Question

Suppose the gravitational force varies inversely as the n^th power of distance. Then the time period of a planet in circular orbit of radius R around the sun will be proportional to –

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

Answer 1

Answer: (A)

$R^{\left( \frac{n + 1}{2} \right)}$

Answer 2

The required centripetal force is obtained from gravitational force

$\frac{mv^{2}}{R} = \frac{GMm}{R^{n}}$ or v = $\sqrt{\frac{GM}{R^{n - 1}}}$

Again, T =$\frac{2\pi R}{v} = 2\pi R\sqrt{\frac{R^{n - 1}}{GM}}$

or T = 2p$\sqrt{\frac{R^{n + 1}}{GM}}$ = 2p$\frac{R^{\frac{n + 1}{2}}}{\sqrt{GM}}$ or T µ $R^{\frac{n + 1}{2}}$