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Question
a satellite is revolving around an unkown planet in a circular orbit with time period T. the orbital radius of satellite is r, and the radius of planet is R. etermine the acceleration due to gravity at its surface and also the mass of the planet
a satellite is revolving around an unkown planet in a circular orbit with time period T. the orbital radius of satellite is r, and the radius of planet is R. etermine the acceleration due to gravity at its surface and also the mass of the planet
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Solution
Let, M be the mass of the planet, m be the mass of the satellite, v be the orbital speed.
The centripetal force is provided by the gravitational force, so,
mv2/r = GMm/r2
=> v2 = (GM/r)
The time period is given by,
T = 2Ï€r/v
=> T2 = (2Ï€r)2/v2
=> T2 = 4Ï€2r2/(GM/r)
=> T2 = 4Ï€2r3/(GM)
=> M = 4Ï€2r3/(GT2)
The acceleration due to gravity at the surface is given by,
g = GM/R2 = G[4Ï€2r3/(GT2)]/R2
=> g = 4Ï€2r3/(RT)2
Let, M be the mass of the planet, m be the mass of the satellite, v be the orbital speed.
The centripetal force is provided by the gravitational force, so,
mv2/r = GMm/r2
=> v2 = (GM/r)
The time period is given by,
T = 2Ï€r/v
=> T2 = (2Ï€r)2/v2
=> T2 = 4Ï€2r2/(GM/r)
=> T2 = 4Ï€2r3/(GM)
=> M = 4Ï€2r3/(GT2)
The acceleration due to gravity at the surface is given by,
g = GM/R2 = G[4Ï€2r3/(GT2)]/R2
=> g = 4Ï€2r3/(RT)2
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