State the parallel axis theorem and use it to calculate the moment of inertia of a uniform hemisphere of mass $m$ and radius $a$ about an axis through its centre of mass and parallel to the base.

[You may assume that the centre of mass is located at a distance $\frac{3}{8}$ a from the flat face of the hemisphere, and that the moment of inertia of a full sphere about its centre is $\frac{2}{5} M a^{2}$, with $M=2 m$.]

The hemisphere initially rests on a rough horizontal plane with its base vertical. It is then released from rest and subsequently rolls on the plane without slipping. Let $\theta$ be the angle that the base makes with the horizontal at time $t$. Express the instantaneous speed of the centre of mass in terms of $b$ and the rate of change of $\theta$, where $b$ is the instantaneous distance from the centre of mass to the point of contact with the plane. Hence write down expressions for the kinetic energy and potential energy of the hemisphere and deduce that

$$\left(\frac{d \theta}{d t}\right)^{2}=\frac{15 g \cos \theta}{(28-15 \cos \theta) a}$$