Gravitation

Gravitation is the force of mutual attraction between any two objects with mass. It is the weakest of the four fundamental forces of nature, but because it always attracts (never repels) and its range is infinite, it dominates on the scale of planets, stars and galaxies. This chapter starts historically: Kepler found the pattern in planetary motion from data, and Newton then found the single force law explaining that pattern — the same force that makes an apple fall and keeps the Moon in orbit.

Keep the big picture in mind: one inverse-square law (F ∝ 1/r²) explains free fall near Earth's surface, the variation of g with height and depth, planetary orbits, escape speed and satellites. Almost every numerical here applies F = GMm/r² or energy conservation using gravitational potential energy.

Johannes Kepler analysed Tycho Brahe's naked-eye observations of planetary positions and found three empirical laws, all with the Sun at one focus, not the centre.

Law of orbits: every planet moves in an elliptical orbit with the Sun at one of the two foci.

Law of areas: the line joining a planet to the Sun sweeps out equal areas in equal intervals of time — dA/dt = constant. This follows from angular momentum conservation, because the gravitational force on the planet is always directed along the line to the Sun (a central force) and so exerts no torque about it. Since L = mvr sinθ is constant and dA = ½r(v dt) sinθ, the areal velocity dA/dt = L/2m is constant, so a planet moves faster near the Sun (perihelion) and slower far from it (aphelion).

Law of periods: T² ∝ a³, where a is the semi-major axis of the orbit; for near-circular orbits this becomes T² ∝ r³. This law is what let Newton deduce that the force must vary as 1/r², shown next.

Newton combined Kepler's law of periods with the idea of centripetal force to propose that every particle in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them, directed along the line joining them.

Derivation sketch (F ∝ 1/r² from T² ∝ r³): for a planet of mass m in a circular orbit of radius r, gravity supplies the centripetal force: F = mv²/r = m(2πr/T)²/r = 4π²mr/T². Using T² = kr³, F = 4π²mr/(kr³) = (4π²/k)(m/r²), i.e. F ∝ 1/r². By Newton's third law the force must also be proportional to the Sun's mass M, giving the complete law.

In vector form, for point masses m₁ and m₂ separated by r₁₂ (pointing from 1 to 2), the force on mass 1 is F₁₂ = −G m₁m₂ r̂₁₂/r² — the minus sign shows it is attractive. The law obeys superposition: the net force on any mass due to several others is the vector sum of the pairwise forces.

G is a universal constant — the same everywhere in the universe, unlike g which changes from place to place. Its value, 6.674 × 10⁻¹¹ N m² kg⁻², was first measured by Henry Cavendish using a torsion balance. Two small spheres are attached to the ends of a light rod suspended by a fine fibre; two large spheres brought close to them make the tiny gravitational attraction twist the fibre through a measurable angle. Knowing the fibre's torsional constant, the restoring torque and hence G can be calculated.

G is extremely small, which is why gravitational attraction between everyday objects is undetectably tiny; only when at least one mass is planet-sized does gravity become the dominant force we feel.

Treating the Earth as a uniform sphere of mass M_E and radius R_E, the force it exerts on a small mass m on its surface is F = GM_Em/R_E². By Newton's second law this gives acceleration g = F/m = GM_E/R_E², the acceleration due to gravity. It is independent of the falling body's mass m — all bodies fall with the same g near Earth's surface (ignoring air resistance), which is why a feather and a coin fall together in vacuum.

Substituting values gives g ≈ 9.8 m s⁻². Since Earth is an oblate spheroid (flattened at the poles), R_E is slightly smaller at the poles, so g is slightly larger there (≈9.83 m s⁻²) than at the equator (≈9.78 m s⁻²).

Above the surface (height h): at height h, distance from Earth's centre becomes (R_E + h), so g_h = GM_E/(R_E+h)² = g(1 + h/R_E)⁻². For h ≪ R_E, the binomial approximation gives g_h ≈ g(1 − 2h/R_E) — g decreases with height.

Below the surface (depth d): only the mass within radius (R_E − d) contributes at depth d, since the shell outside that radius exerts zero net force inside it (shell theorem). For uniform density ρ, enclosed mass M' = (4/3)πρ(R_E−d)³, and g_d = GM'/(R_E−d)² works out to g_d = g(1 − d/R_E) — g falls linearly with depth, reaching zero at the centre (d = R_E).

