Gravitational Fields Field Guide

01Two bodies pull each other: F = GMm/r²

Every two masses attract. The force on each body has the same size (Newton’s third law), F = GMm/r², with r measured between the centres. Double r and F falls to a quarter. Change M and both arrows change together. The small body accelerates more only because its mass is smaller.

02Field strength: the force per unit mass

Gravitational field strength is the force per unit mass experienced by a small point mass at that point. Walk the test mass away from the planet: g = GM/r² falls as 1/r². At r = 2R it is a quarter of the surface value. Choose another planet to see how M and R both matter.

03Field lines: always towards the mass

Field lines show the direction of the force on a small mass. They point towards masses, never cross, and are closer together where the field is stronger. Around one sphere they are radial. With two masses the lines bend and there is a neutral point where none pass, closer to the smaller mass. Close to the surface of a planet the field is nearly uniform.

04Where the pulls cancel

On the line between two masses each one pulls a test mass towards itself. The net field is zero where the pulls have the same size: GM/x² = Gm/(d − x)², so x/(d − x) = √(M/m). Move the probe and watch the two arrows. The zero point is always closer to the smaller mass. For the Earth and the Moon (M/m = 81) it is at 0.90 d.

05Kepler’s first and second laws

A planet moves on an ellipse with the Sun at one focus (first law). The line from the Sun sweeps equal areas in equal times (second law): each shaded sector takes the same time, so the planet is fast near the Sun (short, wide sectors) and slow far away (long, thin ones). r_Pv_P = r_Av_A.

06Newton’s cannon: the total energy decides the path

Fire a ball sideways from a mountain on the Earth (no air). The sign of its total energy E_T = E_k + E_p decides the path. E_T < 0: bound: it falls back to the ground, or moves in a circle (at the circle speed) or an ellipse, and we see it again. E_T = 0 (at the escape speed for the mountain top): a parabola; it just escapes and never returns. E_T > 0: a hyperbola; it escapes with speed to spare. Press “Show all five paths” to compare them.

07Circular orbits: v = √(GM/r) and T² ∝ r³

Gravity provides the centripetal force: GMm/r² = mv²/r, so v = √(GM/r). With v = 2πr/T this gives T² = 4π²r³/(GM). Two satellites go round together: the outer one is slower and takes longer. Every satellite of the Earth sits on the same straight line of T² against r³, with gradient 4π²/GM.

08Weightless, not without gravity

In a space station 400 km up, g is still about 8.7 N kg⁻¹. The astronaut and the station fall around the Earth together, with the same acceleration, so the floor does not push on the astronaut: the scale reads zero and the astronaut feels weightless. Compare with standing on the ground, and with a lift falling freely.

09The potential well: why V is always negative

Gravitational potential is the work done per unit mass in bringing a small mass from infinity to that point. Zero is at infinity. Gravity always attracts, so to take a mass from any point out to infinity we must do positive work on it: the mass starts below zero. V = −GM/r is negative everywhere and rises towards zero as r grows. Press “Bring it in from infinity”: gravity does the work, and the potential energy falls below zero.

10Reading the field from a V–r graph

The field strength is minus the gradient of the V–r graph: g = −ΔV/Δr. Move the probe between a planet and its moon and watch the green tangent. Where the graph is steep the field is strong. At the top of the hump the tangent is horizontal: zero gradient, so the net field is zero, the same point as in section 04. The field always points “downhill”.

11Equipotentials and the work done

The dashed circles are equipotentials drawn every 1 × 10⁷ J kg⁻¹. They cross the field lines at 90° and get further apart as the field weakens. Move a 100 kg probe from A to B. Work done against the field (by an external force) = mΔV. Work done by the field = −mΔV. Put A and B on the same equipotential: ΔV = 0, so no work is done.

12Energy of an orbit, and what drag does

For a circular orbit E_k = GMm/2r, E_p = −GMm/r and the total E_T = −GMm/2r. Press “Air resistance on”: drag takes energy, the total becomes more negative, so the orbit radius decreases and the speed increases. E_p falls twice as fast as E_k rises.

13Escape speed: a total energy of zero

Escape speed is the minimum speed a body must have at the surface to reach infinity, with no further energy supplied. It just escapes when its total energy is zero: ½mv² − GMm/R = 0, so v_esc = √(2GM/R) = 11.2 km s⁻¹ for the Earth, whatever the mass. Launch below it and the probe stops and falls back.

14Practice and self-check

Before the exam, can you…

• state the definitions of gravitational field strength, gravitational potential and escape speed word for word?
• use F = GMm/r² and g = GM/r² with r from the centre?
• find where the net field is zero, from an equation and from a V–r graph?
• state Kepler’s three laws with the words ellipse, focus, equal areas and cube?
• derive T² = 4π²r³/(GM) and use it to find a mass or compare two orbits?
• explain why V and E_p are always negative?
• use W = mΔV, both as work against the field and as work done by the field?
• derive the escape speed from energy and explain what drag does to an orbit?