Physics - Gravitation Question with Solution | TestHub

The time period of a satellite orbiting just above the surface of a planet is given by $T=2\pi \sqrt{\frac{R^3}{GM}}$, where $R$ is the radius and $M$ is the mass of the planet.

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

Substituting $M$ in the expression for $T$, we get:

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

$T=2\pi \sqrt{\frac{3}{4\pi G\rho }}$

Squaring both sides, we get:

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

$T^2=\frac{3\pi }{G\rho }$

$T^2\rho =\frac{3\pi }{G}$

Answer: $\frac{3\pi }{\mathrm{G}}$