Gravitational force: F = G·m₁·m₂/r²

Newton's universal law of gravitation (1687) is a cornerstone of physics: every pair of masses in the universe attracts each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between their centers. F = G·m₁·m₂/r², where G = 6.674×10⁻¹¹ N·m²/kg² is the universal gravitational constant. This calculator solves F, m₁, r, or planet surface gravity g, with presets for Earth, Moon, Sun, ISS, and common pairs.

Fundamental formulas

• F = G·m₁·m₂ / r² — attractive force
• g = G·M / R² — surface gravity (M mass, R radius)
• v_orb = √(G·M/r) — circular orbital velocity
• v_esc = √(2·G·M/r) — escape velocity
• T² = (4π²/G·M)·r³ — Kepler's third law

The gravitational constant G

G = 6.67430 × 10⁻¹¹ N·m²/kg² (CODATA 2018). One of the least precisely known fundamental constants: ~2×10⁻⁵ relative uncertainty. First accurately measured by Henry Cavendish in 1798 with a torsion balance, only 1% off modern value.

G is tiny: gravity is the weakest of four fundamental forces. Electric force between two electrons is 10⁴⁰ times stronger than their gravitational attraction. Yet gravity dominates astronomically because electric charges in large masses neutralize, while mass is always positive (only attractive).

Key celestial body data

Practical numerical examples

Example 1: person's weight on Earth

70 kg person at surface (r = 6,371,000 m from center):

• F = G·M_earth·m / r² = 6.674e-11 × 5.972e24 × 70 / (6.371e6)² = 686 N
• Or: F = m·g = 70 × 9.81 = 686.7 N

Example 2: Earth-Moon attraction

• F = 6.674e-11 × 5.972e24 × 7.342e22 / (3.844e8)² = 1.98×10²⁰ N
• Keeps Moon in orbit, causes Earth tides

Example 3: Sun-Earth attraction

• F = 3.54×10²² N
• About 180× Earth-Moon force. Keeps Earth orbiting Sun.

Example 4: two people attracting

Two 70 kg people at 1 m:

• F = 6.674e-11 × 70 × 70 / 1² = 3.27×10⁻⁷ N
• Weight of 33 micrograms — less than a dust speck!

Example 5: ISS at 400 km altitude

r = 6771 km:

• g_ISS = 8.69 m/s² (only 11% less than surface!)
• Astronauts float because they're in perpetual free fall around Earth, not absence of gravity
• Orbital velocity: 7.67 km/s
• Period: 92 minutes

Gravity as acceleration

On Earth, surface gravity is g = 9.81 m/s². Formula:

g = G·M_earth / R² = 9.82 m/s²

Weight P = m·g. On Moon g = 1.62 m/s²: Apollo astronauts jumped 2-3 m easily.

g variation with altitude

g(h) = g₀ · (R/(R+h))²

• Everest (8,848 m): 9.77
• Airliner (10 km): 9.78
• ISS (400 km): 8.69
• Geostationary (35,786 km): 0.22

g variation with latitude

• Equator: 9.78
• Poles: 9.83
• Difference ~0.5%

Circular orbital velocity

v_orb = √(G·M/r)

• ISS: 7.67 km/s
• GPS: 3.87 km/s
• Moon (around Earth): 1.02 km/s
• Earth (around Sun): 29.78 km/s

Escape velocity

v_esc = √(2·G·M/r) = √2 · v_orb

• Earth surface: 11.19 km/s
• Moon: 2.38 km/s
• Mars: 5.03 km/s
• Jupiter: 59.5 km/s
• Solar System (from Earth): 42.1 km/s

Tides and differential gravity

Moon pulls closer Earth side harder than far side. This differential gravity creates two tidal bulges. Earth rotates under bulges → two high and two low tides daily.

Tide amplitude ∝ ρ·r_body³/d³. Sun is 27M× more massive than Moon but 400× farther: its tidal effect is only 46% of lunar.

