# Homework Help: Oscillations of an Exercise Ball

1. Oct 2, 2006

### TacoTycoon

Hey all,

First I wanted to say hello, as I am new to this forum. I'm a third-year physics student at the University of Toronto.

Anyhow, the question goes like this:

We have an exercise ball (one of those large, inflatable ones) with a diameter of 0.7m. If you've ever seen one of these, you'll know that the material isn't constant throughout the ball, and at the point where you blow the ball up the mass is heavier. Thus, the ball will always try to roll to the equilibrium position where that point is below the absolute centre.

The question is: find the equation of motion for the oscillations of the ball if you displace it slightly from it's equilibrium. The ball will rock back and forth, as shown in this video:

What you're doing by having this 'extra mass' at a point on the ball is shifting the centre of mass away from the centre of the ball.

I believe that it's the force of gravity acting on the centre of mass which is the 'restoring' force of the oscillation, and the rotational momentum of the ball which is the 'inertial' counter-force. I haven't been able, however, to model this mathematically. Does anyone have any ideas? Thanks!

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Edit: Just read the rules of the homework forum, and saw that I had to post my work:

For the restoring force:

F=ma, this will be gravity. I pictured the centre of mass similarly to a pendulum on a string, and if we take theta to be the angle the ball has rotated from equilibrium, then:

F=-mgsin(theta)

For the inertial force:

F=ma
F=I(d^2/dt^2)(theta) which is the term describing torqe (the d thing means second deriv)

so the overall equation would be: I(d^2/dt^2)(theta)+mgsin(theta)=0

I know this is non-linear, but the question never asks us to solve the equation of motion. Sorry for my notation, I have to get used to writing math online.

Last edited: Oct 2, 2006