§ 13.1 Gravitational field
Key ideas
- A gravitational field is a region where a mass feels a force; the field strength g is the force per unit mass.
- Field lines point towards the mass; for a point or spherical mass they are radial, closer together where the field is stronger.
Equations
g = F / mfield strength = force per unit massN kg⁻¹
Fig. 1 · A radial gravitational field: every field line points towards the centre of the mass, and the lines crowd together near the surface where the field is strongest.
Watch out: gravitational field lines always point towards mass; there is no gravitational repulsion, so they never point outwards (unlike electric field lines, which can do either).
§ 13.2 Gravitational force between point masses
Key ideas
- A uniform sphere behaves as a point mass at its centre, so r is measured centre to centre.
- Newton's law of gravitation: the attractive force obeys an inverse-square law.
- For a circular orbit, gravity provides the centripetal force, which links the orbital radius and period (Kepler's third law).
- A geostationary satellite orbits over the Equator, west to east, with a 24-hour period, so it stays above one point on the ground.
Equations
F = G m₁ m₂ / r²G = 6.67 × 10⁻¹¹ N m² kg⁻²N
GM = 4π² r³ / T²from gravity = centripetal force for a circular orbitm³ s⁻²
Fig. 2 · A circular orbit: setting the gravitational pull GMm/r² equal to the centripetal force mv²/r gives the orbital speed and, with v = 2πr/T, Kepler's third law.
Watch out: r is the distance to the centre, not the height above the surface. A satellite 300 km up orbits at r = R_Earth + 300 km ≈ 6.7 × 10⁶ m, not 3 × 10⁵ m.
§ 13.3 Gravitational field of a point mass
Key ideas
- Combining g = F/m with Newton's law gives the field of a point mass, an inverse-square law.
- Near the Earth's surface r barely changes over everyday heights, so g is approximately constant at about 9.81 N kg⁻¹.
Equations
g = G M / r²field strength of a point (or spherical) massN kg⁻¹
Watch out: g falls as 1/r², not 1/r. Doubling the distance from the centre quarters the field strength; tripling it cuts the field to a ninth.
§ 13.4 Gravitational potential
Key ideas
- Gravitational potential φ is the work done per unit mass in bringing a small test mass from infinity to that point.
- It is negative everywhere: infinity is the zero, and the attractive field does work on a mass moving inwards, so the potential drops below zero.
- The potential energy of a mass m is Ep = mφ.
Equations
φ = −G M / rpotential, with zero set at infinityJ kg⁻¹
Ep = −G M m / rgravitational potential energy of two massesJ
Fig. 3 · The potential well: φ is large and negative close to the mass and rises towards zero at infinity, so work must be done against the field to climb out.
Watch out: moving away from a mass increases potential (it becomes less negative). A more negative φ means deeper in the well, so −3 × 10⁷ J kg⁻¹ is a higher potential than −6 × 10⁷ J kg⁻¹.