r If you did Problem 4.41 you met the virial theorem for a circular orbit of a particle in a central force with U = kr". Here is a more general form of the theorem that applies to any periodic orbit of a particle. (a) Find the time derivative of the quantity G = r - p and, by integrating from time 0 to I, show that G(t) — G(0) = t 2(T) + (F-r) where F is the net force on the particle and (f) denotes the average over time of any quantity f. (b) Explain why, if the particle’s orbit is periodic and if we make I sufficiently large, we can make the left—hand side of this equation as small as we please. That is, the left side approaches zero as t —> 00. (c) Use this result to prove that if F comes from the potential energy U = kr", then (T) = n (U ) /2, if now (f) denotes the time average over a very long time.