Gravitational Fields Questions
Q6.
Communications satellites are usually placed in a geo-synchronous orbit.
(a) State two features of a geo-synchronous orbit.
It has a period of 24 hours or its period equals the period of the Earth's rotation 
It remains in a fixed position relative to surface of Earth
It has an equatorial orbit
It has the same angular speed as the Earth (2 MAX)
(2 marks)
(b) Given that the mass of the Earth is 6.00 × 1024 kg and its mean radius is 6.40 × 106 m,
(i) show that the radius of a geo-synchronous orbit must be 4.23 × 107 m,
GMm/r2 = mv2/r
GM/r =v2
v = 2πr/T
GM/r = 4π2r2/T2
GMT2 = 4π2r3
r3 = GMT2/4π2
r3 = (6.67 x 10-11 x 6.00 × 1024 x (24 x 602)2)/4π2
r3 = 7.57 x 1022
r = 4.23 x 107m
(ii) calculate the increase in potential energy of a satellite of mass 750 kg when it is raised from the Earth's surface into a geo-synchronous orbit.
V = -GM/r
ΔV = -GM x (1/R - 1/r)
ΔV = -GM x ( -1/(6.40 × 106) - 1/(4.23 × 107))
ΔV = -GM x 1.32 × 10-7
ΔV = -6.67 x 10-11 x 6.00 × 1024 x 1.32 × 10-7
ΔV = -5.31 x 107 J/kg
ΔW = mΔV = 750 x -5.31 x 107 = -3.98 x 1010 J
Therefore there is an increase of 3.98 x 1010 J
(6 marks)
(Total 8 marks)
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