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# Period of a Planet

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

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

August 15, 2017