Question

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

Answer 1

Expression for critical velocity:

Let, 

M = mass of the earth 

R = radius of the earth 

h = height of the satellite from the earth’s surface 

m = mass of the satellite 

v_c = critical velocity of the satellite in the given orbit 

r = (R + h) = radius of the circular orbit

For the circular motion of the satellite, the necessary centripetal force is given as, 

F_CP = \(\frac{mv^2_c}{r}\)   ..........(1)

The gravitational force of attraction between the earth and the satellite is given by, 

F_G = \(\frac{GMm}{r^2}\)   ...........(2)

Gravitational force provides the centripetal force necessary for the circular motion of the satellite.

∴ F_CP = F_G

∴ \(\frac{mv^2_c}{r}\) = \(\frac{GMm}{r^2}\)   ….[From equations (1) and (2)] 

∴ \(v^2_c\) = \(\frac{Gm}{r}\)

∴ v_c = \( \sqrt{\frac{GM}{r}}\)    .......(3)

But, r = R + h

∴ v_c = \( \sqrt{\frac{GM}{r+h}}\)    .......(4)

Also, GM = g_h (R + h)² 

where, g_h is acceleration due to gravity at height ‘h’ above earth’s surface.

∴ v_c = \( \sqrt{g_h(R+H)}\)    .......(5)

Equations (4) and (5) represent critical velocities of satellite orbiting at a certain height above the earth’s surface.