Gravitation and Fields

🎯 Learning Objectives

• Describe the field concept and apply Newton's law of gravitation F = GMm/r^2 as an inverse-square law (AO1/AO2) - AQA 7408 3.7.2.1, OCR A H556 5.4
• Calculate gravitational field strength g = GM/r^2 and use g = F/m (AO2) - AQA 7408 3.7.2.2
• Use gravitational potential V = -GM/r, potential energy mV, and the field-potential gradient relation (AO2) - AQA 7408 3.7.2.3
• Derive orbital speed and period for circular orbits, explain Kepler's third law, geostationary satellites and total orbital energy (AO2/AO3) - AQA 7408 3.7.2.4
• Derive escape velocity and analyse orbital data with uncertainty (AO3)

📋 Key Definitions and Core Concepts

📌 Key Points

Fields and Newton's law of gravitation

A force field is a region in which a body experiences a non-contact force - a useful bookkeeping idea that lets us assign a value (field strength) to every point in space (AQA 7408 3.7.1). Gravity is always attractive and acts on mass (AQA 7408 3.7.2.1; OCR A H556 5.4; Edexcel 9PH0 Topic 12).

Newton's law of gravitation: F = GMm/r^2 - the force between two point (or spherically symmetric) masses is proportional to each mass and to the inverse square of their separation. G = 6.67 × 10^-11 N m^2 kg^-2. Why inverse square? The field lines spread out radially, crossing spheres of area 4πr^2 - the same 'flux' crosses a sphere twice the radius, spread over four times the area, so the force per unit mass falls as 1/r^2. The same geometry underlies Coulomb's law and the intensity of light - a favourite comparison question.

Derivation of field strength: field strength is force per unit mass, g = F/m. Substitute Newton's law: g = GMm/r^2 ÷ m = GM/r^2 - independent of the test mass m, which is why all objects fall with the same acceleration in a vacuum. At Earth's surface (M = 5.97 × 10^24 kg, R = 6.37 × 10^6 m): g = 6.67 × 10^-11 × 5.97 × 10^24/(6.37 × 10^6)^2 = 9.81 N/kg. Higher altitude: g falls as 1/r^2 from the centre - at height h, use r = R + h.

Gravitational potential and potential energy

Gravitational potential at a point is the work done per unit mass bringing a small mass from infinity to that point: V = -GM/r (AQA 7408 3.7.2.3). Negative because gravity attracts - you do negative work (the field does the work) moving a mass in from infinity; V = 0 at infinity, the zero of potential by convention. Gravitational potential energy of a mass m at that point: E_p = mV = -GMm/r.

Derivation of V = -GM/r: the work done moving mass m from radius r_1 to r_2 against the force F = GMm/r^2 is W = ∫F dr = ∫GMm/r^2 dr from r_1 to r_2 = GMm(1/r_1 - 1/r_2). Bringing the mass from infinity (r_1 = ∞) to r (r_2 = r): W = -GMm/r - the potential energy. Per unit mass: V = -GM/r.

Field strength is the potential gradient: g = -dV/dr - the field points down the potential 'slope' (from high to low potential, i.e. towards the mass where V is more negative). Graphically, V against r for a point mass is a curve rising from -GM/r at small r towards zero; the gradient at any r gives -g. Moving between two points: ΔW = mΔV - work done equals mass times the change in potential. Equipotential surfaces are spheres of constant V: moving along one does no work (ΔV = 0), and field lines meet them at right angles - the two facts tested in diagram questions.

Orbits, satellites and Kepler's laws

For a circular orbit, gravity supplies the centripetal force (AQA 7408 3.7.2.4; OCR A H556 5.2/5.4):

Derivation of orbital speed: GMm/r^2 = mv^2/r. Cancel m and one r: v^2 = GM/r, so v = √(GM/r) - orbital speed falls with orbit radius (outer satellites move slower). Orbital angular speed: v = ωr gives ω = √(GM/r^3).

Derivation of Kepler's third law: period T = 2πr/v = 2πr/√(GM/r) = 2π√(r^3/GM). Square: T^2 = 4π^2r^3/GM - T^2 ∝ r^3, so a log-log or T^2 against r^3 plot of planetary data gives a straight line - how Kepler's law is verified and G or the central mass found. The constant T^2/r^3 = 4π^2/GM is the same for all satellites of one central body.

