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If you did .41 you met the virial theorem for a circular

Chapter 8, Problem 8.17

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QUESTION:

If you did 4.41 you met the virial theorem for a circular orbit of a particle in a central force with U = krn. 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 t, show that G (t) G(0) =2(T) + (F r) t 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 t 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 oo. (c) Use this result to prove that if F comes from the potential energy U = krn, then (T) = n(U)12, if now (f) denotes the time average over a very long time.

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QUESTION:

If you did 4.41 you met the virial theorem for a circular orbit of a particle in a central force with U = krn. 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 t, show that G (t) G(0) =2(T) + (F r) t 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 t 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 oo. (c) Use this result to prove that if F comes from the potential energy U = krn, then (T) = n(U)12, if now (f) denotes the time average over a very long time.

ANSWER:

Step 1 of 4

The gravitational constant in terms of position and the momentum vector is,

                                                               

So,

                                               

                                                             

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