Gravitational Field of a Point Mass (CIE A Level Physics)

The gravitational field strength on the surface of a particular moon is 3.2 N kg^–1. The moon orbits a planet of similar density, but the diameter of the planet is 2.6 times greater than the moon. 

Calculate the gravitational field strength at the surface of the planet when the effect of each body on the other is negligible. 

Fig. 1.1 shows a graph with a horizontal axis representing the distance from the centre of a celestial body in terms of the moon's radius r_m.

Fig. 1.1

Complete and label the vertical axis values and sketch the graph to show gravitational field strength against distance from the centre for the moon in part a) in Fig. 1.1. 

Sketch the graph of gravitational field strength against distance from the centre of the planet on Fig. 1.1. 

Fig. 1.2 shows the relative positions of the planet P and the moon m. 

Fig. 1.2

Find a ratio of the distance from the neutral point to the planet's centre and the distance from the neutral point to the moon's centre. 

The mass of the Sun is 3.33 × 10⁵ times greater than the mass of the Earth.

Calculate the gravitational force between the Sun and the Earth.

Use the following data:

• Distance between the Earth and the Sun = 1.47 × 10¹¹ m
• Mass of the Earth = 5.97 × 10²⁴ kg

The Earth is 1.7 times larger in diameter than Mars and is 1.5 times closer to the Sun.

The average gravitational force between Mars and the Sun is 1.64 × 10²¹ N.

Determine the ratio

Suggest why, for small changes in height near the surfaces of Earth and Mars, the ratio in (b) would remain approximately constant.

The Mariana Trench is the deepest known point on the Earth's surface. It is thought to be at a depth of 11 km below sea level.

Suggest and explain whether the gravitational field strength at the Mariana Trench would be greater than, or less than, the gravitational field strength at sea level.

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

Write an expression 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.

Show that a planet with the same gravitational field strength at its surface and a radius 3 times greater than that of the Moon must be 9 times more massive.

Fig. 1.1 shows how the gravitational field strength above the surface of the Moon varies with distance from its centre. 

Fig. 1.1

On Fig. 1.1, sketch the variation of distance with gravitational field strength for the planet in (c).