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.
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.