(a) Show that the principal moments of inertia for an infinitesimally thin uniform rectangular sheet of mass $M$ with sides of length $a$ and $b$ (with $b<a$ ) about its centre of mass are $I_{1}=M b^{2} / 12, I_{2}=M a^{2} / 12$ and $I_{3}=M\left(a^{2}+b^{2}\right) / 12$.

(b) Euler's equations governing the angular velocity $\left(\omega_{1}, \omega_{2}, \omega_{3}\right)$ of the sheet 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

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

A possible solution of these equations is such that the sheet rotates with $\omega_{1}=\omega_{3}=0$, and $\omega_{2}=\Omega=$ constant.

By linearizing, find the equations governing small motions in the neighbourhood of this solution that have $\left(\omega_{1}, \omega_{3}\right) \neq 0$. Use these to show that there are solutions corresponding to instability such that $\omega_{1}$ and $\omega_{3}$ are both proportional to exp $(\beta \Omega t)$, with $\beta=\sqrt{\left(a^{2}-b^{2}\right) /\left(a^{2}+b^{2}\right)} .$