Gravitational Fields

Gravitational fields

A gravitational field is a region where a mass experiences a force. Fields can be represented by field lines pointing towards the mass. Around a point (or spherical) mass the field is radial; near a planet's surface it is approximately uniform.

Newton's law of gravitation

Every mass attracts every other mass:

F = Gm₁m₂ ÷ r²

where G = gravitational constant (6.67 × 10⁻¹¹ N m² kg⁻²), r = separation of the centres. It's an inverse-square law — double the distance, quarter the force.

Gravitational field strength (g)

Force per unit mass:

g = F ÷ m       and for a radial field   g = GM ÷ r²

Units: N/kg (equivalent to m/s²). At Earth's surface g ≈ 9.81 N/kg. g also follows the inverse-square law with distance from the centre.

Gravitational potential (V)

The gravitational potential at a point is the work done per unit mass to bring a mass from infinity to that point:

V = −GM ÷ r

It is always negative (fields are attractive; potential is zero at infinity and decreases as you approach the mass). The potential difference relates to the work done moving a mass: W = mΔV.

Field strength is the negative gradient of the potential: g = −ΔV/Δr.

Orbits

For a satellite in a circular orbit, gravity provides the centripetal force:

GMm ÷ r² = mv² ÷ r      →      v = √(GM ÷ r)

This leads to Kepler's third law: T² ∝ r³.

• A geostationary satellite orbits above the equator with a period of exactly 24 hours, staying above the same point (used for communications).
• Escape velocity = √(2GM/r) — the minimum speed to escape a body's gravitational field.

Worked example

Find the gravitational field strength at the surface of a planet of mass 6.0 × 10²⁴ kg and radius 6.4 × 10⁶ m.

• g = GM/r² = (6.67×10⁻¹¹ × 6.0×10²⁴) ÷ (6.4×10⁶)² = 4.0×10¹⁴ ÷ 4.1×10¹³ = 9.8 N/kg. ✓

Common mistakes

• Forgetting gravitational potential is always negative.
• Using r as the distance from the surface instead of from the centre.
• Not squaring r in the inverse-square laws.

Exam tips

• Learn both inverse-square laws: F = Gm₁m₂/r² and g = GM/r².
• Use V = −GM/r and W = mΔV for potential/energy problems.
• Derive orbital speed by equating gravity to the centripetal force; know geostationary orbits.

Key facts to remember

• Newton's law: F = Gm₁m₂/r² (inverse square); g = GM/r² (N/kg), radial field.
• Gravitational potential V = −GM/r (always negative); g = −gradient of V; W = mΔV.
• Orbits: gravity = centripetal force → v = √(GM/r), T² ∝ r³; geostationary period = 24 h.