Gravitational Force Calculator

The law of universal gravitation

Any two bodies with mass attract each other. According to Newton's law the force is proportional to the product of the masses and inversely proportional to the square of the distance between them:

F = G × m₁ × m₂ / r²

G is the gravitational constant, 6.6743 × 10⁻¹¹ m³/(kg·s²), and r is the distance between the centres of the bodies. Double the distance and the force drops to a quarter.

Why do we not notice it day to day?

G is a very small number. Two 9000 kg masses 10 m apart attract each other with only 5.4 × 10⁻⁵ N, the weight of an object of 5.5 milligrams. The force becomes noticeable only when one of the bodies is as large as a planet.

Gravitational acceleration

On the surface of a planet of mass M and radius R, the force on an object of mass m can be written as m × g, with g = G × M / R². For the Earth this gives 6.6743 × 10⁻¹¹ × 5.9722 × 10²⁴ / (6.371 × 10⁶)² = 9.82 m/s². On the Moon it is 1.62 m/s², roughly one sixth of the value on Earth. At a height h above the surface it falls off as g = G × M / (R + h)²: at 400 km, where the space station orbits, it is still 8.69 m/s².

Example

Taking the mean distance between the Earth (5.9722 × 10²⁴ kg) and the Moon (7.346 × 10²² kg) as 384 400 km, the force between them is 1.98 × 10²⁰ N.

Limits

The formula is for point masses or spherical bodies. The tool does not include the rotation of the planet, so the result can differ slightly from the measured acceleration of gravity. Masses and radii of the bodies are NASA Planetary Fact Sheet values.