# Coupled Oscillator

1. Feb 17, 2009

### roeb

1. The problem statement, all variables and given/known data
A thin hoop of radius R and mass M oscillates in its own plane with one point of the hoop fixed. Attached to the hoop is a small mass M constrained to move (in a frictionless manner) along the hoop. Consider only small oscillations, and show that the eigenfrequencies are blah blah blah (two eigenfrequencies).

2. Relevant equations

3. The attempt at a solution

My difficulties are in setting up this problem. I believe that I am picturing the system correctly, but I can't quite figure out how to do it. I need to find the Lagrangian of the system first, but I am having a hard time with the kinetic energy part.

The oscillation of the hoop if I am not mistaken will be like that of a pendulum.

Hoop: T = 1/2*Iw^2 = 1/2 m * R^2 * $$\omega ^2$$
Small Mass: T = 1/2 mR^2 $$\theta '$$

I have set up theta as the angle between the center of the hoop and the position of the small mass. The problem is that I don't quite know how to get the second generalized coordinate -- I am assuming there are two generalized coordinates because this is a coupled oscillator problem and I am given two eigenfrequencies.

I am tempted to say that $$\omega = R \theta '$$ but that doesn't yield a correct answer and I don't think it's right to begin with...

Any help?