Gravity

Newton's universal law

Every two masses attract each other with a force:

F = G · m₁ · m₂ / r²

where:

• m₁, m₂ — the two masses (kg).
• r — distance between centers (m).
• G = 6.674 × 10⁻¹¹ N·m²/kg² — the gravitational constant.

The force is attractive (always pulling them together) and along the line connecting their centers. By Newton's third law, the force on object 1 from object 2 equals minus the force on 2 from 1.

Gravity at Earth's surface

For an object of mass m near Earth's surface (Earth mass M, radius R):

F = GMm/R² = m·g

where g = GM/R² ≈ 9.81 m/s² is the surface gravitational acceleration.

Location	g (m/s²)	Why
Equator (sea level)	9.78	Earth bulges; centripetal effect; further from center
North/South pole	9.83	Closer to center; no centripetal reduction
Mt. Everest summit	9.77	Higher altitude → larger r → smaller g
ISS orbit (408 km up)	8.69	Still significant gravity; "floating" is free-fall
GPS orbit (20,200 km up)	~0.56	Far from Earth; ~6% of surface g
Earth's center	0	Mass surrounds you symmetrically — no net force

Surface gravity of celestial bodies

Body	g (m/s²)	Relative to Earth
Sun	274	27.9× Earth
Earth	9.81	1×
Mars	3.71	0.38
Mercury	3.70	0.38
Moon	1.62	0.165
Jupiter (cloud tops)	24.79	2.53
Saturn (cloud tops)	10.44	1.06
Pluto	0.62	0.063
Phobos (Mars's moon)	0.0057	0.0006
Sirius B (white dwarf)	~3,000,000	~300,000×
Neutron star (typical)	~10¹²	~10¹¹×

Orbits — gravity as centripetal

For a circular orbit, gravity provides the centripetal force:

GMm/r² = mv²/r
v_orbit = √(GM/r)

Period from v = 2πr/T:

T = 2π·√(r³/(GM))

This is Kepler's third law — T² ∝ r³.

Escape velocity (just barely escapes to infinity):

v_escape = √(2GM/r) = √2 · v_orbit

For Earth surface: v_escape ≈ 11.2 km/s. For Sun: 617 km/s. For black holes: c (speed of light, by definition).

Einstein's reframing

In general relativity (1915), gravity is NOT a force. Mass and energy curve spacetime, and free-falling objects follow the "straightest possible" paths (geodesics) in that curved geometry.

Predictions matching Newton:

• Apples falling, planets orbiting — basically Newtonian motion with tiny corrections.

Predictions where Einstein differs from Newton:

• Mercury's perihelion precession — Mercury's elliptical orbit precesses by 43"/century more than Newton predicts. Einstein got it exactly.
• Light bending — light passing the Sun bends by twice Newton's prediction. Confirmed by Eddington (1919) during a solar eclipse.
• Time dilation — clocks in stronger gravity run slower. GPS satellites correct for this (~38 µs/day).
• Gravitational waves — ripples in spacetime curvature, detected by LIGO (2015).
• Black holes — extreme spacetime curvature; not even light escapes from inside the event horizon.

JavaScript — gravity calculations

// Newton's law of universal gravitation
const G = 6.674e-11;  // N·m²/kg²

function gravitationalForce(m1, m2, r) {
  return G * m1 * m2 / (r * r);
}

// Surface gravity for a body of mass M and radius R
function surfaceG(M, R) {
  return G * M / (R * R);
}

// Earth
const M_EARTH = 5.972e24;
const R_EARTH = 6.371e6;
console.log(`Earth surface g: ${surfaceG(M_EARTH, R_EARTH).toFixed(2)} m/s²`); // 9.82

// Moon
console.log(`Moon surface g: ${surfaceG(7.342e22, 1.737e6).toFixed(2)} m/s²`); // 1.62

// Orbital velocity for circular orbit
function orbitalVelocity(M, r) {
  return Math.sqrt(G * M / r);
}

// Escape velocity
function escapeVelocity(M, r) {
  return Math.sqrt(2 * G * M / r);
}

console.log(`ISS orbit: ${orbitalVelocity(M_EARTH, R_EARTH + 408e3).toFixed(0)} m/s`); // ~7,668
console.log(`Earth escape: ${escapeVelocity(M_EARTH, R_EARTH).toFixed(0)} m/s`);       // ~11,186

// Tidal force (gradient of gravity) — differential gravity from object of mass M at distance r over distance dr
function tidalForce(M, m, r, dr) {
  // Differentiate F = GMm/r² → dF/dr = -2GMm/r³, so Δ ≈ 2GMmdr/r³
  return 2 * G * M * m * dr / (r * r * r);
}

// Moon's tidal pull on a 1 m water column (Earth's radius gives dr ≈ Earth's diameter for full tidal range)
const moonTidal = tidalForce(7.342e22, 1, 384400e3, 6.371e6);
console.log(`Moon tidal force per kg: ${(moonTidal*1e7).toFixed(2)}×10⁻⁷ N`);

// Time to fall from height h (no air resistance)
function fallTime(h, g = 9.81) {
  return Math.sqrt(2 * h / g);
}

console.log(`Fall 100m: ${fallTime(100).toFixed(2)} s`); // ~4.5 s
console.log(`Fall 1000m: ${fallTime(1000).toFixed(2)} s`); // ~14.3 s

Where gravity shows up

• Astronomy. All planetary, stellar, and galactic dynamics. Solar system structure, exoplanet detection, galaxy rotation curves.
• Space mission design. Trajectory optimization, gravitational slingshots, orbit transfers (Hohmann), launch windows.
• Geodesy. Mapping Earth's gravity (gravity anomalies indicate dense or sparse subsurface — used for oil exploration).
• Engineering — structures. Buildings, bridges, dams — must withstand gravitational loading. Static analysis of all weight-bearing structures.
• Sports physics. Projectile trajectories (basketball, football, golf), pendulums (clocks, swings).
• Cosmology. Universe expansion (gravity vs dark energy), structure formation (galaxy clustering), dark matter inferred from rotation curves.
• Geophysics. Plate tectonics, mountain building, volcanic eruptions — all driven by gravitational potential energy.

Common mistakes

• Confusing weight with mass. Weight = mg (force). Mass is intrinsic. On Mars, weight is 38% of Earth's; mass is unchanged.
• Thinking g is constant everywhere. g varies with altitude (g goes as 1/r² with distance from Earth's center), latitude (Earth's bulge + rotation), and local geology.
• Forgetting the inverse-square law. Doubling distance from a planet quadruples the reduction in force (and acceleration).
• Treating gravity as a force in general relativity. In GR, gravity is geometry, not force. For everyday physics, Newton's force picture works perfectly.
• Misapplying near vs far-from-Earth formulas. Use F = mg only near Earth's surface (h ≪ R_Earth). For higher altitudes, use F = GMm/r² with full r = R + h.
• Confusing G (universal) with g (local). G is a fundamental constant; g is the local acceleration on a specific body. g = GM/R² for that body.