Question

A satellite of mass $m$ is revolving around the Earth of mass $M$ in circular orbit of radius $r.$ The time period of revolution relates with the ($r$) as - ($Gravitationalforce=\frac{GMm}{r^2}$)

• A. $T\propto r^2$
• B. $T^2\propto r^2$
• C. $T^2\propto r^3$
• D. $T^3\propto r^2$

Answer 1

The correct option is C $T^2\propto r^3$

We can write the orbital velocity $v_o$ of the satellite (mass $m$) in a circular orbit of radius $r$ around the earth (mass $M$) as,

$v_o=\sqrt{\frac{GM}{r}}$

The time period $T$ of this satellite is given as,

$T=\frac{2\pi r}{v_o}$

substituting,

$\therefore T=\frac{2\pi r}{\sqrt{\frac{GM}{r}}}$

$\therefore T=\frac{2\pi }{\sqrt{GM}}{r}^{3/2}$

$\therefore T\propto r^{3/2}$

$\therefore T^2\propto r^3$