2.1 Show that the critical velocity of a body revolving in a circular orbit very closed to the surface of planet of radius ‘ R ’ and mean density ρ is Vc = 2 R √ [ (G ρ π ) / 3 ]

                      

Solution- 

Critical Velocity (V_c )

                                The minimum velocity required to revolve in a circular orbit around a planet is called critical velocity.

V_c = √ [ ( G M ) / R ]

_ _ _ _ _  ( 1 )

Where ,

G = Gravitational Constant

M = Mass of planet

R = Radius of planet

The density ( ρ ) of a body can be define as mass enclosed per unit volume

Therefore,

ρ = M / V

i.e.

M = ρ V

Volume of spherical planet is

V = ( 4 / 3 ) π R ³

The mass of planet can be given as

M = ρ ( 4 / 3 ) π R ³

Putting this value in equation ( 1 ) we get,

V_c = √ [ ( ( 4 / 3 ) G ρ π R ³ ) / R ]

V_c = 2 √ [ ( G ρ π R ² ) / 3 ]

V_c = 2 R √ [ ( G ρ π ) / 3 ]