Gravitational field strength

Gravitational field strength, g, is the gravitational force acting on each kilogram of mass at a point in a gravitational field. It is measured in newtons per kilogram (N/kg). Near the Earth's surface it is about 9.8 N/kg, so every kilogram is pulled down with a force of 9.8 N. It links mass and weight through W = mg, and at A-level it is calculated from g = GM/r².

What is gravitational field strength?

Every object with mass has a gravitational field around it: a region where other masses feel a force of attraction. The gravitational field strength at a point tells you how strong that pull is, per kilogram:

Its unit is N/kg. Because a force on a mass gives an acceleration, g is also the acceleration of a falling object when air resistance can be ignored, so 9.8 N/kg is the same as 9.8 m/s².

Gravitational field strength on Earth and other worlds

These values of surface gravity are from NASA's planetary fact sheet. A more massive planet usually has a stronger field, but its size matters too: Mars and Mercury have the same surface value even though Mars has nearly twice the mass, because Mars is also larger.

Weight, mass and W = mg

Mass measures how much stuff an object contains, in kilograms, and it is the same everywhere. Weight is the force of gravity on that mass, in newtons, and it depends on where you are. The two are linked by:

AQA GCSE Physics defines weight as the force acting on an object due to gravity, and says that in any calculation the value of g will be given. Use the value your paper gives you.

Example 1: weight on Earth and the Moon

A 65 kg astronaut has a weight on Earth of 65 × 9.8 = 637 N. On the Moon, where g = 1.6 N/kg, her weight is 65 × 1.6 = 104 N. Her mass is still 65 kg.

Example 2: finding mass or g

• A bag weighs 196 N on Earth. Its mass is m = W ÷ g = 196 ÷ 9.8 = 20 kg.
• A 40 kg probe weighs 300 N on a moon. The field strength there is g = W ÷ m = 300 ÷ 40 = 7.5 N/kg.

Sort quantities into mass and weight in this question from the Laws of Motion course.

Gravitational field lines

Field lines map out which way a mass would be pulled. For gravity they always point towards the mass causing the field, because gravity only ever attracts.

Around a planet the field is radial. The lines get further apart as you move away, which shows the field getting weaker. Close to the surface, over a small area, the lines are almost parallel and evenly spaced, so the field is treated as uniform: the same strength everywhere. That is why g is taken as a constant 9.8 N/kg for everyday heights.

g = GM/r² (A-level)

Newton's law of gravitation gives the force between two masses M and m whose centres are a distance r apart:

G is the gravitational constant, 6.67 × 10⁻¹¹ N m² kg⁻². Dividing by m gives the field strength at distance r from the centre of a mass M:

AQA A-level Physics covers both g = F/m and g = GM/r² in section 3.7.2.2, which is A-level only, and its data booklet gives G and the Earth's and Sun's masses and radii.

Example 3: g at the Earth's surface

Using the data booklet values for the Earth, M = 5.97 × 10²⁴ kg and r = 6.37 × 10⁶ m:

Example 4: g higher up

At 400 km above the surface, r = 6.37 × 10⁶ + 4.00 × 10⁵ = 6.77 × 10⁶ m:

That is still about 89% of the surface value. Objects in orbit at that height feel weightless because they are falling around the Earth, not because gravity has gone.

Example 5: the force between the Sun and the Earth

With the Sun's mass 1.99 × 10³⁰ kg, the Earth's mass 5.97 × 10²⁴ kg and a separation of 1.50 × 10¹¹ m (one astronomical unit):

You can try your own masses and distances in the Gravitational Force Calculator.

The inverse square law

Here x is the distance from the Earth's centre in Earth radii, starting at the surface (x = 1), and y is g in N/kg. Double the distance and g falls to a quarter: 9.81 ÷ 4 = 2.45 N/kg. Triple it and g falls to a ninth, 1.09 N/kg. This is the inverse square law.

Check your understanding.

Common mistakes

• Mixing up mass and weight. Mass in kg never changes; weight in N depends on g.
• Giving weight in kilograms. Weight is a force, so its unit is the newton.
• Using height instead of distance from the centre in g = GM/r². Add the planet's radius to the height.
• Forgetting to square r. It is an inverse square law.
• Thinking g is zero in orbit. It is a little smaller, not zero.
• Using the wrong value of g. Use the one given on your paper.

Practise with a short timed paper

Gravity and weight come up in GCSE Physics, and gravitational fields in A-level Physics. The daily GCSE Physics Higher and A-level Physics: Year 2 papers are short timed papers of 8 questions, with a new one every day.

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