# Virial theorem for an inverse-square law force

Calculate the time-average of i) potential and ii) kinetic energy of a particle orbiting on ellipse in an inverse-square-law force field f =(k/r^2) (K<0)

Express your answers in terms of a ( semi-major axis of the ellipse) and k (constant in the force given)

have no idea how to do this question

can any i help me?

## Answers and Replies

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vela
Staff Emeritus
Homework Helper
The time-average of the potential energy is given by

$$\langle V \rangle = \frac{1}{T}\int_0^T V(r(t))\,dt$$

where T is the period of the orbit. You should know what V(r) is since you're given the force. Presumably, you know the equation for the orbit. You probably have it as r(θ) whereas you want r(t), so you may have to do some math to come up with the correct integral.

Similarly, to find the average kinetic energy, you need to evaluate the integral

$$\langle K \rangle = \frac{1}{T}\int_0^T \frac{1}{2}mv(t)^2 \,dt$$

By the way, what course is this question for? It seems a little advanced for an introductory physics course.