**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 × 10^{24} kg and its mean radius is 6.40 × 10^{6} m,

(i) show that the radius of a geo-synchronous orbit must be 4.23 × 10^{7} m,

GMm/r^{2} = mv^{2}/r

GM/r =v^{2}

v = 2πr/T

GM/r = 4π^{2}r^{2}/T^{2}

GMT^{2} = 4π^{2}r^{3}

r^{3} = GMT^{2}/4π^{2}

r^{3} = (6.67 x 10^{-11} x 6.00 × 10^{24} x (24 x 60^{2})^{2})/4π^{2}

r^{3} = 7.57 x 10^{22}

r = 4.23 x 10^{7}m

(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 × 10^{6}) - ^{1}/(4.23 × 10^{7}))

ΔV = -GM x 1.32 × 10^{-7}

ΔV = -6.67 x 10^{-11} x 6.00 × 10^{24} x 1.32 × 10^{-7}

ΔV = -5.31 x 10^{7} J/kg

ΔW = mΔV = 750 x -5.31 x 10^{7} = -3.98 x 10^{10} J

Therefore there is an increase of 3.98 x 10^{10} J

**(6 marks) **

**(Total 8 marks) **