First slide

Universal law of gravitation

Question

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

Easy

A

Rⁿ
B

$R^{(n + 1) / 2}$
C

$R^{(n - 1) / 2}$
D

$R^{- n}$

Solution

$T = \frac{2 πR}{V}$ ……………..(1)
Further $E = \frac{1}{2} {mv}^{2} = \frac{GMm}{R^{n - 1}}$
or $V = {[\frac{2 GM}{R^{n - 1}}]}^{1 / 2}$ ……..(2)
From eqs. (1) and (2)
$\begin{array}{l} T = \frac{2 πR}{\sqrt{(2 GM / R^{n - 1})}} \\ = \frac{2 π}{\sqrt{2 GM}} \times R^{(n + 1) / 2} \propto R^{(n + 1) / 2} \end{array}$

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