A LevelPhysicsAQARevision NotesFields & Their ConsequencesGravitational FieldsNewton's Law of Gravitation

Newton's Law of Gravitation (AQA A Level Physics): Revision Note

Exam code: 7408

Written by: Katie M

Reviewed by: Tim

Updated on 1 October 2026

Guided study available on this topic

Understand this topic with deeper explanations and understanding checks.

Newton's law of gravitation

The gravitational force between two bodies outside a uniform field, e.g. between the Earth and the Sun, is defined by Newton’s law of gravitation
Newton’s law of gravitation states that:

The gravitational force between two point masses is proportional to the product of the masses and inversely proportional to the square of their separation

In equation form, this can be written as:

$F = \frac{G m_{1} m_{2}}{r^{2}}$

Where:
- $F$ = gravitational force between two masses (N)
- $G$ = Newton’s gravitational constant
- $m_{1}$ and $m_{2}$ = the two point masses (kg)
- $r$ = distance between the centre of the two masses (m)

Two spherical masses, m₁ and m₂, separated by centre-to-centre distance r. Equal gravitational-force arrows, Fg, point towards each other from the centres of the masses. — **The gravitational force between two masses outside a uniform field is defined by Newton’s law of gravitation**

Newton’s law of gravitation applies to orbiting bodies, e.g. planets orbiting the Sun
Although stars and planets are very large, they can be considered to be point masses as:
- they are approximately uniform spheres
- their separation is much larger than their radii

The $\frac{1}{r^{2}}$ relation is called the inverse square law
This means that when a mass is twice as far away from another, the gravitational force reduces by a quarter, i.e. ${(\frac{1}{2})}^{2} = \frac{1}{4}$

Worked Example

A satellite of mass 6500 kg is orbiting the Earth at 2000 km above the Earth's surface. The gravitational force between them is 37 kN.

Calculate the mass of the Earth.

Radius of the Earth = 6400 km.

[3]

Answer:

Step 1: List the known quantities

Mass of satellite, $m_{1} = 6500 kg$
- $m_{1}$ and $m_{2}$ can be either way around
Distance of satellite above Earth's surface = 2000 km
Gravitational force, $F_{G} = 37 kN$
Radius of Earth = 6400 km

Step 2: State the equation for Newton's law of gravitation and rearrange for the mass of the Earth

$F_{G} = \frac{G m_{1} m_{2}}{r^{2}}$

$m_{2} = \frac{r^{2} F_{G}}{G m_{1}}$

Step 3: Calculate the distance, $r$

$r$ is the distance between the centre of the Earth and the satellite
$r$ = distance of satellite above Earth's surface + radius of Earth

The Earth with a satellite above its surface, showing r measured from the Earth's centre as the radius of Earth plus the satellite's height.

$r = 2000 + 6400 = 8400 \times 10^{3} m$ [1 mark]

Step 4: Substitute the known values into Newton's law of gravitation to calculate the mass of the Earth

$m_{2} = \frac{{(8400 \times 10^{3})}^{2} \times (37 \times 10^{3})}{(6.67 \times 10^{- 11}) \times 6500}$ [1 mark]

$m_{2} = 6.0 \times 10^{24} kg (2 s . f .)$ [1 mark]

Examiner Tips and Tricks

A common mistake is to forget to add together the distance from the surface of the planet and its radius when you work out $r$ . The distance $r$ is measured from the centre of the mass, which is from the centre of the planet.

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