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

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