Gravitational force calculator

Choose what to solve for, then fill in the other three. Masses are in kilograms and the force in newtons; the distance can be in kilometres or metres. Very big and very small numbers can be typed as 5.97e24 or 5.97 × 10^24.

The presets fill in the Earth and the Moon, the Earth and the Sun, and two 70 kg people standing 1 m apart, using NASA's masses and average distances. Drag either mass in the drawing to change the distance, and the calculator switches to solving for the force.

F = G × m₁ × m₂ ÷ r². F is the force each mass pulls the other with, m₁ and m₂ are the two masses, and r is the distance between their centres. G is the gravitational constant, 6.674 30 × 10⁻¹¹ N m²/kg², from the CODATA values published by NIST.

The force is the same on both. The Earth pulls the Moon with 1.982 × 10²⁰ N and the Moon pulls the Earth with the same 1.982 × 10²⁰ N. The Earth hardly moves because it has 81 times the Moon's mass.

Rearranged for the other three: m₁ = F × r² ÷ (G × m₂), m₂ = F × r² ÷ (G × m₁), and r = √(G × m₁ × m₂ ÷ F).

The distance is squared, so double it and the force drops to a quarter. Triple it and the force is a ninth. Halve it and the force is four times as strong. The arrows in the drawing follow this as you drag.

The same law gives surface gravity. Put Earth's mass and radius into G × M ÷ r² and you get 6.674 × 10⁻¹¹ × 5.972 × 10²⁴ ÷ (6.371 × 10⁶)² = 9.82 m/s², the figure NASA gives for Earth.

Isaac Newton set out the law in the Principia, published on 5 July 1687. He wrote it as a proportion and never used a constant G. The first measurement of gravity between masses in a laboratory was Henry Cavendish's torsion balance experiment of 1798, which he used to find the density of the Earth; the value of G was worked out from his results later.