Gravitational Potential (AQA A Level Physics)

Calculate the gravitational potential of a satellite at a point 2500 km above the Earth’s surface.

What is the effect, if any, on the values of the gravitational potential energy,  and the gravitational potential, V, if a body of mass m placed in a gravitational field is halved?

Figure 1 shows two of the orbits, P and D of two of Mars’ moons, Phobos and Deimos respectively that are in circular orbit above Mars, M. 

Figure 1

The gravitational potential due to Mars of each of these orbits is: 

      Orbit P: –7.19 MJ kg^–1

      Orbit D:  –2.13 MJ kg^–1 

The mass of Mars is 0.107 times the mass of Earth. 

Calculate the radius, from the centre of Mars, of orbit P.

Calculate difference in distance between orbits P and D.

Figure 1 shows the gravitational equipotential lines around a planet. The gravitational potential at point A is –40 MJ kg^–1. 

Figure 1

Determine which of the following potentials equate to points B, C and D. 

            (i)         –20 MJ kg^–1

            (ii)        –40 MJ kg^–1

            (iii)       –50 MJ kg^–1

A 400 kg probe is sent to the asteroid Ceres to collect rock samples before returning to Earth. Ceres has a gravitational field strength, g on its surface is –0.27 N kg^–1. 

The gravitational potential, V, at the surface of the asteroid is –1.33 × 10⁵ J kg^–1. 

Calculate the gravitational potential 35 km above the surface of the asteroid.

The radius of the asteroid is 470 km. 

Calculate the mass of the asteroid Ceres.

Calculate the work done by the probe as it travels from the surface to a point 2000 m above the surface.

The graph in Figure 1 shows how the gravitational potential varies with distance in the region above the surface of the Earth. R is the radius of the Earth. At the surface of the Earth, the gravitational potential is −63 MJ kg^–1. 

Figure 1

Explain what is meant by the gravitational potential and why the values on Figure 1 are negative.

Use Figure 1 to determine the increase in gravitational potential between a distance 2R and 3R from the centre of the Earth.

Using Figure 1, calculate the decrease in potential energy of a 1500 kg satellite has it is brought to the surface of the Earth from a circular orbit of 3R.

By use of Figure 1, calculate the gravitational field strength at distance 2R from the centre of the Earth.

The gravitational field associated with a planet is radial, as shown in Figure 1, but near the surface it is effectively uniform, as shown in Figure 2. 

Figure 1

Figure 2

Sketch a graph to show how the gravitational potential, V, associated with the planet varies with distance r for: 

(i)         Figure 1 

(ii)        Figure 2

Explain how the graphs in part (a) would change for a planet with the same radius, but larger mass.

Titan is the largest moon of Saturn. 

Using the following data to calculate the gravitational potential at the surface of Saturn. 

   Mass of Titan = 2.38 × 10^–4 × mass of Saturn

   Radius of Titan = 0.0455 × radius of Saturn

   Gravitational potential at surface of Titan = –3.50 MJ kg^–1

Sketch a graph on Figure 3 to indicate how the gravitational potential varies with distance along a line outwards from the surface of the Earth to the surface of the Moon. 

Label any values on the axes. 

Figure 3

Table 1 shows how the gravitational potential varies for three points about the centre of Earth. 

Table 1

Show that the data suggest that the potential is inversely proportional to the distance from the centre of the Earth.

Figure 1 shows two points, P and Q, at distances r and 2r from the centre of Earth. 

Figure 1

The gravitational potential at P is −36 kJ kg⁻¹. 

Show that the work done on a 12 kg mass when it is taken from P to Q is 216 kJ.

Mercury has a diameter approximately 0.38 that of a planet Y, and a mass 0.06 that of planet Y. The gravitational potential at the surface of planet Y is −50 MJ kg^–1. 

Calculate the approximate value of the gravitational potential at the surface of Mercury. .

Which planet, Mercury or planet Y requires more work to be done to move a satellite of mass 200 kg from its surface to a distance point? Explain your answer.

The International Space Station (ISS) orbits the Earth at a height of 400 km above the Earth’s surface. The Soyuz spacecraft is used to transport astronauts to the ISS. At lift-off, the spacecraft, with all its contents and fuel, has a mass of 308 000 kg .

Calculate the work done in taking the Soyuz spacecraft from the Earth’s surface to the ISS. 

Assume that the mass of the Soyuz spacecraft remains constant.

In reality it takes less work than your answer to part (a) to get the Soyuz rocket from the surface of the Earth to the ISS. 

Discuss a possible reason for the difference in values of work done.

Calculate the work done in taking the Soyuz spacecraft from the Earth’s surface to a point where the Earth’s gravitational effect is negligible.

Explain why there is no work done by the ISS when it maintains a constant orbit around the Earth.

A space shuttle of mass 2 × 10⁶ kg is travelling from the Earth to the moon.  It accelerates uniformly from launch at 5.25 m s^–2.It has enough propellant to provide thrust for the first 124 seconds. 

Calculate the work done by the rocket during the first 124 seconds after launch. Assume the mass of the space shuttle does not change during launch.

The Moon has a diameter approximately 27% that of the Earth, and a mass of 1.2% that of the Earth. 

Calculate the gravitational potential at the surface of the moon in terms of the gravitational potential on the surface of the Earth.

Figure 1 below shows the gravitational field strength lines between the Earth and the moon.  Point P is the neutral point between the Earth and the moon where there is no resultant gravitational field.

Figure 1

On Figure 1 sketch the equipotential lines between the Earth and the moon.  

The graph in Figure 1 shows how the amount of work done on a spacecraft relates to the gravitational potential above the Moon’s surface during a launch from the surface of the Moon.   

Figure 1

Use Figure 1 to calculate the mass of the spacecraft.

Using the graph in Figure 1, calculate the work done on the spacecraft to raise it to a point where the gravitational potential is half of the value at the surface of the Moon. 

Calculate the height of the spacecraft above the surface of the Moon when it is raised it to a point where the gravitational potential is half of the value at the surface of the Moon, as in part (b). 

   The Moon has a mass of 7.37 × 10²² kg

A binary planet system consists of two stars, A and B, as shown in Figure 1.  A has a mass of mass 4.0 × 10³⁰ kg and B has a mass of 8.0 × 10³⁰ kg.  The centre of the stars is separated by a distance of 2 × 10¹¹ m.

Figure 1

Calculate the gravitational potential at the midpoint between the stars.

The amount of energy required to send a space probe of mass 1800 kg from the surface of star A to the midpoint between stars A and B is 4.2 × 10¹¹ J.

Calculate the gravitational field strength on the surface of star A.

Star A is drifting away from star B. 

Calculate how far star A will have drifted if its gravitational potential energy decreases by 10% of its initial value.

Mars is 1.885 times smaller than the Earth. Table 1 shows how the gravitational potential varies for three points above the surface of Mars.

                           Table 1 

Use the data to determine the gravitational field strength on the surface of Mars.

The Mars Global Surveyor is a robotic space probe with a mass of 1030 kg.  In 1996 it took 1.29 × 10⁹ J of energy to launch the probe from the surface of Mars into an orbit where it maintains a constant height above the surface. 

Calculate the height of the Mars Global Surveyor above the surface of Mars 

The Mars Global Surveyor has a fuel reserve of 1.5 × 10⁶J to allow it to be moved to different orbital heights.

Determine whether this fuel reserve would be sufficient for the Mars Global Surveyor to escape its orbit calculated in (b) to a point where it is no longer influenced by the gravitational attraction of Mars.