2012: Problem 10

Topic: Rigid Bodies
Concepts: Moments of inertia, Rolling down inclined plane

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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 object has moment of inertia

$$I=\beta mr^2$$

Objects A, B, and D are all solid balls with $\beta=2/5$ so they reach the bottom at the same time. Object C is a hollow sphere with $\beta=2/3$ so it has a smaller angular acceleration than a solid sphere. Hence, it takes longer to reach the bottom. Thus,

$$T_C>T_A=T_B=T_D$$

so the answer is D.

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