Gravitational Potential (Cambridge (CIE) A Level Physics)

Mars is considered to be an isolated sphere of radius R with its mass concentrated at its centre. 

The variation of the gravitational potential φ with distance x from the centre of Mars is shown in Fig 1.1.

Fig 1.1

 Explain why the values for φ in Fig 1.1. are negative.

The radius R of Mars is 3400 km.

By considering the gravitational potential at Mars' surface, determine a value for its mass.

A meteorite is at rest at infinity. The meteorite travels from infinity towards Mars. 

Calculate the speed of the meteorite when it is at a distance of 3R above Mars' surface. Explain your working.

Mars has two moons, Phobos and Deimos, as shown in Fig 1.2.

Fig 1.2

Calculate the work done to move a satellite of 210 kg from the orbit of Deimos to the orbit of Phobos.

Orbital radius of Phobos = 9400 km

Orbital radius of Deimos = 2.3 × 10⁴ km

Work done = ............................. MJ 

The gravitational potential at a point A about the surface of a planet is –7.2 MJ kg^–1. 

For a point B above the surface of the planet, the gravitational potential is –9.1 MJ kg^–1.

 State, with a reason, whether point A or B is further away from the planet.

A rocket is landing onto the surface of Mercury.

Table 1.1 gives data for the speed of the rocket at two heights about the surface of Mercury, after the rocket engine has been switched off.

Table 1.1

Mercury may be assumed to be a uniform sphere of radius R, with its mass M concentrated at its centre. The rocket, after the engine has been switched off, has mass m. The change in gravitational potential energy of the rocket between the two heights is ΔE_p.

Show that the mass M of Mercury is given by the expression

Write an expression, in terms of m, v₁ and v₂ for the change in kinetic energy of the rocket.

Venus may be assumed to be an isolated sphere of radius 6100 km with its mass, M, at 4.9 × 10²⁴ kg concentrated at its centre. 

An object is projected vertically from the surface of Venus so that it reaches an altitude of 7.2 × 10⁴ km.

Explain why the equation MgΔh where g is the gravitational field strength of Venus and Δh is the height of the object, cannot be used to calculate the object's gravitational potential energy.

The change in the gravitational potential of a different object is 4.2 MJ kg^–1. This object ends up in orbit around Venus.

Calculate the radius of orbit of this object.

Calculate the change in work done per unit mass of the object projected vertically in part (a).

 Change in work done per unit mass = .................... MJ kg^–1

Show that the initial speed of the object in (a) is 1.1 km s^–1, assuming air resistance is negligible. 

A spherical planet has a mass M and radius R. The planet may be considered to have all its mass concentrated at its centre.

A rocket, of mass m is launched from the surface of the planet.

The rocket engines are stopped when the rocket is at a height R above the surface of the planet, as shown in Fig 1.1.

Fig. 1.1

The change in gravitational potential energy of the rocket as it travels from a height R to a height 4R above the planet's surface is equal to   

Determine the value of x

At a distance R away from the surface of the planet the rocket has a velocity 2v and at a distance of 4R away from the planet a velocity v. 

Show that the expression for the thermal energy lost  to the atmosphere of the planet as the rocket moves from R to 4R is given by

The rocket travels towards a moon orbiting the planet

The distance from the centre of the planet to the centre of the moon is 200R. The moon has a mass  where M is the mass of the planet.  

Determine the distance, in terms of R, from the centre of the planet at which 

Give your answer to the nearest whole number.

On the axes of Fig. 1.2, sketch a graph to show the variation of the gravitational potential with distance along a line between the surface of the planet and the surface of the moon.

Fig. 1.2

The Earth may be assumed to be a uniform sphere of radius R = 6.38 × 10⁶ m, with its mass M concentrated at its centre.

A satellite of mass m orbits the Earth at a height h₁ above the Earth's surface.

Derive an expression for the total energy of the satellite in terms of G, M, m, h₁ and R. 

The satellite moves to a distance h₂ away from the surface of the Earth.  

Derive an expression for the change in gravitational potential energy as the satellite moves from h₁ to h₂.

When the satellite reaches a height h₂ above the surface of the Earth, it begins to drift out of its orbit. 

Determine the distance moved by the satellite once it has drifted to the point the gravitational potential energy has increased by 20% of its initial value at h₁. 

Assume that the satellite moves along the same line connecting points h₁, h₂ and the centre of the Earth. 

Mass of the Earth = 5.97 × 10²⁴ kg

Mass of the satellite = 1720 kg

Height h₁ = 2.23 × 10⁶ m

Height h₂ = 3.05 × 10⁶ m

A space shuttle of mass 3 × 10⁶ kg is travelling from the moon back to Earth. It accelerates uniformly from launch at 2.3 m s⁻². It has enough propellant to provide thrust for the first 155 seconds. 

The mass of the moon is 7.35 × 10²² kg and the mean radius is 1740 km. 

Calculate the work done by the rocket during the first 155 seconds after launch. State any assumptions you have made.

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

Show that the gravitational potential at the surface of the Earth is about 24 times greater than the gravitational potential at the surface of the moon.