Question

Imagine a light planet revolving around a very massive star in a circular orbit of radius R with a period of revolution T. If the gravitational force of attraction between the planet and the star is proportional to $R^{-5/2}$ , then

• A. $T^2$ is proportional to $R^2$
• B. $T^2$ is proportional to $\, R^{7/2}$
• C. $T^2$ is proportional to $R^{3/2}$
• D. $T^2$ is proportional to $R^{3.75}$

Answer 1

Answer: (B)

$T^2$ is proportional to $\, R^{7/2}$

Answer 2

$\frac{mv^2}{R} \propto R^{-5/2}$
$\therefore \, \, \, \, \, \, \, \, v \propto R^{-3/4}$
Now , $\, \, \, \, \, \, \, \, \, T=\frac{2 \pi R }{v} \, \, \, or \, \, \, T^2 \propto \bigg(\frac{R}{v}\bigg)^2$
$or \, \, \, \, \, \, \, T^2 \propto \bigg( \frac{R}{R^{-3/4}} \bigg)^2 \, \, \, or \, \, T^2 \propto R^{7/2}$