Gravitational Fields Questions
Q4.
(a)
By considering the force equation for a satellite of mass m in an orbit of radius r around a planet of mass M, show that the orbital time period T of the satellite does not depend upon m.
![](equations.png)
GMm/r2 = mv2/r ![](../../../graphics/symbols/nuclides/ticksmall.png)
GM/r= v2
'm' cancels out
Now v = 2πr/T ![](../../../graphics/symbols/nuclides/ticksmall.png)
so, GM/r = 4π2r2/T2
T2= 4π2r3/GM
∴ T = 2π(r3/GM)0.5
m does not figure in the expression so T is independednt of m QED ![](../../../graphics/symbols/nuclides/ticksmall.png)
(3 marks)
(b) One of the moons of Jupiter, Ganymede, is the largest satellite in the solar system.
![](jupiter.png)
Its orbital period is equal to 7.15 Earth days and the radius of its orbit is 1.07 × 106 km. Calculate
(i) the angular speed of Ganymede in its orbit,
ω = 2π/T
ω = 2π/(7.15 x 24 x 602)
ω = 1.02 x 10-5 rad s-1![](../../../graphics/symbols/nuclides/ticksmall.png)
(ii) the centripetal acceleration of Ganymede in its orbit,
r = 1.07 × 106 km = 1.07 × 109 m.
a = ω
2
r = (1.02 × 10-5)2 × 1.07 × 109
a = 0.111 m s-2![](../../../graphics/symbols/nuclides/ticksmall.png)
(iii) the mass of Jupiter.
centripetal acceleration = g = GM/r2![](../../../graphics/symbols/nuclides/ticksmall.png)
M = gr2/G
M = 0.111 x (1.07 × 109)2/
6.67 × 10-11
M = 1.91 x 1027 kg
T2= 4π2r3/GM )
M = 4π2r3/GT2![](../../../graphics/symbols/nuclides/ticksmall.png)
M = 4π2(1.07 × 109)3/(6.67 × 10-11(7.15 x 24 x 602)2)![](../../../graphics/symbols/nuclides/ticksmall.png)
M = 1.90 x 1027 kg![](../../../graphics/symbols/nuclides/ticksmall.png)
(5 marks)
(Total 8 marks)