Question

Suppose the gravitational force vaires inversly as the $n$th power of the distance. Then, the time period of a planet in a circular orbit of radius $R$ around the Sun will be proportional to

• A. $R^n$
• B. $R^{\frac{(n+1)}{2}}$
• C. $R^{\frac{(n-1)}{2}}$
• D. $R^{-n}$

Answer 1

The correct option is B $R^{\frac{(n+1)}{2}}$

We have $\frac{GMm}{R^n}=\frac{m{v}^2}{R}$
$\Rightarrow \frac{1}{v^2}=\frac{R\times R^n}{GM}$
$or,\frac{1}{v}\propto R^{\left(\frac{n-1}{2}\right)}\text{and}T=\frac{2\pi R}{v}$
$\text{or,}T\propto R.R^{\left(\frac{n-1}{2}\right)}\propto R^{(1+\frac{n-1}{2})}\propto R^{\left(\frac{n+1}{2}\right)}$
$\therefore T\propto R^{\left(\frac{n+1}{2}\right)}$