Homework Help: Tough oscillation question

1. Jun 20, 2011

AriAstronomer

1. The problem statement, all variables and given/known data
A solid sphere (radius = R) rolls without slipping in a
cylindrical trough (radius = 5R), as shown in Figure
P13.56. Show that, for small displacements from equilib-
rium perpendicular to the length of the trough, the
sphere executes simple harmonic motion with a period
T = 2Pi √28R/5g.

2. Relevant equations

3. The attempt at a solution
It's essentially a pendulum type problem except a ball is rolling instead of just moving. Once we get w were set:
Using physical pendulum, w = root(mgd/I):
d = distance to CM of system = 5R-R = 4R = CM of ball.
I = I was thinking that the axis of rotation is at 4R from the CM of the ball, Icm of ball = 2/5MR^2, so I was thinking I = 2/5MR^2 + M(4R)^2, but this doesn't look like it's going to put me in the right direction.

Any thoughts?

2. Jun 20, 2011

ehild

The ball rotates around its own CM while rolling in the trough and the CM performs circular motion around the axis of the cylinder. The angular speed of rotation is different from the angular frequency of the SHM.

ehild