocity as $0.4666$ RPS with an unknown $t_2$ results in the simplifications shown in Eqn. \ref{finalEqn}

\begin{eqnarray}
\left(\frac{1}{2}\left(0.086 \text{kg}\right)\left(\left(0.075438 \text{mm} \right)^2 - \left(69.09 mm\right)^2\right)\right)\left(0.46666 \text{s}^{-1}\right) \nonumber \\ = \left(10.16 \text{mm}\right)\left(0.086\text{kg}\right)\left(9.81 \frac{\text{m}}{\text{s}^2}\right)\ \label{finalEqn}
\end{eqnarray}


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