Question

Suppose the gravitational force varies inversely as the nth power of distance. Then, the time period of a planet in circular orbit of radius r around the sun will be proportional to

Moderate

Solution

As, gravitational force provides the necessary centripetal force
$\frac{m v^{2}}{r} \propto r^{- n}$
$\Rightarrow v \propto r^{(1 - n) / 2}$
Also, $T = \frac{2 π r}{v} or T \propto r v^{- 1}$
or $T \propto r \cdot r^{(n - 1) / 2}$
or $T \propto r^{(n + 1) / 2}$

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