Gravitational Fields (AQA A Level Physics)

The Moon can be assumed to be a uniform sphere of radius r. 

Write an equation for the mean density of the Moon in terms of r, G and g.

The Moon has a radius of 1700 km and a mean density of 3.34 g cm^–3. 

Show that the gravitational field strength on the moon is about 1.6 N kg^–1.

Planet P has mass M and radius R. Planet Q has a radius 4R. The values of the gravitational field strengths at the surfaces of P and Q are the same.

Determine the mass of planet Q in terms of M.

Figure 1 shows how the gravitational field strength above the surface of planet P varies with distance from its centre. 

Figure 1

The values of the gravitational field strengths at the surfaces of P and another planet R are the same. 

 Draw on the diagram the variation of the gravitational field strength above the surface of planet R with a radius 3R over the range shown.

During a Solar eclipse, the Moon passes exactly in front of the sun causing a shadow to fall on Earth in certain locations. At this position, the distance between the Moon and the Sun is around 400 times the distance between the Moon and the Earth. 

Show that the gravitational force of the Sun on the Moon is approximately twice the gravitational force of the Earth on the Moon.

            Distance from the Earth to the Moon = 380 000 km

Tides vary in height with the relative positions of the Earth, the Sun and the Moon which change as the Earth and the Moon move in their orbits. Two possible configurations are shown in Figure 1. 

Figure 1

Consider a 1 kg mass of sea water at position P. This mass experiences forces ,  and  due to its position in the gravitational fields of the Earth, the Moon and the Sun respectively. 

Draw labelled arrows on both diagrams in Figure 1 to indicate the three forces experienced by the mass of sea water at P.

State and explain which configuration A or B of the Sun, the Moon and the Earth will produce a low tide at position P.

Calculate the magnitude of the gravitational force experienced by 2.7 kg of sea water on the Earth’s surface at P, due to the Sun’s gravitational field.

            Radius of the Earth’s orbit = 1.5 × 10¹¹ m   

In 1774, Nevil Maskelyne carried out an experiment near the mountain of Schiehallion in Scotland to determine the density of the Earth. 

Figure 1 shows two positions of a pendulum hung near to, but on opposite sides of, the mountain. The centre of mass of the mountain is at the same height as the pendulum. 

Figure 1

Explain why the pendulums point towards the mountain and how Maskelyne took measures to cancel systematic errors.

Figure 2 shows measurements made with the right-hand pendulum in Figure 1. 

Figure 2

The mountain can be modelled as a cone with a flat base radius of 1.3 km. The mass of the mountain is 2 × 10¹² kg. 

Show that the height of the mountain is about 0.50 km high. 

            Volume of a cone =  

            Density of rock = 2.5 × 10³ kg m^–3

Figure 2 shows the right-hand pendulum bob lying on a horizontal line that also passes through the centre of mass of the mountain. The bob is 1.4 km from the centre of the mountain and it hangs at an angle of 0.0011° to the vertical. 

Calculate the gravitational force of the mountain on the bob.

Hence, or otherwise, calculate the tension in the rope.

A student wants to redo the experiment which Henry Cavendish performed between 1797 and 1798 to investigate Newton’s gravitation law. 

The experiment consists of two unequal uniform lead spheres. The radius of the larger sphere is 150 mm and that of the smaller sphere is 30 mm.

Calculate the mass of the smaller sphere. 

            Density of lead = 11.3 × 10³ kg m^–3

Calculate the mass of the larger sphere if the gravitational force between the spheres when their surfaces are in contact is 1.5 × 10^–7 N.

The student wants to investigate how the radius of the spheres affects the force between gravitational force between them. 

Describe and explain the change in gravitational force if the radii of both spheres are halved.

Two difference leads spheres are now used where the smaller sphere has a mass equal to  that of the larger sphere. They are separated at a distance d between their centres as shown in Figure 1. 

X is the point at which the resultant gravitational field due to the two spheres it zero. 

Find the distance of X from the centre of the larger sphere, in terms of d. 

Figure 1

Figure 1 shows a small horizontal region on the surface of the earth. X is a point underneath the surface which contains material of higher density than the material surrounding it.

On Figure 1, draw the gravitational field lines on the surface of the Earth. 

Figure 1

By considering the gravitational force acting on a mass, m, at the surface of the Earth, write an equation for the mass, M, of the Earth in terms of the radius of the earth, R, the gravitational field strength, g, and the gravitational constant, G.

Calculate the mass of Venus and express its mass as a percentage of the mass of the Earth. 

