A ``flywheel" is a device that spins around real fast on an almost frictionless bearing so that it can maintain a high angular velocity. The velocity at the tips of the flywheel can be really high and so the kinetic energy of the system can be quite enormous. This device is used to store energy. It'd be nice to be able to figure out the total kinetic energy of a flywheel as a function of its angular velocity, its mass and its shape. So let's try to figure out the kinetic energy in a rotating rigid body.
The way we're solve this problem is to divide and conquer. We know
what the kinetic energy of a point particle of mass m moving at velocity
. It's 
. Now what we have is a collection of
particles. For N particles, the kinetic energy K is
but now the velocities of each particles are related because they're
part of the same rigid body, 
. 
is the the
distance between a point on the body and the axis of rotation. Plugging
this in we have
The term in the parenthesis is called the ``moment of inertia"
It doesn't depend on the angular velocity of the object, but just the masses that it's made up of, and their distance away from the axis of rotation. It might be complicated to compute, and we'll discuss what it is for some simple shapes pretty soon. It's useful in rotational dynamics because it plays the same role that we've seen mass play in our previous studies of dynamics. This is what I mean. In terms of the moment of inertia the kinetic energy is
If we replace I by m and 
by v we get the good old formula 
.
So we can add this to our list of equations (eqns. 1.17-1.20) that are similar to what we've already seen, but with a few name changes.