1. Show that none of the principal moments of inertia can exceed the sum of the other two.

2.
A uniform rod of length *b* stands vertically upright on a rough floor and then tips over. What is the rod's angular velocity when
it hits the floor?

3.
Find the frequency of small oscillations for a thin
homogeneous plate if the motion takes place in the plane of the plate and if
the plate has the shape of an equilateral triangle and is suspended

(a)
from the midpoint of one side.

(b)
from one apex.

4.
A physical or compound pendulum is a rigid body that
oscillates due to its own weight about a horizontal axis that does not pass
through the center of mass of the body (see Figure).

Assume the pendulum
of mass *M* is released from rest from an
angle *q*_{0}. Determine the angular velocity *w* as function of the angle *q*. Note: do not assume that the angles are small.

Practice the material and concepts discussed in Chapter 11 and 12.