Black holes and Schwarzschild radius

A black hole is so compact that escape velocity exceeds light speed. Critical radius:

r_s = 2·G·M/c²

• Sun: r_s = 2.95 km
• Earth: r_s = 0.89 cm
• Human 70 kg: r_s = 10⁻²⁵ m
• Sagittarius A* (4M solar masses): r_s = 12 million km

Cavendish experiment (1798)

Henry Cavendish measured G with torsion balance: small masses on horizontal suspended rod, large masses nearby. Gravitational torque twists suspension wire; measuring angle and torsion constant gives G.

Original result: G = 6.74×10⁻¹¹ (1% from modern value). Cavendish first "weighed the Earth", calculating mean density 5.48 g/cm³.

Kepler's third law

From F = G·M·m/r² = m·v²/r (centripetal force): v² = G·M/r. With v = 2π·r/T:

T² = (4π²/G·M) · r³

Square of orbital period proportional to cube of semi-major axis.

Applications

Astronomy and astronautics

• Satellite orbit calculations (ISS, GPS, geostationary)
• Interplanetary missions (gravity slingshots)
• Exoplanet discovery (radial velocity)
• Galactic dynamics and dark matter

Geodesy

• Earth shape (geoid)
• Sea level altitude
• Gravimetry for mineral/oil search

Fundamental physics

• Einstein's general relativity tests
• Gravitational waves (LIGO 2015)
• Dark matter and energy

Engineering

• Construction weight tolerances
• Precision scales
• Aerospace

Daily life

• Object weight (m·g)
• Free fall time (h = ½gt²)
• Potential energy m·g·h

Gravity as spacetime curvature (Einstein)

Newtonian gravity works perfectly in daily life and solar system but fails at very large scales or speeds near c. Einstein (1915, General Relativity): gravity is NOT a force but spacetime curvature caused by mass-energy.

Tested consequences:

• Mercury precession: 43"/century unexplained by Newton → predicted exactly by Einstein
• Light deflection: photons bent by Sun (verified 1919 eclipse)
• Gravitational redshift: light from strong fields has lower frequency
• Time dilation: clocks in strong gravity tick slower (GPS corrects)
• Gravitational lensing: massive galaxies bend light
• Gravitational waves: first detected by LIGO (2015)

FAQ

What is the gravitational force formula?

F = G·m₁·m₂/r², G = 6.674×10⁻¹¹ N·m²/kg², m in kg, r in meters. Result in newtons.

Why is gravity weaker than other forces?

G is extremely small. Gravity between two protons is 10³⁶× weaker than electrostatic repulsion. Gravity dominates astronomically because large masses neutralize electric charges, while mass is always attractive.

Are ISS astronauts "weightless"?

No. At 400 km altitude gravity is 8.69 m/s² (only 11% less than surface). Astronauts float because in perpetual free fall around Earth, falling sideways at 7.67 km/s fast enough to "miss" Earth.

Earth escape velocity?

11.19 km/s (40,270 km/h) from surface. Minimum speed to escape gravity field. Rockets burn several minutes to reach it.

Does gravity depend on mass or weight?

On mass (intrinsic, kg). Weight is consequence: P = m·g. On Moon your mass is same but weight 1/6 because g=1.62.

How was G first measured?

Cavendish (1798) with torsion balance. Small spheres on suspended rod, large spheres nearby attract gravitationally. Wire torsion gives G. Only 1% off modern value.

How big is force between two people?

Tiny. Two 70 kg people at 1 m: about 3×10⁻⁷ N — less than dust speck weight. Why we don't feel terrestrial object attractions.

Key numerical data

• G = 6.67430×10⁻¹¹ N·m²/kg²
• g_Earth = 9.80665 m/s² (standard)
• Earth mass = 5.972×10²⁴ kg
• Earth radius (mean) = 6,371 km
• Moon mass = 7.342×10²² kg (1/81 Earth)
• Earth-Moon distance = 384,400 km (mean)
• Sun mass = 1.989×10³⁰ kg (333,000 Earths)
• 1 AU = 149,597,870,700 m (exact, 2012)

Common errors

• Confusing mass and weight: mass constant, weight depends on local g
• Forgetting r squared: doubling distance reduces force 4×, not 2×
• Using surface distance: r is between centers
• Applying to non-spherical objects: formula for point masses or spherical symmetry
• Neglecting relativity in strong fields: need general relativity near black holes

Updated July 2026