Question

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

Answer 1

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Refer to the attachment for the solution

Answer 2

Critical Velocity (Vc )

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

Vc = √ [ ( 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,

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

Vc = 2 √ [ ( G ρ π R 2 ) / 3 ]

Vc = 2 R √ [ ( G ρ π ) / 3 ]