Physics 222 Research Project Notes & LaTeX Learning Experience

phys222-project-notes.tex

\documentclass{article}
\usepackage{amsmath}


\title{Damped Oscillations Research}
\author{ Mitchell Pell\\ pell.mitchell@gmail.com }
\date{02/21/2017}

\begin{document}

\maketitle

\section{ A Learning Experience }

This document is a rough set of notes for a research project and a learning experience for LaTeX. You can access the source .tex file at \\
https://www.mp-works.us/

\section{ Derivations }
This section contains my notes on deriving the needed equations for my research project.
\subsection{ Netwon's and  Kirchhoff's Equations }


% Table of variables
%-------------------------------------------------------------------------------------------------------------------------------------------------------
\subsubsection{Variables}

\begin{tabular}{ |p{4cm}|p{4cm}|p{4cm}|  }
 \hline
 \multicolumn{3}{|c|}{Variables} \\
 \hline
SMD System & RLC System & Pendulum System \\

 \hline
 \[ x\] Linear displacement  & \[ q\] Charge  & \[ s\] Arc displacement    \\
 \[ F\] Force  & \[ V\] Voltage  & \[ F\] Force    \\
 \[ b\] Damping coefficient  & \[ R\] Resistance  & \[ b\] Damping coefficient    \\
 \[ m\] Mass  & \[ L\] Inductance  & \[ m\] Mass    \\
 \[ k\] Spring constant  & \[ C\] Capacitance  & \[ \frac{g}{l} = \frac{[gravity]}{[length]}\] Pendulum constant  \\
 \hline
\end{tabular}


% Related Equations
%-------------------------------------------------------------------------------------------------------------------------------------------------------
\clearpage
\subsubsection{Spring Mass Dampener Equations}
Starting with Newton's law of motion applied to a Spring, Mass, Dampener system.

\begin{equation}
m\bigg[\frac{d^2x}{dt^2}\bigg] = - kx - b\bigg[\frac{dx}{dt}\bigg]
\end{equation}
solving for 0 we get,
\begin{equation}
0 = m\bigg[\frac{d^2x}{dt^2}\bigg] + kx + b\bigg[\frac{dx}{dt}\bigg]
\end{equation}

\subsubsection{RLC Circuit Equations}
Starting with Kirchhoff's voltage law applied to an LRC circuit. 

\begin{equation}
L\bigg[\frac{d^2q}{dt^2}\bigg] =  \frac{1}{C}q - R\bigg[\frac{dq}{dt}\bigg]
\end{equation}
solving for 0 we get,
\begin{equation}
0 = L\bigg[\frac{d^2x}{dt^2}\bigg] + \frac{1}{C}q + R\bigg[\frac{dq}{dt}\bigg]
\end{equation}

\subsubsection{Pendulum Equations}
Starting with Newton's law of motion applied to a Pendulum system.

\begin{equation}
m\bigg[\frac{d^2s}{dt^2}\bigg] = - s\Big[\frac{g}{l}\Big]- b\bigg[\frac{ds}{dt}\bigg]
\end{equation}
\begin{align*}
\sin{\theta} \approx \theta \indent \{ \theta | \theta << 1 \} 
\end{align*}
solving for 0 we get,
\begin{equation}
0 = m\bigg[\frac{d^2s}{dt^2}\bigg] + s\Big[\frac{g}{l}\Big] + b\bigg[\frac{ds}{dt}\bigg]  
\end{equation}

% Pendulum Math
%-------------------------------------------------------------------------------------------------------------------------------------------------------
\clearpage
\subsection{ Pendulum Calculations }

% Start
% - - - - - - - - - - - -- - - - - - - -- - - - - -- - - -- - - - -

\begin{align*}
m\ddot{s} = - b\dot{s} - mgl\sin{\theta}
\end{align*}

Solve for 0.

\begin{equation}
0 = m\ddot{s} + b\dot{s} + mgl\sin{\theta}
\end{equation}

% Restrict to small angles
% - - - - - - -- - - -- - - - - -- - - - - - - -- - - - - - - - - -
Restrict maximum displacement angle.

\begin{align*}
\sin{\theta} \approx \theta \indent \{ \theta | \theta << 1 \} 
\end{align*}

\begin{equation}
0 = m\ddot{s} + b\dot{s} + mgl\theta
\end{equation}

% Swich to change in angle with respect to time
% - - - - - -- - - - - -- - - - - -- - - - - -- - - - - -- - - - -
Represent as change of angle with respect to time.


\begin{align*}
s &= l\theta \\
0 &= ml\ddot{\theta} + lb\dot{\theta} + mgl\theta
\end{align*}

\begin{align*}
\bigg[\frac{ds}{dt}\bigg] = l\bigg[\frac{d\theta}{dt}\bigg] &\indent& \bigg[\frac{d^2s}{dt^2}\bigg] = l\bigg[\frac{d^2\theta}{dt^2}\bigg]
\end{align*}

\begin{equation}
0 = m\ddot{\theta} + b\dot{\theta} + m\frac{g}{l}\theta
\end{equation}

% Substitute
% - - - - - - -- - - - - -- - - - - -- - - - - -- - - - - - - - - -
Devide by mass and substitute to get into a quadratic form.

% Sub Differential

\begin{align*}
0 &= \ddot{\theta} + \frac{b}{m}\dot{\theta} + \frac{g}{l}\theta \\
\alpha &= \bigg[\frac{d\theta}{dt}\bigg] \\
0 &= \alpha^2\theta + \frac{b}{m}\alpha\theta + \frac{g}{l}\theta
\end{align*}

\begin{equation}
0 = \theta\bigg( \alpha^2 + \frac{b}{m}\alpha + \frac{g}{l} \bigg)
\end{equation}

% Sub gamma and omega

\begin{align*}
\gamma &= \frac {b}{m}  ,& \omega_0^2 &= \frac{g}{l} \\
\end{align*}

\begin{equation}
0 = \theta\bigg( \alpha^2 + \gamma\alpha + \omega_0^2  \bigg)
\end{equation}

% Find the roots
% - - - - - - - - - - - - -- - - - - - -- - - - - -- - - - -- - - -
\clearpage
\setcounter{subsection}{1}
\subsection{ Pendulum Calculations Cont. }

\begin{align*}
0 = \theta\bigg( \alpha^2 + \gamma\alpha + \omega_0^2  \bigg)
\end{align*}

Find the roots using the quadratic formula.

\begin{equation}
\alpha_\pm = \frac{ \frac{1}{2} }{ \frac{1}{1} }
\end{equation}



% Resources
%-------------------------------------------------------------------------------------------------------------------------------------------------------
\clearpage
\begin{thebibliography}{9}
 
\bibitem{knuthwebsite} 
Morin: Oscillations,
\\\texttt{http://www.people.fas.harvard.edu/\~{}djmorin/waves/oscillations.pdf}
\end{thebibliography}


\end{document}