Gravitational Fields

Describe gravitational fields around masses and calculate field strength using Newton's law of gravitation.

About 3 min read
  • Define gravitational field strength
  • Use Newton's law of gravitation
  • Calculate field strength around a point mass
  • Compare uniform and radial fields
Before you start

You can use F = ma and standard form. You also know that weight is the gravitational force on a mass.

What is a gravitational field?

A gravitational field is a region where a mass experiences a force. Field lines show the direction of the force on a small test mass. Near the surface of a planet the lines look almost parallel, so the field is nearly uniform. Further away, the lines point towards the centre, so the field is radial.

Definition: gravitational field strength

The gravitational force per unit mass at a point, measured in N/kg.

Newton's law of gravitation

Every pair of masses attracts each other. The force is proportional to the product of the masses and inversely proportional to the square of the distance between their centres.

Maths
F = Gm₁m₂ ÷ r²
g = F ÷ m
g = GM ÷ r²

G is the gravitational constant, 6.67 × 10⁻¹¹ N m² kg⁻².

Field strength at Earth's surface

Earth has mass 6.0 × 10²⁴ kg and radius 6.4 × 10⁶ m. Calculate g at its surface.

Maths
g = GM ÷ r²
g = (6.67 × 10⁻¹¹ × 6.0 × 10²⁴) ÷ (6.4 × 10⁶)²
g ≈ 9.8 N/kg

Answer: about 9.8 N/kg

Measure r from the centre

For a satellite, r is the planet's radius plus the height above the surface.

Exam focus
Say it like this“r is measured from the centre of the planet to the satellite.”
Not like this“r is the height above the surface.” — Forgetting the planet's radius gives a very wrong answer.
Common mistakeForgetting to square r, or using height above the surface instead of distance from the centre.

Practice

A 2.0 kg mass has a weight of 3.2 N on the Moon. Find g.
g = F ÷ m = 3.2 ÷ 2.0 = 1.6 N/kg.
The distance between two masses doubles. What happens to the gravitational force?
The force becomes one quarter, because F is proportional to 1 ÷ r².
Why is the field close to Earth's surface treated as uniform?
Over a small height, the field lines are nearly parallel and g is almost constant.
Going further
A satellite orbits at 2R from Earth's centre, where R is Earth's radius. Find g there if g = 9.8 N/kg at the surface.
g is proportional to 1 ÷ r². Doubling r gives g = 9.8 ÷ 4 = 2.45 N/kg.

Quick check

Gravitational fields quick check

What is the unit of gravitational field strength?

It is defined as force divided by mass at a point.

If r triples, what happens to g around a point mass?

The formula divides by r², and 3² equals 9.

What shape are field lines around an isolated planet?

A test mass feels a force towards the planet's centre from every direction.
Key takeaways
  • A gravitational field is a region where masses feel a force.
  • g is the force per unit mass.
  • Force follows an inverse square law.
  • Measure r between centres of mass.
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