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hbweb500

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## Homework Statement

"A small ball with radius r and uniform density rolls without slipping near the bottom of a fixed cylinder of radius R. What is the frequency of small oscillations, assuming r<<R?"

http://img89.imageshack.us/img89/8614/helpwu8.png

## Homework Equations

[tex]

F=ma

[/tex]

[tex]

\tau=I\alpha \\

[/tex]

[tex]

I=\frac{2}{5}mr^2 \\

[/tex]

[tex]

a = \ddot{\theta} R= \alpha r

[/tex]

## The Attempt at a Solution

The solution I can get, I just have a few questions about the details.

First, writing out the forces on the CM of the ball: the force of friction points in the same direction as the velocity vector, due to the non-slipping constraint. The magnitude of the ball's velocity is decreasing the greater the angle between it and the bottom of the cylinder, and thus the angular frequency of its rotation must also decrease. Thus the friction force is opposite the rotation.

The other relevant force is obviously gravity, scaled by a sine of theta:

[tex]

F_{\hat{\theta}}= F_f - mg sin\theta = ma

[/tex]

And there is a also a torque equation. Relative to the CM of the ball, we have a torque exerted by friction. Thus:

[tex]

-r F_f = \frac{2}{5}mr^2 \alpha

[/tex]

Combining these two equations, and assuming that, for small theta, [itex]\theta\approx sin\theta[/itex], we get the differential equation:

[tex]

\ddot{\theta}+\frac{5g}{7R}\theta =0

[/tex]

[tex]

\omega = \sqrt{\frac{5g}{7R}}

[/tex]

__What I can't understand...__**1. The Work Done By Friction**

The bottom of the ball is instantaneously at rest with respect to the bottom of the cylinder. The force of static friction, then, exerts a force over zero distance, and thus does no work. However, it also is responsible for the change in angular frequency. Without the friction, the ball would not rotate.

The rotational kinetic energy of the ball is changing due to the friction force, but the friction force does no work. I don't get this.

**2. Non-inertial Frame?**

The CM of the ball is accelerating, yet the torque is still calculated using it as an origin. How is this legal?

If it matters, this is problem 8.13 in Morin's Introduction to Classical Mechanics.

Thanks for your help!

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