Newton's Law of Universal Gravitation
Every mass attracts every other mass with a force proportional to their masses and inversely proportional to the square of the distance between them.
Formula
F = G × m₁ × m₂ / r²
G = 6.674×10⁻¹¹ N·m²/kg²
m₁, m₂ = masses (kg)
r = distance between centres (m)Surface Gravity
g = G × M / R²
Earth: M=5.97×10²⁴ kg, R=6.37×10⁶ m
g = 6.674×10⁻¹¹ × 5.97×10²⁴ / (6.37×10⁶)² = 9.81 m/s²Surface Gravity on Other Planets (m/s²)
- Moon: 1.62 (16.5% of Earth)
- Mars: 3.72 (38%)
- Venus: 8.87 (90%)
- Jupiter: 24.8 (2.5×)
- Saturn: 10.4 (1.06×)
Escape Velocity
v_e = √(2GM/R)
Earth: v_e = √(2×6.674×10⁻¹¹×5.97×10²⁴/6.37×10⁶) = 11.2 km/s
Moon: v_e = 2.38 km/sCalculate gravitational force: Free Gravity Calculator
Gravitational Force Quick-Reference Table
| Object 1 | Object 2 | Distance | Force (N) |
|---|---|---|---|
| Earth (5.97×10²⁴ kg) | 1 kg mass | 6,371 km (surface) | 9.81 |
| Earth | Moon (7.34×10²² kg) | 384,400 km | 1.98×10²⁰ |
| Sun (1.99×10³⁰ kg) | Earth | 149.6×10⁹ m | 3.52×10²² |
| Two 1 kg spheres | 1 kg | 1 m | 6.67×10⁻¹¹ |
| Two people (70 kg each) | 70 kg | 1 m | 3.27×10⁻⁷ |
How Newton's Law of Gravitation Works
Newton's Law of Universal Gravitation: F = Gm₁m₂/r², where G = 6.674×10⁻¹¹ N·m²/kg² is the gravitational constant, m₁ and m₂ are the masses, and r is the centre-to-centre distance. Force is attractive and acts along the line joining the two bodies. It falls off with the square of distance: double the distance, quarter the force.
On Earth's surface, g = GM_E/R_E² ≈ 9.81 m/s², where M_E = 5.97×10²⁴ kg and R_E = 6.371×10⁶ m. Gravitational force governs planetary orbits, tidal effects, satellite trajectories, and the large-scale structure of the universe. Einstein's general relativity describes gravity as spacetime curvature, but Newton's law remains accurate enough for all terrestrial and most solar-system calculations.
Common Mistakes
- Using diameter instead of radius: r in F = Gm₁m₂/r² is the distance between centres, not surface-to-surface. For a 1 m diameter sphere, r = 0.5 m from centre.
- Confusing weight and mass: Weight = mg (a force, in newtons); mass = amount of matter (in kg). Your mass is constant; your weight changes on the Moon (g ≈ 1.62 m/s²).
- Wrong G value: G = 6.674×10⁻¹¹ N·m²/kg² — a very small number. Errors by factors of 10 in G produce wildly wrong answers.
Frequently Asked Questions
Weight = m × g_planet. Mars: g = 3.72 m/s² (38% of Earth). A 70 kg person weighs 686 N on Earth but only 260 N on Mars. Jupiter's surface gravity is 24.8 m/s² — 2.53× Earth's — so the same person would weigh 1,736 N there.
Astronauts in low Earth orbit are in free fall — both they and the spacecraft fall toward Earth at the same rate (9 m/s²), so there is no contact force between them and the floor. This is weightlessness (zero apparent weight), not zero gravity. The ISS at 400 km altitude experiences about 90% of Earth's surface gravity.
Force is F = Gm₁m₂/r² (a vector, points toward the other mass). Gravitational potential energy is U = −Gm₁m₂/r (a scalar, always negative). Force equals −dU/dr. Near Earth's surface, U ≈ mgh is the linear approximation; the full expression is needed for satellite trajectories and escape velocity calculations.