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Physics 222 Research Project Notes & LaTeX Learning Experience
\documentclass{article}
\usepackage{amsmath}
\title{Damped Oscillations Research}
\author{ Mitchell Pell\\ [email protected] }
\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}
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