Question

Show that the critical velocity of a body revolving very close to the surface of a planet of radius R and mean density p is $\sqrt{\frac{\pi \rho G}{3}}$

Answer 1

We know, minimum velocity required to revolve in a circular orbit around a planet is called critical velocity

e.g., $v_c\sqrt{\frac{GM}{R}}$

where M is the mass of planet , R is the radius of that planet.

we know, mass = volume × density

so,

$M=\frac{4}{3}\pi R^3\times p$

$v_c\sqrt{\frac{G\frac{4}{3}\pi R^{3}p}{R}}$

so, $2R\sqrt{\frac{G\pi p}{3}}$

hence, the critical velocity of a body revolving in circular orbit very close to the surface of a planet of radius R and mean density ρ is $\sqrt{\frac{\pi \rho G}{3}}$