Gravitational fields

Physics · Fields. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

Every mass attracts every other mass. Newton's law says the force between two point masses is proportional to the product of the masses and inversely proportional to the square of the distance between them. A uniform sphere acts as if all its mass were at its centre, so distances are measured from centre to centre.

Gravitational field strength g at a point is the force per unit mass on a small test mass placed there. Field lines show the direction of this force. Around a sphere they are radial and point inwards. Near the surface they are parallel and equally spaced.

Kepler's laws describe orbits. For a circular orbit, gravity provides the centripetal force.

Rules to remember

Gravitational force = G × mass 1 × mass 2 ÷ distance² (F = Gm₁m₂/r²), always attractive
Field strength = force ÷ mass = G × mass of the body ÷ distance² (g = F/m = GM/r²), unit N kg⁻¹
Kepler's first law: each planet moves in an ellipse with the Sun at one focus
Kepler's second law: the line from the Sun to a planet sweeps out equal areas in equal times, so the planet is fastest when closest to the Sun
Kepler's third law: period² is proportional to mean orbital radius³ (T² ∝ r³)
Circular orbit: GMm/r² = mv²/r, which gives v = √(GM/r) and T² = 4π²r³ ÷ (GM)
Field lines closer together mean a stronger field; radial lines spread out with distance

Common mistake

Students use the height above the surface as r. The distance r is measured from the centre of the planet, so a height of one radius above the surface means r = 2R and g falls to one quarter, not one half.

Worked example

A planet has mass 6.4 × 10²³ kg and radius 3.4 × 10⁶ m. What is the gravitational field strength at its surface? (G = 6.67 × 10⁻¹¹ N m² kg⁻²)

Use g = GM ÷ r², with r equal to the radius of the planet.
GM = 6.67 × 10⁻¹¹ × 6.4 × 10²³ = 4.27 × 10¹³.
r² = (3.4 × 10⁶)² = 1.156 × 10¹³ m².
g = (4.27 × 10¹³) ÷ (1.156 × 10¹³) = 3.7 N kg⁻¹.

Answer: 3.7 N kg⁻¹

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D.1 Gravitational Fields

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