Gravitational Fields

This guide covers Topic 13, Gravitational fields, in full — subtopics 13.1 Gravitational field, 13.2 Gravitational force between point masses, 13.3 Gravitational field of a point mass and 13.4 Gravitational potential — from Cambridge International AS & A Level Physics 9702, 2025–2027 series. This is A Level content, building on circular motion from Topic 12 for orbital applications.

Before studying this

This resource assumes centripetal acceleration and force from Motion in a Circle, and gravitational potential energy in a uniform field from Work, Energy and Power.

Syllabus coverage

CAMBRIDGE INTERNATIONAL AS & A LEVEL PHYSICS 9702 — A Level, Topic 13

13.1 Gravitational field — understanding the concept of a gravitational field as a field of force, and representing it with field lines.

13.2 Gravitational force between point masses — recalling and using Newton’s law of gravitation, F = Gm₁m₂/r², for the force between two point masses.

13.3 Gravitational field of a point mass — defining gravitational field strength g at a point as the force per unit mass acting on a small test mass at that point; recalling and using g = GM/r² for the field of a point mass (or outside a spherical mass); understanding why g at the surface of the Earth can be approximated using this formula.

13.4 Gravitational potential — defining gravitational potential at a point as the work done per unit mass in bringing a small test mass from infinity to that point; recalling and using ϕ = -GM/r; understanding the term “gravitational potential energy” as the product of mass and gravitational potential.

Gravitational field

A gravitational field is a region of space in which a mass experiences a force due to the presence of another mass. It is represented using field lines, which point in the direction of the force on a small test mass, and are drawn closer together where the field is stronger.

Newton’s law of gravitation

The gravitational force of attraction between two point masses m₁ and m₂, separated by distance r, is:

F = Gm₁m₂ / r²

where G is the gravitational constant (6.67 × 10⁻¹¹ N m² kg⁻²). This is an inverse square law: doubling the separation reduces the force to a quarter of its original value.

Gravitational field strength

Gravitational field strength g at a point is the force per unit mass acting on a small test mass placed there:

g = F/m

For a point mass M (or a uniform sphere, measured from outside it), this can be derived directly from Newton’s law of gravitation: the force on a small test mass m at distance r from M is F = GMm/r², so

g = F/m = GMm/r² ÷ m = GM/r²

— the test mass cancels, leaving g = GM/r² as a property of the field itself (M and r), independent of whatever test mass is used to probe it.

Worked example. The mass of the Earth is 5.97 × 10²⁴ kg and its radius is 6.37 × 10⁶ m. The gravitational field strength at its surface:

g = GM/r² = (6.67 × 10⁻¹¹ × 5.97 × 10²⁴) / (6.37 × 10⁶)²
g ≈ 9.8 m s⁻²

This matches the familiar value used throughout AS Level mechanics, showing that g = 9.81 m s⁻² is not an arbitrary constant but a direct consequence of Newton’s law of gravitation applied at the Earth’s surface.

Gravitational potential

Gravitational potential ϕ at a point is the work done per unit mass in bringing a small test mass from infinity (where potential is defined as zero) to that point:

ϕ = -GM / r

The negative sign reflects that gravity is attractive: work must be done against the field to move a mass away from a large mass, so potential increases (becomes less negative) with distance and is zero only at infinity. Multiplying gravitational potential by a mass m gives the gravitational potential energy of that mass at that point, E = mϕ, which written out in full — the form given on the data/formula sheet — is:

E_p = -GMm / r

Orbits

For a satellite in a circular orbit, gravity supplies the entire centripetal force needed to keep it on its curved path:

GMm/r^2 = mv^2/r     ->     v = sqrt(GM/r)

substituting v = 2*pi*r/T:

T^2 = (4*pi^2/GM) r^3     (Kepler's third law)

The orbiting mass m cancels from both sides, so orbital speed and period depend only on the central mass M and the orbital radius r — every satellite at a given radius above a given planet keeps the same period, regardless of its own mass.

A geostationary orbit requires three conditions together: a period of exactly 24 hours, orbiting west to east, and directly above the Equator, at a radius of about 42 000 km from Earth’s centre. All three are needed for the satellite to stay fixed above the same point on the ground as the Earth turns beneath it — giving only one of them is a common way to lose marks.

Escape velocity (beyond the syllabus — background only, not examinable)

Escape velocity is not part of the Cambridge 9702 Gravitational Fields content; it is included here only as an extension of the energy ideas above. Escape velocity is the minimum launch speed at which an object’s total energy — kinetic plus gravitational potential — just reaches zero, allowing it to reach infinity with no speed left over:

1/2 m v^2 = GMm/r     ->     v_esc = sqrt(2GM/r)

Escape velocity is independent of the escaping object’s own mass, since m cancels from both sides of the energy equation, leaving only the central mass M and the starting distance r.

Common mistakes

• Using g = GM/r² inside a uniform sphere or for a non-point/non-spherical mass — this formula only applies outside a spherical mass, or for genuine point masses.
• Forgetting the negative sign in ϕ = -GM/r, or misinterpreting what it means — potential is defined as zero at infinity and negative everywhere else, not the other way round.
• Confusing gravitational potential (ϕ, per unit mass, J kg⁻¹) with gravitational potential energy (E = mϕ, in joules) — check the units a question is asking for.
• Treating g as a fixed constant (9.81 m s⁻²) rather than a quantity that varies with r — it decreases with height and with distance from other planets.

Quick revision checklist

• Gravitational field lines and the concept of a field of force
• F = Gm₁m₂/r² for the force between point masses
• g = GM/r² for the field strength of a point mass, derived from g = F/m and Newton’s law of gravitation, and why g ≈ 9.81 m s⁻² at Earth’s surface
• ϕ = -GM/r for gravitational potential, and E_p = mϕ = -GMm/r for potential energy

Related resources

• Motion in a Circle — the previous A Level topic, used for orbital mechanics
• Temperature — the next A Level topic
• Cambridge AS & A Level Physics hub

Written against Cambridge International AS & A Level Physics 9702, 2025–2027 series. Always check the current syllabus for your examination year.