Question

Derive an expression for critical velocity of a satellite revolving around the earth in a circular orbit.

Answer 1

Consider a satellite of mass m revolving round the Earth at a, height 'h' above the surface of the Earth.

Let M be the mass and R be the radius of the Earth.

The satellite is moving with velocity V and the radius of the circular orbit is $r=R+h$. Centripetal force = Gravitational force

$\therefore \frac{M{v}_c^2}{r}=\frac{GMm}{r^2}$

$\therefore v_c^2=\frac{GM}{r}$

$\therefore v_c=\sqrt{\frac{GM}{R+h}}$

This is the expression for critical velocity of a satellite moving in a circular orbit around the Earth,

We know that,

$g_h=\frac{GM}{(R+h{)}^2}$

$GM=g_h(R+h{)}^2$

Substituting in equation (1), we get

$\therefore v_c=\sqrt{\frac{g_h(R+h{)}^2}{R+h}}$

$\therefore v_c=\sqrt{g_h(R+h)}$

where $g_h$ is the acceleration due to gravity at a height above the surface of the Earth.