Physics - Gravitation Question with Solution | TestHub

For a satellite revolving close to the surface of a planet, the orbital radius is approximately equal to the radius of the planet $R$.

The orbital velocity is given by $v=\sqrt{\frac{GM}{R}}$.

The time period of revolution is $T=\frac{2\pi R}{v}=2\pi \sqrt{\frac{R^3}{GM}}$.

The mass of the planet in terms of its density $\rho$ is $M=\frac{4}{3}\pi R^3\rho$.

Substituting the value of $M$ in the expression for $T$:

$T=2\pi \sqrt{\frac{R^3}{G\left(\frac{4}{3}\pi R^3\rho \right)}}=2\pi \sqrt{\frac{3}{4\pi G\rho }}=\sqrt{\frac{3\pi }{G\rho }}$.

The frequency of revolution is $n=\frac{1}{T}=\sqrt{\frac{G\rho }{3\pi }}$.

Answer: $\sqrt{\frac{G\rho }{3\pi }}$