Hint 7

The principal axes are ${\bf e}_3$, normal to the plate, and any pair of perpendicular axes lying in the plane of the plate. The principal moment about the ${\bf e}_3$ axis is

$$I_3={Ma^2\over 6},$$

and, by the perpendicular axes theorem, the principal moments about the other two axes are

$$I_1=I_2={1\over 2}I_3.$$

(a) Since $I_1=I_2$, the Euler's equation for $\omega_3$ becomes

$$\dot\omega_3=0.$$

Hence $\omega_3$ must be a constant.

(b) The other two equations simplify to

$$\dot\omega_1+\omega_2\omega_3 = 0$$

$$\dot\omega_2-\omega_3\omega_1 = 0$$

Note that these equations are a special case of the ones for the freely rotating symmetric top. They can be solved by differentiating one equation with respect to $t$ and using the other to get a second-order differential equation for either $\omega_1$ or $\omega_2$. Alternatively we can define a complex ``coordinate"

$$\eta=\omega_1+{\rm i}\omega_2.$$

This satisfies the first-order differential equation

$$\dot\eta-{\rm i}\omega_3\eta=0,$$

which has solutions of the form

$$\eta(t)=A\exp[{\rm i}(\omega_3 t+\phi)],$$

where $A$, $\phi$ and $\omega_3$ are constants. Interpret this result in terms of the precession of $\hbox{{\boldmath$\omega$ }}$.