Radius of Venus = 6.05 × 10⁶ m

Gravitational field strength at Venus’s surface = 8.80 N kg^–1

Although Venus has no moons, the European Space Agency (ESA) launched Venus Express, an artificial satellite, to observe its atmosphere. At its closest, Venus Express had an altitude of 250 km above Venus’s surface. 

Calculate the gravitational field strength of Venus Express at this altitude.

The gravitational field strength on the surface of the moon is 1.63 N kg^–1.

Assuming that the moon is a uniform sphere of radius 1.74 ×10⁶ m, calculate the mass of the moon.

Figure 1 shows the orbital paths of the International Space Station (ISS) and the moon around the Earth. 

Figure 1

The ISS orbits the Earth at an average distance of 408 km from the surface of the Earth. The average distance between the centre of the Earth and the centre of the Moon is 3.80 × 10⁸ m.  

Using your answer from (a), calculate the maximum gravitational field strength which could be experienced by the ISS. 

You can assume that both the Moon and the ISS can be positioned at any point on their orbital path.

Show that for any planet or radius, r, the relationship between the gravitational field strength and the planets density can be expressed as: 

         g = Gπrρ     [Equation 1] 

where ρ is the density of the planet and G is the Gravitational constant.

Two planets, X and Y have the same mass.  Planet X has a radius R and the gravitational field strength on its surface is g.  

The radius of Planet Y is twice that of planet X and the gravitational field strength at the surface of Planet Y is a quarter the value of the gravitational field strength on Planet X. 

Use Equation 1 to determine the volume of Planet Y, V_Y, in terms of the volume of Planet X, V_X.

 An object weighs 100 N at a distance of 200 km above the centre of a small planet. 

Sketch a graph to show the relationship between the gravitational force, F, between two masses and the distance, r, between them. 

Use the information provided to mark some values on the graph.

A student obtains the following data about Earth: 

• Distance along surface from North Pole to equator, d = 10 000 000 m
• Gravitational field at the Earth's surface, g = 9.81 N kg^–1

Using only the data above, calculate the mass of the Earth.

A rocket is sent from the Earth to the moon, as shown in Figure 1 below.  The moon has a radius of 1.74 × 10⁶ m and the gravitational field strength on its surface is 1.62 N kg^–1.  

The distance between the centre of the Earth and the centre of the moon is 385 000 km. 

Figure 1

Using your answer to part (b), calculate the distance above the Earth’s surface where there is no resultant gravitational field strength acting on the rocket.  

The radius of the Earth is 6370 km.

The rocket will require a different amount of fuel to get to the moon than it will to return to the Earth. 

Explain which journey will require the most fuel.

The gravitational field strength on the surface of a particular planet is 1.6 N kg^–1. The planet orbits a star of similar density, but the diameter of the star is 100 times greater than the planet. 

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

Two planets P and Q are in concentric circular orbits about a star S, as shown in Figure 1. The radius of the orbit of P is R and the radius of orbit of Q is 2R. 

Figure 1

The gravitational force between P and Q is F when angle SPQ is 90°, as shown in Figure 1.

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.

On the diagram below sketch the gravitational field lines between the two planets. 

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 the Sun and Mercury is F when they are closest together. 

Write an expression for the gravitational attraction, in terms of F, when they are furthest apart.

Mercury has a mass of 3.30 × 10²³ kg and a mean diameter of 4880 km.  A rock is thrown vertically upwards from its surface with a velocity of 10 m s^–1.  

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

Venus is approximately 5.00× 10¹⁰ m from Mercury and has a mass of 4.87× 10²⁴ kg.  A satellite of mass 1.05× 10⁴ kg is momentarily at point P, which is1.34 × 10¹⁰ m from Mercury.

The angle between the satellite, Mercury and the Venus is 90^o, as shown in Figure 1.

Figure 1

Calculate the magnitude of the resultant force experienced by the satellite when it is 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 150 mm and a mass of 160 kg. The smaller sphere has radius of 50 mm. 

The surfaces of the two lead spheres are in contact with each other as shown in Figure 1.  An iron sphere of mass 20 kg and radius 60 mm is positioned to the left of the larger sphere so that the centre of mass of all 3 spheres lies on the same straight line.  

Figure 1

Calculate the distance between the surface of the iron sphere and the surface of the largest lead sphere which would result in no resultant 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 now removed. The separation distance between the surface of the spheres is r.  

On the axis below draw a graph showing the variation of gravitational field strength, g, between the iron sphere and the larger lead sphere. 

You do not need to include values on the graph.