Question

If G is the universal gravitational constant and $\rho$ is the uniform density of a spherical planet. Then shortest possible period of rotation around a planet can be

• A. $\sqrt{\frac{\pi G}{2\rho }}$
• B. $\sqrt{\frac{3\pi G}{\rho }}$
• C. $\sqrt{\frac{\pi }{6G\rho }}$
• D. $\sqrt{\frac{3\pi }{G\rho }}$

Answer 1

The correct option is D

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

The fastest possible rate of rotation around a planet is that for which the gravitational force on material at the equator just barely provides the centripetal force needed for the rotation.Let M be the mass of the planet R its radius and m the mass of a particle on its surface.Then

$MR{\omega }_{max}^2=\frac{GMm}{R^2}$

or ${\omega }_{max}=\sqrt{\frac{GM}{R^3}}=\sqrt{\frac{G\left(\frac{4}{3}\pi R^3\rho \right)}{R^3}}$

= $2\sqrt{\frac{\pi G\rho }{3}}=\frac{2\pi }{T_{min}}$

$\therefore$ $T_{min}=\sqrt{\frac{3\pi }{G\rho }}$