Gravitational potential

Mimas and Enceladus are moons of Saturn.

Mimas orbits at a height of 1.28 × 10⁸ m above the surface of Saturn. The orbital period of Mimas is 0.94 Earth days.

Enceladus orbits at a height of 1.80 × 10⁸ m above the surface of Saturn.

Calculate, in Earth days, the orbital period of Enceladus.

Titan is the largest moon of Saturn.

Figure 2 shows the variation of gravitational potential with distance from the centre of Saturn in the direction of Titan.

The dashed line shows the gravitational potential due to the mass of Saturn alone.

The solid line shows the total gravitational potential due to Saturn and Titan. The solid line ends at the surface of Titan.

A student suggests that, over the range of distances shown in Figure 2, the gravitational field strength due to the mass of Saturn alone is approximately constant.

Explain why the suggestion is correct.

At a point P, the resultant gravitational field strength due to Saturn and Titan is zero.

Determine, using the solid line in Figure 2, the distance from Saturn to P.

The centre of Titan is 12.22 × 10⁸ m from the centre of Saturn.

Show that the gravitational potential due to the mass of Titan at a distance of 12.08 × 10⁸ m from Saturn is about −7 × 10⁵ J kg⁻¹. Go on to determine the mass of Titan.

Describe two properties of a radial gravitational field.

A space probe is launched from the Earth’s surface.

Figure 3 shows how the gravitational force acting on the space probe varies with height above the Earth’s surface.

State the physical significance of the shaded area in Figure 3.

At the Earth’s surface,

• the gravitational field strength of the Sun is g_S

• the gravitational field strength of the Earth is g_E.

distance from the Earth to the Sun = 1.50 × 10¹¹ m

Explain why g_S is more important than g_E in predicting the motion of the space probe as it escapes from the Solar System.

The space probe eventually reaches a point where the gravitational influence of the Solar System is negligible. The probe is unpowered as it approaches an isolated interstellar body X. The gravitational field of X changes the kinetic energy of the space probe.

Table 2 shows the distance of the space probe from the centre of mass of X and the speed for two positions A and B of the space probe.

The space probe has a mass of 4.9 × 10⁴ kg.

Calculate the mass of X.

The lines in Figure 4 show the shape of the gravitational field around two stars G and H.

Compare, with reference to Figure 4, the masses of G and H.

X and Y are two points in the field.

Annotate Figure 4 to show the field direction at X and the field direction at Y.

A spherical asteroid P has a mass of 2.0 × 10²⁰ kg.

The gravitational field strength at its surface is 0.40 N kg⁻¹.

Calculate the radius R of P.

Sketch, on Figure 5, the variation of the gravitational field strength g with distance r. The distance r is measured from the centre of P.

Explain what is represented by the area under the graph between r = R and r = 2 R on Figure 5.

Asteroid P approaches the two stars G and H. Figure 6 shows one position of P close to H.

The gravitational force on P from G is 6.38 × 10¹² N.

Calculate the magnitude of the acceleration of P.

Explain why P cannot have a circular orbit around H.

The Global Positioning System (GPS) uses satellites to support navigation on Earth.

One GPS satellite is in a circular orbit at a height h above the surface of the Earth. The Earth has mass M and radius R.

Calculate the orbital period of the satellite when h equals 2.02 × 10⁷ m.

Figure 3 shows the orbital plane of the satellite inclined at an angle to the equator. X, Y and Z are locations on the Earth.

X is at the North Pole, Y is on a high mountain and Z is on the equator.

The satellite is to be launched from one of the locations.

State and explain which launch site X, Y or Z minimises the amount of fuel required to send the satellite into its orbit.

The satellite has a mass of 1630 kg.

Calculate the gravitational potential energy of the satellite when in the orbit in Question 02.2.

A different satellite is in a higher circular orbit.

Explain how the linear speed of this satellite compares with the linear speed of the satellite in Question 02.1.

The diagram shows gravitational equipotentials. Adjacent equipotentials are separated by an equal gravitational potential difference V.

Which point has the greatest gravitational field strength?

The diagram shows equipotential lines for a uniform gravitational field. The lines are separated by 20 m.

An object of mass 4 kg is moved from P to Q.

What is the work done against gravity to move the object?

The graph shows how the gravitational potential V varies with the vertical distance d from the surface of the Earth.

What does the gradient of the graph represent at the surface of the Earth?

Define gravitational potential at a point.

Figure 2 shows the positions of equipotential surfaces at different distances from the centre of the Moon.

Explain how the equipotential surfaces in Figure 2 show that the gravitational field is not uniform.

Calculate, using Figure 2, the escape velocity at the surface of the Moon.

radius of Moon = 1.74 × 10⁶ m

The graph shows how the gravitational potential varies with distance between two planets, K and L, that have the same radius.

Which statement is correct?

The diagram shows equipotential lines near a group of asteroids.

Which arrow shows the direction of the gravitational field at X ?

Planet N has a gravitational potential V at its surface. Planet M has double the density and double the radius of planet N. Both planets are spherical and have uniform density.

What is the gravitational potential at the surface of planet M ?

A spacecraft of mass 1.0 × 10⁶ kg is in orbit around the Sun at a radius of 1.1 × 10¹¹ m The spacecraft moves into a new orbit of radius 2.5 × 10¹¹ m around the Sun.

What is the total change in gravitational potential energy of the spacecraft?

Derive an expression to show that for satellites in a circular orbit

where T is the period of orbit and r is the radius of the orbit.

Pluto is a dwarf planet. The mean orbital radius of Pluto around the Sun is 5.91 10⁹ km compared to a mean orbital radius of 1.50 10⁸ km for the Earth.

Calculate in years the orbital period of Pluto.

A small mass released from rest just above the surface of Pluto has an

Assume Pluto has no atmosphere that could provide any resistance to motion.

Calculate the mass of Pluto.

Give your answer to an appropriate number of significant figures.

Figure 8 shows the variation in gravitational potential with distance from the centre of Pluto for points at and above its surface.

A meteorite hits Pluto and ejects a lump of ice from the surface that travels vertically at an initial speed of 1400 m s¹.

Determine whether this lump of ice can escape from Pluto.