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

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

Answer 1

Answer: (A)

The necessary centripetal force required for a planet to move round the sun
= gravitational force exerted on it
$\frac{m v^{2}}{R}=\frac{G M_{e} m}{R^{n}}$
or $v=\left(\frac{G M}{R^{n-1}}\right)^{\frac{1}{2}}$
as $T=\frac{2 \pi R}{v}=2 \pi R \times\left(\frac{R^{n-1}}{G M}\right)^{\frac{1}{2}}$
$T=2 \pi\left[\frac{R^{\frac{(n+1)}{2}}}{\left(G M_{e}\right)^{\frac{1}{2}}}\right]$
$\therefore T \propto R^{(n+1) 2}$