(

*Source: Herbert Goldstein, Classical Mechanics - Chapter 01*)

Kinetic Energy:

Where T

_{1}equals the kinetic energy of the center of mass, and T

_{2}is the kinetic energy about the center of mass. Keep these two parts seperate!

Solve for T

_{1}first, its the easiest:

Solve for T

_{2}, realizing that the rigid rod is not restricted to just the X-Y plane. Don't forget the Z-axis!

Solve for v

^{2}about the center of mass. The angle will be the angle in the x-y plane, while the angle will be the angle from the z-axis.

If and then x = l/2 so:

If and then y = l/2 so:

If = 0, then z = l/2 so:

Find v

^{2}:

Square each:

Now add, striking out the middle terms:

Pull the first and third terms inside the brackets together, and pull the second and fourth terms together as well:

Now that we finally have v

^{2}we can plug this into T

_{2}

It was important to emphasize that T

_{1}is the kinetic energy of the total mass around the center of the circle while T

_{2}is the kinetic energy of the masses about the center of mass.