Question

A planet revolves around a massive star in a circle of radius $R$ with a period of revolution $T$ . Given the gravitational force acting between the planet and the star is proportional to $R^{-5/2}$ and if subsequently $T$ is proportional to $R^{\frac{m}{n}}$ , then find the value of $m+n$ ( $m$ and $n$ are smallest integer values).

Answer 1

Answer: 11

According to the question, the gravitational force between the planet and the star,
$F\propto \frac{1}{R^{5/2}}$
$\Rightarrow F=\frac{GMm}{R^{5/2}}$ ,
where $M$ and $m$ be masses of star and planet, respectively. For the motion of a planet in a circular orbit,
$F_c=F_g$
$\Rightarrow mR{\omega }^2=\frac{GMm}{R^{5/2}}$
$\Rightarrow mR{\left(\frac{2\pi }{T}\right)}^2=\frac{GMm}{R^{5/2}}$
$\Rightarrow \frac{4{\pi }^2}{T^2}=\frac{GMm}{R^{7/2}}$
$\Rightarrow T^2=\frac{4{\pi }^2}{GM}{R}^{7/2}$
$\Rightarrow T^2\propto R^{7/2}$
$\Rightarrow T\propto R^{7/4}$
$\Rightarrow m+n=11$