This unit is ≈16% of the A-Level Physics, across 6 lessons. Full syllabus
Lesson 1 of 6 · Fields and their consequences
Gravitational fields, potential and orbits
10 min read · about 2 h with practice3 quick checks≈3% of the testStretch: Stretch: harder material that separates the top grades
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Gravitational fields is the first full “field” topic at A-level and the template for electric fields, so the same definitions, graphs and energy ideas come back later. Exams test it with quick ratio and graph MCQs, 3–6 mark structured calculations on orbits, potential and energy, the derivation of T2∝r3, and extended responses comparing geostationary and low orbits. The top marks go to learners who handle the negative potential and orbital energy cleanly.
By the end you’ll be able to
Use F = GMm/r² and g = GM/r², and sketch g against r inside and outside a planet
Define gravitational potential, use V = −GM/r and ΔW = mΔV, and relate g to the potential gradient
Derive T² = (4π²/GM)r³ from circular orbits and apply Kepler’s third law
Calculate orbital speed, orbital energy and escape velocity, and explain why energy is needed to change orbit
Describe geostationary and low polar orbits and their uses
What the exam asks
Force and field:F=r2Gm, and ; field-line diagrams; how varies with .
vii.Check your understanding
3 questions on gravitational fields, potential and orbits. Every option is explained once you answer.
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PromptCard 1 of 3
Define gravitational field strength.
1
m2
g=mF
g=r2GM
g
r
Potential: its definition, V=−rGM, why it is negative, ΔW=mΔV, equipotentials, g=−ΔrΔV, and areas under g–r and F–r graphs.
Orbits: orbital speed and period, the derivation of T2=GM4π2r3, Kepler’s laws, and finding the mass of a planet or star from orbital data.
Energy: kinetic, potential and total energy of a satellite, the energy needed to change orbit, and escape velocity.
Applications: geostationary (synchronous) orbits and low polar orbits, and what each is used for.
Core ideas
Newton’s law of gravitation
F=r2Gm1m2G=6.67×10−11N m2kg−2
Gravity is a universal, always attractive force between all masses. r is the centre-to-centre separation. Outside a uniform sphere, the sphere acts as if all its mass were concentrated at its centre. Gravity is extremely weak: two 1 kg masses 1 m apart attract with about 7×10−11 N, which is why only very large masses produce noticeable fields.
Gravitational field strength
Gravitational field strength at a point is the force per unit mass on a small test mass placed there: g=mF, in N kg−1 (equivalent to m s−2). For a point or spherical mass,
g=r2GM
Field lines show the direction of the force on a small mass. They are radial and inward around a planet. Close to the surface, over small heights, they are almost parallel and equally spaced, so the field is uniform.
g against r: outside a planet, g∝r21. At r=2R, g is a quarter of its surface value. Inside a uniform planet, g increases linearly from zero at the centre to its maximum at the surface.
Field strength is a vector. Where two bodies’ fields overlap, add them as vectors. Between the Earth and the Moon there is a point where the two fields cancel.
Gravitational potential
Gravitational potential at a point is the work done per unit mass to bring a small test mass from infinity to that point. Potential is zero at infinity.
V=−rGMunit: J kg−1
Why negative: gravity is attractive, so the field does work on a mass as it falls in from infinity. Work must be done on the mass to take it back out to V=0. So every point at a finite distance has a potential below zero.
Vincreases (towards zero) as r increases. Potential is a scalar, so the potentials due to several masses simply add.
Work to move a mass:ΔW=mΔV. The gravitational potential energy is Ep=mV=−rGMm. Near the surface this reduces to ΔEp=mgΔh.
Equipotentials are surfaces of equal potential. They are perpendicular to field lines, and no work is done moving along one. Around a planet they are concentric spheres, and equal steps of V get further apart as r increases.
Linking the graphs
Graph
Gradient
Area under it
V against r
ΔrΔV=−g, so g is the magnitude of the gradient
—
g against r
—
ΔV between two radii
F against r
—
work done = change in potential energy
Ep against r
magnitude equals the force F
—
The V–r graph is a negative curve rising towards zero; its gradient gets smaller as r increases because g weakens.
