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Points of mass joined by rigid rod

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Express the kinetic energy in generalized coordinates for two points of mass joined by a rigid weightless rod
Classical Mechanics - Generalized coordinates for two points of mass joined by a rigid weightless rod. Classical Mechanics - Generalized coordinates for two points of mass joined by a rigid weightless rod.
Two points of mass m are joined by a rigid weightless rod of length l, the center of which is constrained to move on a circle of radius a. Express the kinetic energy in generalized coordinates.
(Source: Herbert Goldstein, Classical Mechanics - Chapter 01)
 
Kinetic Energy:
kinetic energy
Where T1 equals the kinetic energy of the center of mass, and T2 is the kinetic energy about the center of mass. Keep these two parts seperate!

Solve for T1 first, its the easiest:
Solve for T1

Solve for T2, realizing that the rigid rod is not restricted to just the X-Y plane. Don’t forget the Z-axis!
Solve for T2

Solve for v2 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 v2:





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 v2 we can plug this into T2
T


It was important to emphasize that T1 is the kinetic energy of the total mass around the center of the circle while T2 is the kinetic energy of the masses about the center of mass.
 
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