ExampleOfGeneralizedCoordinatesForConstrainedMotionOnAHorizontalCircle 2008-07-21 00:51:29 2008-07-21 00:51:29 Example generalized coordinates for constrained motion kinetic energy equation had a typo, forgot the dot on theta % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let a particle of mass $m$, constrained to move on a smooth horizontal circle of radius $a$, be given an initial velocity $V$, and let it be resisted by the air with a force proportional to the square of its velocity. Here we have one degree of freedom. Let us take as our coordinate $q_{1}$ the angle $\theta$ which the particle has described about the center of its path in the time $t$. For the kinetic energy $$ T=\frac{m}{2}a^{2}\theta^{2}, $$ and we have $$ \frac{\partial T}{\partial \dot{\theta}}=ma^{2}\dot{\theta}. $$ \quad Our differential equation is $$ ma^{2}\ddot{\theta}\delta\theta=-ka^{2}\dot{\theta}^{2}a\delta\theta, $$ which reduces to $$ \ddot{\theta}+\frac{k}{m}a\dot{\theta}^{2}=0, $$ or $$ \frac{d \dot{\theta}}{dt}+\frac{k}{m}a\dot{\theta}^{2}=0. $$ Separating the variables, $$ \frac{d \dot{\theta}}{\dot{\theta}^{2}}+\frac{k}{m}a dt=0. $$ Integrating, $$ -\frac{1}{\dot{\theta}}+\frac{k}{m} a t= C =-\frac{a}{V}. $$ $$ \frac{1}{\dot{\theta}}=\frac{ma+k V a t}{m V}, $$ $$ \frac{d\theta}{dt}=\frac{mV}{ma+kVat}, $$ $$ \theta=\frac{m}{ka}\log\left[m+kVt\right]+C, $$ $$ \theta=\frac{m}{ka}\log\left[1+\frac{kVt}{m}\right]; $$ and the problem of the motion is completely solved.