(a) Show that the principal moments of inertia of a uniform circular cylinder of radius $a$, length $h$ and mass $M$ about its centre of mass are $I_{1}=I_{2}=M\left(a^{2} / 4+h^{2} / 12\right)$ and $I_{3}=M a^{2} / 2$, with the $x_{3}$ axis being directed along the length of the cylinder.

(b) Euler's equations governing the angular velocity $\left(\omega_{1}, \omega_{2}, \omega_{3}\right)$ of an arbitrary rigid body as viewed in the body frame are

$$\begin{aligned} &I_{1} \frac{d \omega_{1}}{d t}=\left(I_{2}-I_{3}\right) \omega_{2} \omega_{3} \\ &I_{2} \frac{d \omega_{2}}{d t}=\left(I_{3}-I_{1}\right) \omega_{3} \omega_{1} \end{aligned}$$

and

$$I_{3} \frac{d \omega_{3}}{d t}=\left(I_{1}-I_{2}\right) \omega_{1} \omega_{2}$$

Show that, for the cylinder of part $(\mathrm{a}), \omega_{3}$ is constant. Show further that, when $\omega_{3} \neq 0$, the angular momentum vector precesses about the $x_{3}$ axis with angular velocity $\Omega$ given by

$$\Omega=\left(\frac{3 a^{2}-h^{2}}{3 a^{2}+h^{2}}\right) \omega_{3}$$