(a) Show that the principal moments of inertia for the oblate spheroid of mass $M$ defined by

$$\frac{\left(x_{1}^{2}+x_{2}^{2}\right)}{a^{2}}+\frac{x_{3}^{2}}{a^{2}\left(1-e^{2}\right)} \leqslant 1$$

are given by $\left(I_{1}, I_{2}, I_{3}\right)=\frac{2}{5} M a^{2}\left(1-\frac{1}{2} e^{2}, 1-\frac{1}{2} e^{2}, 1\right)$. Here $a$ is the semi-major axis and $e$ is the eccentricity.

[You may assume that a sphere of radius a has principal moments of inertia $\frac{2}{5} M a^{2}$.]

(b) The spheroid in part (a) rotates about an axis that is not a principal axis. Euler's equations governing the angular velocity $\left(\omega_{1}, \omega_{2}, \omega_{3}\right)$ 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 $\omega_{3}$ is constant. Show further that the angular momentum vector precesses around the $x_{3}$ axis with period

$$P=\frac{2 \pi\left(2-e^{2}\right)}{e^{2} \omega_{3}}$$