# Coupled oscillation - ball on a balanced rod

1. May 22, 2009

### SbCl3

1. The problem statement, all variables and given/known data

This system consists of a rod balanced on a fulcrum, with a ping pong ball able to slide on the rod. There's a special point some point away from the center where, once the ball is released, will stop moving once it has gotten the same distance on the opposite side of the rod, and the system will continue forever (air resistance and friction are to be ignored).

For the system shown in the attached image (also http://img39.imageshack.us/img39/479/coupledoscillation.png" [Broken]), find an equation for the motion. Also find an expression for the length X0 away from the center such that the ball will oscillate back and forth forever. Here are the variables that need to be related:

M - mass of beam
m - mass of ball
L - length of beam
X0 - Initial position of the ball such that the system oscillates forever
X - position of the ball (at fulcrum X = 0, at right end X = L/2)
$$\Theta$$0 - Initial angle of the beam (a very small angle)
h - beam end's initial height

The center of the beam is at the fulcrum, perfectly balanced
Also, m is much less than M, so the moment of inertia I remains constant throughout.

3. The attempt at a solution
Because of the small angles, it can be assumed that $$sin(\Theta) \approx \Theta$$, and $$cos(\Theta) \approx 1$$.
IRod = 1/12*M(L2)

My equations:
http://is.gd/CIQ7 [Broken] (conservation of energy)
http://is.gd/CIT4 [Broken] (motion of ball)
http://is.gd/CIUh [Broken] (torque, motion of rod)

But I'm struggling to relate these equations to get a solution for Xo.
My initial conditions:
at t = 0, v = 0
at t = 0, x = $$X_0$$, and $$\Theta=\Theta_0$$

#### Attached Files:

• ###### coupled_oscillation.png
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Last edited by a moderator: May 4, 2017