Suppose that we have a uniform rod inside a circle. It's free to slide with its ends on the inside of the circle, and it subtends an angle of 120 degrees at the center of the circle. I'm looking for the Lagrangian of this system.
I = ml^2/12 for the rod about its center
L = T-V
The Attempt at a Solution
Here's the problem. It seems to me that the ends of the rod (near the circle) are moving faster than the middle of the rod... which would mean that I can't use use the center of mass coordinates to find the kinetic energy. So I'm a bit confused about how to establish the correct kinetic energy - this is where I need help (I can get the potential, I think).
Can anyone help?