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Physics 222 Research Project Notes & LaTeX Learning Experience
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\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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