Kevin S. Huang

2011: Problem 25

Topic: Rigid Bodies
Concepts: Inclined plane, Rolling down inclined plane

Solution:

Recall the acceleration of an object rolling without slipping down an inclined plane is given by

a=\frac{g\sin\theta}{1+\beta}

where the moment of inertia of the object is $I=\beta mr^2$ . Recall the acceleration of a block sliding down an inclined plane is given by

a=g(\sin\theta-\mu\cos\theta)

Since the hollow cylinder and block reach the bottom at the same time, they both have the same acceleration:

\frac{g\sin\theta}{1+\beta}=g(\sin\theta-\mu\cos\theta)

\mu\cos\theta=\sin\theta\left(1-\frac{1}{1+\beta}\right)

For a hollow cylinder, $\beta=1$ :

\mu\cos\theta=\frac{\sin\theta}{2}

\mu=\frac{\tan\theta}{2}

so the answer is C.