Problem 3 (5 pt.) In the last class, we will consider the gyroscopic motion of a heavy symmetrical top whose lowest point is fixed. The effective potential energy of the top is given by
$U_{eff} = \frac{(M_z - M_3 \cos \theta)^2}{2I_1 \sin^2 \theta} + mgl \cos \theta$,
where $M_z$ is the constant angular momentum component along the vertical Z axis (fixed), $M_3$ is the
constant angular momentum component along the $x_3$ axis (the moving axis of symmetry of the top),
$I_1$ is the principal moment of inertia about the axis passing through the lowest point and parallel
to the $x_1$ axis, $m$ is the mass of the top, and $l$ is the distance from the lowest point to the center of
mass of the top (you can see Figure 48 in the Textbook). Find the condition for the rotation of the
top about a vertical axis ($\theta = 0$) to be stable. Hint: For $\theta = 0$, the $x_3$ and Z axes coincide, so that
$M_3 = M_z$. Rotation about this axis is stable if $\theta = 0$ is a minimum of the function $U_{eff}(\theta)$. To find
the minimum, use the small-angle approximation in $U_{eff}(\theta)$: $\sin \theta \approx \theta$ and $\cos \theta \approx 1 - \theta^2/2$.
