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

Mars may be considered to be a uniform sphere of radius 3400 km with its mass M concentrated at its centre. A stone of mass 3.67 kg rests on top of Olympus Mons, the largest volcano on Mars, which is 25 km high.

The gravitational force on the stone is 13.3 N.

Mars spins on its axis with a period of 1 day and 37 mins. 

Calculate M.

Determine the force required to maintain the stone in its circular path.

Derive the equation

where G is Newton's gravitational constant, M is the mass and r is the distance from the mass for the gravitational field strength due to a point mass. Use the equations for Newton's law of gravitation and the gravitational field strength. 

Olympus Mons is an active volcano. Consider the stone from part (a) as a point mass in Mars' gravitational field.

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).

The gravitational field strength on the moon's surface is 1.63 N kg^–1. It has a diameter of 3480 km.

The ISS orbits the Earth at an average distance of 408 km from the surface of the Earth as shown in Fig. 1.1. 

Fig. 1.1

The following data are available:

• Average distance between the centre of the Earth and the centre of the Moon = 3.80 × 10⁸ m
• Mass of the Earth = 5.97 × 10²⁴ kg
• Radius of the Earth = 6.37 × 10⁶ m

Calculate the maximum gravitational field strength g  experienced by the ISS. Assume that both the Moon and the ISS can be positioned at any point on their orbital path.

Show that the gravitational field strength g  is proportional to the radius of a planet r  and its density ρ. 

Two planets X and Y are being compared by a group of astronomers. They have different masses.

Planet X has a density ρ and the gravitational field strength on its surface is g. The density of planet Y is three times that of planet X and the gravitational field strength on its surface is 9 times that of planet X.

Use the equation you derived in part (c) to show that the mass of planet Y is roughly 80 times larger than the mass of planet X.

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.