Question

A stick lying flat on a frictionless horizontal table is fixed in place at one end and free to rotate about that point. Suppose the table and surrounding air constitute a heat bath at temperature \(T\).

(a) Find \(\left\langle\omega^{2}\right\rangle .\) (5 points)

(b) The other end of the stick is connected to a spring which provides a restoring torque \((-c \theta),\) where \(c\) is the stiffness and \(\theta\) the angular displacement away from equilibrium. Find \(\left\langle\theta^{2}\right\rangle .\) (5 points)

(c) In the stick+spring system of part (b), what is the average total energy and what is \(\langle\theta \omega\rangle ?(5\) points \()\)