Question

A geostationary satellite is orbiting the Earth at a height 5R above the surface of Earth, where R is the radius of Earth. Find the time period of another satellite at height of 2R.

Answer 1

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work done on the satellite would be the change in its potential energy as it raised form a height of 'Re' to '2Re'.

now, for a satellite at height 'h' the gravitational potential energy is given as

U = -GmM / (Re+h)

here

m is the mass of the satellite

M is the mass of the Earth

Re is the radius of the Earth

 

thus, for a satellite a height Re, the gravitational potential energy will be

U = -GmM / (Re+Re) = -GmM / 2Re

 

similarly, for a satellite a height 2Re, the gravitational potential energy will be 

U' = -GmM / (Re+2Re) = -GmM / 3Re

thus, the chamge in potential energy would be

work done = dU = U' - U

or

W = -GmM / 3Re  - [-GmM / 2Re] = (GmM/Re). [ 1/2 - 1/3 ] 

thus, we get

W = GmM / 6R