A sphere of uniform density has mass $m$ and radius $a$. Find its moment of inertia about an axis through its centre.

A marble of uniform density is released from rest on a plane inclined at an angle $\alpha$ to the horizontal. Let the time taken for the marble to travel a distance $\ell$ down the plane be: (i) $t_{1}$ if the plane is perfectly smooth; or (ii) $t_{2}$ if the plane is rough and the marble rolls without slipping.

Explain, with a clear discussion of the forces acting on the marble, whether or not its energy is conserved in each of the cases (i) and (ii). Show that $t_{1} / t_{2}=\sqrt{5 / 7}$.

Suppose that the original marble is replaced by a new one with the same mass and radius but with a hollow centre, so that its moment of inertia is $\lambda m a^{2}$ for some constant $\lambda$. What is the new value for $t_{1} / t_{2}$ ?