Orbital energy: KE = ½mv^2 = GMm/2r (substituting v^2 = GM/r); PE = -GMm/r. Total energy = KE + PE = -GMm/2r - negative, meaning the satellite is bound to the planet; lower orbits are more strongly bound (more negative total energy, but faster and higher KE).

Geostationary satellites: period 24 hours, above the equator, same rotation direction - fixed above one point on the surface for communications. Derivation of the orbit radius: T^2 = 4π^2r^3/GM with T = 24 × 3600 = 86,400 s gives r = (GMT^2/4π^2)^(1/3) ≈ 4.23 × 10^7 m - about 36,000 km altitude (6.6 Earth radii). All geostationary satellites share this radius - a standard one-mark fact. Polar orbits (over the poles) scan the whole surface - used for weather and mapping.

Escape velocity and orbital analysis

Escape velocity is the minimum launch speed for a body to just reach infinity with zero speed (all KE spent against gravity).

Derivation: energy conservation. Launch from a planet's surface (radius R) with speed v: KE + PE = ½mv^2 - GMm/R. At infinity: KE = 0 and PE = 0. So ½mv^2 - GMm/R = 0, giving v_esc = √(2GM/R) - independent of the launched mass. For Earth: v_esc = √(2 × 6.67 × 10^-11 × 5.97 × 10^24/6.37 × 10^6) = 11.2 km/s. Comparison: v_esc = √2 × v_orbit at the same radius (since v_orbit = √(GM/r)) - a favourite link. Note also that escape velocity is independent of launch direction in the idealised model (energy, not vector direction, decides) - though launching eastward from the equator borrows Earth's rotation (~465 m/s), a small practical gain rockets exploit.

Energy changes in orbit: a satellite in a decaying orbit (drag from the thin upper atmosphere) loses total energy, so it moves to a lower orbit - where its speed is higher (v = √(GM/r)) even though it lost energy. The apparent paradox resolves through the PE: the drop in PE (-GMm/r becomes more negative) exceeds the gain in KE (GMm/2r), with the difference dissipated as heat by drag. A satellite boosted to a higher orbit by a rocket burn gains PE and loses speed - the standard explanation question.

Analysing orbital data: planetary periods and radii: plot T^2 against r^3 - straight line through the origin with gradient 4π^2/GM, so the central mass M = 4π^2/(G × gradient). Satellite speed data against radius: v^2 against 1/r gives a straight line of gradient GM. This is how the mass of the Sun (from planetary data) or of a planet (from its moons) is found. Uncertainty practice: orbital radii and periods are large, well-measured quantities; the derived mass uncertainty is dominated by the scatter of the line fit and by G's precision; quote the percentage difference of the derived mass from the accepted value in the conclusion.

Comparing gravitational and electric fields

The gravitational and electric field topics are deliberately parallel - exam boards set six-mark synthesis questions on the comparison, so learn the pairs (AQA 7408 3.7.1-3.7.3; OCR A H556 5.4/6.2; Edexcel 9PH0 Topics 7/12):

The pattern: every quantity maps one-to-one (source, force, field strength, potential, gradient, equipotentials), and the differences are exactly two - the sign of the source (making electric forces either attractive or repulsive, and electric potential either sign) and shielding. Both forces are inverse-square because field lines from a point source spread over spheres of area 4πr^2 - the same geometry argument serves both. Uniform versions also pair: a uniform gravitational field near Earth's surface (g constant) and a uniform electric field between parallel plates (E = V/d) - both give parabolic trajectories for projectiles and charged particles respectively, the deepest symmetry of the pair.

💡 Worked Example

❓ Practice Questions & Mark Schemes

🎬 Video Resources

• YouTube: A level physics gravitation gravitational field strength potential orbits geostationary satellites Kepler escape velocity revision

📄 Past Papers & Exam Resources

• AQA find past papers and mark schemes - A-Level Physics 7408 fields papers (gravitational fields section)
• OCR qualifications - OCR A Level Physics A H556 past papers (Module 5 Newtonian world and astrophysics)
• Physics and Maths Tutor physics revision - gravitational fields topic questions and past papers by board

🔗 Further Reading & Resources

• Physics and Maths Tutor - A-Level Physics gravitational fields revision notes and exam questions by board
• AQA - find past papers and mark schemes for A-Level Physics 7408
• OCR - Physics A H556 qualification page with past papers, mark schemes and practical materials

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