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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