# Rigid body problem

I have some doubts about the next excercise:

A bar of length $2a$ and mass $m$ moves freely with both of its extremes on a ring of radius $\sqrt2a$. The ring can rotate freely in a certain diameter, remaining the center fix. Find the equations of motion.

I did it using the balance of angular momentum, but I want to try to do it with the Lagrangian. But I have a problem. The energy of the ring is easy to calculate (just a rotation), but the energy of the bar I am not sure how to do it. ¿Is a translation of the center of mass plus a rotation? ¿Or just translation? ¿Or just a complicated rotation?

I did this picture to illustrate:

Thanks!

• If there is no friction then the rotation of the ring is irrelevant. – John Alexiou Feb 3 '14 at 17:55