Show that the critical velocity of a bodyrevolving in circular orbit v

1.	Show that the critical velocity of a bodyrevolving in circular orbit very close to thesurface of a planet of radius R and mean density
Answer» Critical Velocity (Vc) l 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= 2R√[( Gρ π ) / 3 ]

Show that the critical velocity of a bodyrevolving in circular orbit very close to thesurface of a planet of radius R and mean density

Answer»

Critical Velocity (Vc)

l 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= 2R√[( Gρ π ) / 3 ]

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