Gravitational Force Between Point Masses (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 

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 from the centre of the Earth is 4.2 × 10⁷ m.

 Calculate the mass of the Earth.

The radius of the Earth is 6400 km.

Determine the ratio

One of Jupiter's moons, Europa, has an orbital period of 85 hours and an orbital radius of 670 900 km.

Explain why Europa moves with uniform circular motion.

Show that the orbital speed of Europa is 14 km s^–1.

Determine the mass of Jupiter.

Ganymede is the largest of Jupiter’s Moons. It has an orbital period of 7.15 days and an orbital speed of 10.880 km s^–1.

Calculate the ratio

Scientists want to put a satellite in orbit around planet Venus.

Justify how Newton's law of gravitation can be applied to a satellite orbiting Venus, when neither the satellite, nor the planet are point masses.

The satellite's orbital time, T, and its orbital radius, R, are linked by the equation:

T² = kR³

Venus has a mass of 4.9 × 10²⁴ kg.

Determine the value of the constant k, and give the units in SI base units.

One day on Venus is equal to 116 Earth days and 18 Earth hours. 

Determine the orbital speed of the satellite in m s⁻¹.

The gravitational potential energy between the satellite and Venus is −2.08 × 10⁸ J. 

Calculate the magnitude of the gravitational force keeping the satellite in orbit. Explain your answer. 

A kilogram mass rests on the surface of the Earth as shown in Fig. 1.1. A spherical rock S, whose centre of mass is underneath the Earth's surface at a distance d of 3.5 km, has a radius of 2 km. The density of the rock is 2500 kg m^–3. 

Fig. 1.1

Determine the size of the force exerted on the kilogram mass by the rock S, justifying any approximations.  

The spherical region of rock, S, is now replaced with a different type of rock and there is a hollow, H, made in the sphere such that its diameter is equal to the radius of S, as shown in Fig. 1.2.

The mass of sphere S with the new rock and the hollow is equal to the mass of sphere S in part (a) and is represented by M. 

The centre of the 1 kg mass m still sits at a distance 3.5 km from the centre of S. 

Fig. 1.2

One student assumes that the force F between the mass m and sphere S is equal, despite the presence of the hollow H.

Explain why the student is incorrect in their assumption. 

The new F between S and m with hollow H is times the previous F without hollow H.

Calculate the distance between the new centre of mass of S and the geometric centre of the new shape of S.