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[parent] example of generalized coordinates for constrained motion on a horizontal circle (Example)

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

$\displaystyle T=\frac{m}{2}a^{2}\dot{\theta}^{2}, $

and we have

$\displaystyle \frac{\partial T}{\partial \dot{\theta}}=ma^{2}\dot{\theta}. $

    Our differential equation is

$\displaystyle ma^{2}\ddot{\theta}\delta\theta=-ka^{2}\dot{\theta}^{2}a\delta\theta, $

which reduces to

$\displaystyle \ddot{\theta}+\frac{k}{m}a\dot{\theta}^{2}=0, $

or

$\displaystyle \frac{d \dot{\theta}}{dt}+\frac{k}{m}a\dot{\theta}^{2}=0. $

Separating the variables,

$\displaystyle \frac{d \dot{\theta}}{\dot{\theta}^{2}}+\frac{k}{m}a dt=0. $

Integrating,

$\displaystyle -\frac{1}{\dot{\theta}}+\frac{k}{m} a t= C =-\frac{a}{V}. $

$\displaystyle \frac{1}{\dot{\theta}}=\frac{ma+k V a t}{m V}, $

$\displaystyle \frac{d\theta}{dt}=\frac{mV}{ma+kVat}, $

$\displaystyle \theta=\frac{m}{ka}\log\left[m+kVt\right]+C, $

$\displaystyle \theta=\frac{m}{ka}\log\left[1+\frac{kVt}{m}\right]; $

and the problem of the motion is completely solved.



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Cross-references: motion, differential equation, kinetic energy, square, velocity, mass

This is version 2 of example of generalized coordinates for constrained motion on a horizontal circle, born on 2008-07-21, modified 2008-07-22.
Object id is 291, canonical name is ExampleOfGeneralizedCoordinatesForConstrainedMotionOnAHorizontalCircle.
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Physics Classification: 45.20.Jj (Lagrangian and Hamiltonian mechanics)
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