Question

A satellite is revolving very close to a planet of density $\rho$. The period of revolving of satellite is :

• A. $\sqrt{\frac{3\pi \rho }{G}}$
• B. $\sqrt{\frac{3\pi }{2\rho G}}$
• C. $\sqrt{\frac{3\pi }{\rho G}}$
• D. $\sqrt{\frac{3\pi G}{\rho }}$

Answer 1

The correct option is C $\sqrt{\frac{3\pi }{\rho G}}$

Time period of the satellite is given by

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

where $r$ is the radius of the orbit and $M$ is the mass of the planet

As the satellite is revolving very close to the planet, thus $r=R$

Also $M=\rho \times V=\rho \times \frac{4}{3}\pi R^3$

$\therefore$ $T=2\pi \sqrt{\frac{R^3}{G\times \frac{4\pi }{3}\rho R^3}}=\sqrt{\frac{3\pi }{\rho G}}$