 # 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,

The gravitational pull on the satellite becomes the centripetal force holding the satellite in orbit. So,

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

OR use GMm/r2 = mω2r and T = 2π/ω - it gives the same outcome

(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)