6.2 Newton’s Law of Gravitation

The gravitational field strength on the surface of a particular moon is 2.5 N kg^–1. The moon orbits a planet of similar density, but the diameter of the planet is 50 times greater than the moon. 

Calculate the gravitational field strength at the surface of the planet. 

Two planets P and Q are in concentric circular orbits about a star S. 

The radius of P's orbit is R and the radius of Q's orbit is 2R. The gravitational force between P and Q is F when angle SPQ is 90° as shown. 

Deduce an equation for the gravitational force between P and Q, in terms of F, when they are nearest to each other. 

Planet P is twice the mass of planet Q. 

Sketch the gravitational field lines between the two planets on the image below. 

Label the approximate position of the neutral point. 

The distance between the Sun and Mercury varies from 4.60 × 10¹⁰ m to 6.98 × 10¹⁰ m. The gravitational attraction between them is F when they are closest together. 

Show that the minimum gravitational force between the Sun and Mercury is about 43% of F. 

Mercury has a mass of 3.30 × 10²³ kg and a mean diameter of 4880 km. A rock is projected from its surface vertically upwards with a velocity of 6.0 m s^–1.

Calculate how long it will take for the rock to return to Mercury's surface. 

Venus is approximately 5.00 × 10¹⁰ m from Mercury and has a mass of 4.87 × 10²⁴ kg. A satellite of mass 1.50 × 10⁴ kg is momentarily at point P, which is 1.75 × 10¹⁰ from Mercury, which itself has a mass of 3.30 × 10²³ kg. 

Calculate the magnitude of the resultant gravitational force exerted on the satellite when it is momentarily at point P.

A student has two unequal, uniform lead spheres. 

Lead has a density of 11.3 × 10³ kg m^–3. The larger sphere has a radius of 200 mm and a mass of 170 kg. The smaller sphere has a radius of 55 mm. 

The surfaces of two lead spheres are in contact with each other, and a third, iron sphere of mass 20 kg and radius 70 mm is positioned such that the centre of mass of all three spheres lie on the same straight line. 

Calculate the distance between the surface of the iron sphere and the surface of the larger lead sphere which would result in no gravitational force being exerted on the larger sphere. 

Calculate the resultant gravitational field strength on the surface of the iron sphere. 

The smaller lead sphere is removed. The separation distance between the surface of the iron sphere and the large lead sphere is r. 

Sketch a graph on the axes provided showing the variation of gravitational field strength g between the surface of the iron sphere and the surface of the lead sphere. 

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

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

If the region S consisted of oil of density 900 kg m^–3 instead of rock, the force recorded on the kilogram mass would reduce by approximately 2.9 × 10^–4 N. 

A spherical hollow is made in a lead sphere of radius R, such that its surface touches the outside surface of the lead sphere on one side and passes through its centre on the opposite side. The mass of the sphere before it was made hollow is M. 

Show that the magnitude of the force F exerted by the spherical hollow on a small mass m, placed at a distance d from its centre, is given by: 

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.

A satellite orbits the Earth with mass M above the equator with a period, T equal to 48 hours. The mass of the Earth is 5.972 × 10²⁴ kg. 

Derive an equation for the radius, r of the satellite’s orbit.

The mean radius of Earth is 6.37 × 10⁶ m.

Calculate the height of the satellite above the Earth’s surface.

The Hubble Space Telescope is in orbit around the Earth at a height of 490 km above the Earth’s surface.

Calculate Hubble’s speed.

Calculate the magnitude of the gravitational field on the Hubble Space Telescope at this height above the Earth’s surface.

Europa, a moon of Jupiter, has an orbital period of 85 hours and an orbital radius of  670 900 km.

Outline why Europa moves with uniform circular motion.

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

Deduce 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 orbital radius of Ganymede in terms of billions of km.

Define Newton’s universal law of gravitation.

The diagram shows a satellite orbiting the Earth. The satellite is part of the network of global–positioning satellites (GPS) that transmit radio signals used to locate the position of receivers that are located on the Earth. 

When the satellite is directly overhead the microwave signal reaches the receiver 62 ms after leaving the satellite. 

Calculate the height of the satellite above the surface of the Earth.

Explain why the satellite is accelerating towards the centre of the Earth even though its orbital speed is constant.

The radius of Earth is 6.4 × 10⁶ m.

Calculate the gravitational field strength of the Earth at the position of the satellite.

                      Mass of Earth = 6.0 × 10²⁴ kg

A satellite is in a circular orbit around a planet of mass M.

Sketch arrows to represent the velocity and acceleration of the satellite.

Show that the angular speed, ω is related to the orbital radius r by 

         r =

Because of friction with the upper atmosphere, the satellite slowly moves into another circular orbit with a smaller radius before.

Suggest the effect of this on the satellites angular speed. 

Titus and Enceladus are two of Saturn’s moons. Data about these moons are given in the table.

Moon	Orbit radius / m	Angular speed / rad s-1
Titan	1.22 109
Enceladus	2.38  108	5.31 10-5

Determine the mass of Saturn.