Universal Gravitation (CIE A Level Physics)

Explain how the force(s) on a satellite can result in the satellite being in a circular orbit around a planet.

The Earth and the Moon may be considered to be uniform spheres that are isolated in space. The Earth has mass M, radius R and mean density ρ. 

The Moon, mass m, is in a circular orbit about the Earth with radius nR, as illustrated in Fig. 1.1.

Fig. 1.1

The Moon makes one complete orbit of the Earth in time T.

Show that the mean density ρ of the Earth is given by the expression

where G is the gravitational constant.

The radius R of the Earth is 6.38 × 10⁶ m and the distance between the centre of the Earth and the centre of the Moon is 3.84 × 10⁸ m.

The period T of the orbit of the Moon about the Earth is 27.3 days.

Use the expression in (b) to calculate ρ.

ρ = ................................... kg m^–3 

Mars may be considered to be a uniform sphere of radius 3400 km with its mass M concentrated at its centre. A stone of mass 3.67 kg rests on top of Olympus Mons, the largest volcano on Mars, which is 25 km high.

The gravitational force on the stone is 13.3 N.

Mars spins on its axis with a period of 1 day and 37 mins. 

Calculate M.

Determine the force required to maintain the stone in its circular path.

Derive the equation

where G is Newton's gravitational constant, M is the mass and r is the distance from the mass for the gravitational field strength due to a point mass.

Olympus Mons is an active volcano.

Explain 

A satellite of mass m is in orbit around a planet. 

The mass M of the planet may be considered to be concentrated at its centre.

Show that the time period T of the orbit of the satellite is given by the expression

where R is the radius of the orbit of the satellite and G is the gravitational constant. 

Explain your working.

The radius of a geostationary orbit around Earth is 4.2 × 10⁷ m.

 Calculate the mass of the Earth.

The radius of the Earth is 6400 km.

Determine the ratio