# Period of a Planet

Try this Problem, useful for Physics Olympiad, based on the period of a planet.

The Problem:

Suppose that the gravitational force varies inversely as the $n^{th}$ power of the distance. Then, the period of a planet in circular orbit of radius $R$ around the sun will be proportional to

(A) $R^{\frac{n+1}{2}}$

(B)$R^{\frac{n-1}{2}}$

(C) $R^n$

(D) $R^{n/2}$

Discussion:
The gravitational force can be given as $$\frac{GMm}{R^n}=mR\omega^2$$

Now, we know $\omega=\frac{2\pi}{T}$,

Hence

$$\frac{GMm}{R^n}= mR(\frac{2\pi}{T})^2$$ $$T^2= \frac{4\pi^2R^{n+1}}{GM}$$

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