So g peaks at the surface and falls on either side of it — this is one of the most frequently derived results in the chapter; be ready to derive both cases in full for 5 marks.

Gravitational potential energy is the work done in bringing a mass from infinity to a point in a gravitational field, without change in kinetic energy. With zero potential energy taken at infinity, the potential energy of a mass m at distance r from a mass M is U(r) = −GMm/r — negative because gravity does positive work as the mass is brought in from infinity, so the system's energy is lower than at infinite separation.

Derivation: the external force needed to move mass m quasi-statically against gravity at distance x is F_ext = GMm/x² (outward). Work done moving it from r to infinity is W = ∫_r^∞ GMm/x² dx = GMm[−1/x]_r^∞ = GMm/r. Since U(∞) = 0 and W = U(∞) − U(r), we get U(r) = −GMm/r.

Near Earth's surface, for small h this reduces to the familiar U = mgh (surface as reference), verified by expanding U(R+h) − U(R) with the binomial approximation.

Escape speed is the minimum speed with which a body must be projected from a planet's surface so it just escapes the planet's gravitational pull, reaching infinity with zero kinetic energy left over.

Derivation: total energy at the surface = ½mv_e² − GMm/R. For the body to just escape, total energy at infinity = 0. So ½mv_e² − GMm/R = 0, giving v_e = √(2GM/R) = √(2gR). For Earth, v_e ≈ 11.2 km s⁻¹, independent of the mass and direction of projection of the escaping body.

Escape speed explains why light gas molecules (hydrogen, helium) escape a planet's atmosphere more easily than heavier ones, and why the Moon, with much smaller v_e, has essentially no atmosphere.

A satellite is a body revolving in a stable orbit around a planet under the planet's gravity, which supplies the centripetal force. For mass m in a circular orbit of radius r (= R_E + h), GM_Em/r² = mv_o²/r, giving orbital speed v_o = √(GM_E/r). Close to the surface (r ≈ R_E), v_o = √(gR_E) ≈ 7.9 km s⁻¹ — the minimum speed a projectile needs to become a satellite.

Time period T = 2πr/v_o = 2π√(r³/GM_E) — exactly Kepler's third law for a circular orbit, confirming Newton's law is consistent with Kepler's laws. Note v_o = v_e/√2 at any given radius.

A satellite in circular orbit has kinetic energy K = ½mv_o² = GM_Em/2r and potential energy U = −GM_Em/r. Total mechanical energy E = K + U = GM_Em/2r − GM_Em/r = −GM_Em/2r.

The total energy is negative — the signature of a bound system: the satellite cannot escape to infinity on its own. Note E = −K and U = 2E = −2K, worth memorising for quick numericals. Moving to a higher orbit needs energy input to raise r (E becomes less negative), even though orbital speed and K actually decrease there.

A geostationary satellite orbits the Earth in the equatorial plane, west to east, with period exactly equal to Earth's rotation period (23 h 56 min, taken as 24 h), so it appears fixed relative to a point on the surface. Using T = 2π√(r³/GM_E) with T = 24 h gives orbital radius r ≈ 4.2 × 10⁴ km, i.e. height ≈ 3.6 × 10⁴ km. Used for communication and TV broadcast, since a fixed ground antenna stays pointed at them. Three such satellites, 120° apart, cover almost the whole globe except the poles.

A polar satellite is a low-altitude satellite (around 500-800 km) passing over the poles in a near north-south orbit while the Earth rotates beneath it, scanning the entire surface over successive orbits (period ≈100 minutes). Used for remote sensing, weather monitoring and surveillance.

Weight is the normal reaction force the ground (or a scale) exerts on a body — what a scale actually measures, not the gravitational force itself. A body in free fall, or in an orbiting satellite, accelerates under gravity alone with no other contact force, so the normal reaction on it (and hence its apparent weight) is zero, even though the gravitational force on it is not. This state is weightlessness.

Inside an orbiting satellite, both the satellite and everything inside it (including an astronaut) fall towards Earth with the same acceleration, which supplies exactly the centripetal acceleration for the orbit. Since astronaut and satellite fall together, there is no relative acceleration and hence no contact force between them — the astronaut floats. Earth's gravity there is still nearly as strong as at the surface; it is continuous free fall, not an absence of gravity.