Gravitation— Notes, Formulas & Revision

Newton's law of universal gravitation (1687): every two point masses attract along the line joining them with F = Gm₁m₂/r².

G = 6.674 × 10⁻¹¹ N·m²/kg² — the universal gravitational constant.

Force is always ATTRACTIVE, inverse-square in distance, proportional to product of masses.

Gravity acts equally on both bodies (Newton's third law): each pulls the other with the same magnitude F.

For spherically symmetric mass distributions, you can replace the sphere by a point mass at its centre (shell theorem).

Inside a uniform shell: gravitational field is ZERO. Inside a solid sphere of uniform density: F ∝ r (linear).

Cavendish (1798) measured G using a torsion balance — first laboratory test of gravity.

Universal: same law works between two atoms, Earth-Moon, Sun-Planets, galaxy-galaxy. Galileo's free-fall, Kepler's orbits, and tides all follow from this single equation.

Surface gravity: g₀ = GM/R² ≈ 9.81 m/s² at Earth's surface (R = 6.37 × 10⁶ m, M = 5.97 × 10²⁴ kg).

At height h above surface: g_h = g₀·(R/(R+h))² = g₀/(1 + h/R)². For h ≪ R: g_h ≈ g₀(1 − 2h/R).

At depth d below surface (uniform sphere): g_d = g₀(1 − d/R). Linear decrease; reaches zero at center.

g at the center of Earth = 0 (mass on all sides pulls equally).

g_h falls more slowly than 1/r² near the surface — only because surface gravity is the reference.

Real Earth: g varies from ~9.78 m/s² (equator) to ~9.83 m/s² (poles) due to rotation and oblateness.

Effect of Earth's rotation: g_apparent = g − ω²R cos²(latitude). Smallest at equator.

Altitude effect important for satellites, mountaineering, sensitive gravimetry.

Gravitational potential V at a point is the work done per unit mass in bringing a test mass from infinity to that point (against gravity).

For a point mass M: V(r) = −GM/r. Always negative (potential is zero at infinity, decreases as you approach).

Potential is a SCALAR. Add potentials directly (not vector addition) for multiple sources.

Gravitational field g = −∇V — points 'downhill' on the potential.

Potential difference does work on a moving mass: W = −m·(V_B − V_A). Moving DOWN potential gives KE.

On Earth's surface: V_surface = −GM/R ≈ −6.25 × 10⁷ J/kg.

For a uniform sphere outside: V(r) = −GM/r (like a point mass). Inside: V(r) = −GM(3R² − r²)/(2R³).

Equipotential surfaces are perpendicular to gravitational field lines — flat ground locally, spheres around a point mass.

Escape velocity v_esc is the MINIMUM speed needed to escape a gravitational field, going to infinity with zero residual KE.

Derivation: ½mv_esc² = GMm/R ⇒ v_esc = √(2GM/R) = √(2gR).

Earth: v_esc ≈ 11.2 km/s. Moon: ~2.4 km/s. Sun (at its surface): ~617 km/s.

Relation to orbital velocity: v_esc = √2 · v_orb. So escape = ~1.41× orbital.

Independent of the direction of launch (assumes no atmosphere, no rotation).

Escape velocity does NOT depend on the escaping body's mass. Same v for an atom or a rocket.

If launch speed > v_esc: body follows a hyperbolic trajectory. v = v_esc: parabola. v < v_esc: ellipse (bound).

Above v_esc, body has POSITIVE total energy ⇒ unbound, leaves to infinity.

Orbital velocity v_orb is the speed required for an object to maintain a stable circular orbit around a central body.

Set centripetal force = gravity: mv²/r = GMm/r² ⇒ v_orb = √(GM/r).

At Earth's surface (r ≈ R): v_orb = √(GM/R) = √(gR) ≈ 7.9 km/s (first cosmic velocity).

Period: T = 2πr/v = 2π·√(r³/GM) — Kepler's third law.

Higher orbits ⇒ slower velocity but longer period.

Geosynchronous orbit (T = 24 hours): r ≈ 42,164 km, v ≈ 3.07 km/s.

Total energy in circular orbit: E = −GMm/(2r). Negative ⇒ bound.

Escape velocity = √2 × orbital velocity at the same r.

Kepler's three laws (1609, 1619) describe planetary motion — empirical, derived from Tycho Brahe's data; later explained by Newton's gravity.

1st law (LAW OF ORBITS): each planet moves in an ELLIPSE with the Sun at one FOCUS.

2nd law (LAW OF AREAS): the line joining planet to Sun sweeps out equal AREAS in equal times. Consequence of conservation of angular momentum (no torque from a central force).

3rd law (LAW OF PERIODS): square of orbital period ∝ cube of semi-major axis. T² = (4π²/GM)·a³.

Applies to any small body orbiting a much larger one (planets/Sun, moons/planet, satellites/Earth).

2nd law implies planets move FASTER at PERIHELION (closest to Sun) and SLOWER at APHELION.

Newton showed Kepler's laws follow from the inverse-square attractive force.

Bonus: Kepler's 3rd law lets us measure the mass of the central body if T and a are known.

Newton's Law of Gravitation: Every mass attracts every other mass with force F = Gm₁m₂/r².

The gravitational constant G = 6.674 × 10⁻¹¹ N·m²/kg².

Orbital velocity of a satellite: v₀ = √(GM/r) = √(gR²/r), where r is the orbital radius.

Escape velocity from Earth's surface: vₑ = √(2gR) ≈ 11.2 km/s. It is √2 times the orbital velocity at surface.

Kepler's Third Law: T² ∝ r³ — the square of the period is proportional to the cube of the orbital radius.

Geostationary orbit: T = 24 hours, r ≈ 42,164 km from Earth's center, above the equator.

Total energy of an orbiting satellite: E = −GMm/(2r) — it is negative (bound state).

At the surface: g = GM/R². As you go up: g decreases as GM/(R+h)². Inside Earth: g decreases linearly.

Net gravitational force on a body is the vector sum of forces from each other mass.

Always attractive — each force points from the test mass toward each source.

For continuous distributions, sum becomes an integral.

Field lines point in the direction of the gravitational field g — always toward masses (always attractive).

Density of lines is proportional to |g|. Lines start at infinity and end on masses.

Lines never cross — at each point g has a unique direction.

A satellite in circular equatorial orbit with period T = 24 h appears stationary above one point on Earth.

Required altitude ≈ 35,786 km above Earth's surface (radius from center ≈ 42,164 km).

Used for telecommunications, weather imaging, broadcast TV.