Circular orbits
Gravity provides the centripetal force:
r2GMm=rmv2⇒v=rGM
With v=T2πr, this becomes
T2=GM4π2r3
This is Kepler’s third law: T2∝r3 for every body orbiting the same central mass. The satellite’s own mass cancels, so orbital speed and period depend only on M and r. Higher orbits are slower and take longer. Measuring r and T for any orbiting body gives the central mass: M=GT24π2r.
Kepler’s other two laws (required by OCR A, Eduqas and WJEC): planets move in ellipses with the Sun at one focus, and a line from the Sun to a planet sweeps out equal areas in equal times, so a planet moves fastest when it is closest to the Sun.
Energy of an orbiting satellite
Ek=21mv2=2rGMmEp=−rGMmEtotal=−2rGMm
The total energy is negative: the satellite is bound. It becomes less negative in a higher orbit.
To move from radius r1 to a larger r2, energy ΔE=2GMm(r1 must be supplied. The satellite ends up slower: Ep rises by twice the amount supplied, and Ek falls by the amount supplied.
Drag paradox: air resistance on a low satellite removes energy, so Etotal becomes more negative. The orbit shrinks, and the satellite speeds up.
Escape velocity
The escape velocity is the minimum speed an object must be given at the surface to reach infinity, with no further propulsion and ignoring air resistance. Its kinetic energy must equal the work needed to reach V=0:
21mv2=RGMm⇒vesc=R2GM=2gR
For the Earth this is about 11.2km s−1. It is independent of the object’s mass and is 2 times the speed of a (hypothetical) orbit at the surface. Rockets do not need to reach it, because they keep supplying energy as they climb.
Geostationary and low orbits
Geostationary
Low (often polar) orbit
Period
24 h (one sidereal day, 23 h 56 min)
about 90–100 minutes
Radius / height
r≈4.2×107 m, about 36 000 km above the surface
a few hundred km up
Plane and direction
above the equator, moving west to east with the Earth’s rotation
over or near the poles; the Earth turns beneath
Main uses
communications, satellite TV, weather images of a fixed region
Link to kinetic theory: at the lunar daytime temperature (about 400 K), nitrogen molecules have crms≈600m s−1. That is a quarter of the Moon’s escape velocity, compared with about one-twentieth of the Earth’s. Enough molecules in the high-speed tail escape that the Moon cannot keep an atmosphere.
J kg
−1
m−1
g
g=−ΔrΔV, so the field strength has magnitude 2.45N kg−1, directed towards the Earth. Check: 229.81=2.45 ✓.
r3
v=T2πr
ω=T2π
Ratios:g∝r2M, V∝−rM, v∝r1, T∝r3/2. For planets of equal density, M∝R3, so surface g∝R.
Energy changes between two radii: use ΔEp=GMm(r11−r21), not mgh, whenever h is not small compared with R.
Graphs: potential from the area under g–r; field strength from the gradient of V–r. Use trapezia for areas and state the unit of one grid square.
Logarithmic data (AQA MS 3.11): a graph of lgT against lgr is a straight line of gradient 1.5 if T2∝r3.
5
31km s−1
7.7km s−1
ΔV
positive
2
g=mF=r2GM
g∝r21
V=−rGM: the work per unit mass to bring a small mass from infinity; zero at infinity, negative everywhere else; a scalar.
ΔW=mΔV; g=−ΔrΔV; ΔV is the area under the g–r graph; no work is done along an equipotential.
Orbits: v=rGM and T2=GM4π2r, independent of the satellite’s mass.
Ek=2rGMm, Ep=−rGMm, Etotal=−2rGMm; a higher orbit has more total energy but a lower speed.
vesc=R2GM, about 11.2km s−1 for the Earth.
Geostationary: equatorial, west to east, T=24 h, r≈4.2×107 m. Low polar: about 90 min, whole-Earth coverage, high resolution.