Question

Which one of the following spheres will roll down an inclined plane slowest?
a) A solid copper ball.
b) A hollow copper sphere of the same diameter as above with mercury to compensate for weight.
c) A hollow copper ball filled with a lead ball inside to compensate for weight.
d) A hollow copper ball without weight compensation.

Answer 1

option (D)

Answer 2

If you think about the moment of inertia which is , \[I=\sum_0^N m_ir_i^2\] Then between a) and b) since copper is heavier than mercury (right?) b) will have a greater I. This eliminates a). Likewise lead is heavier than mercury so this eliminates c). d) has no interior so it will have the smallest moment of inertia, I.

To see how this relates to speed to go down the hill,
\[\tau=I\alpha=\sum F_\perp(R)=Mg\sin\theta = \sum_0^N m_i g\sin\theta =K\implies \alpha=\frac{K}{I}\]
Alpha is just the angular acceleration. The fastest ball will have biggest alpha. So this is d) if you think about it long enough.