Newton’s law of universal gravitation

A force that acts at a distance

In the 17th century, Isaac Newton realised that the force that causes an apple to fall and the force that keeps the Moon in orbit are one and the same: gravity. Two bodies with mass attract each other, even without contact and even in a vacuum.

The expression for the force

For two point masses m1 and m2, separated by a distance d (between their centres), the gravitational force is given by: F = G * m1 * m2 / d^2

where:

• F is in newtons (N)
• m1 and m2 are in kilograms (kg)
• d is in metres (m)
• G is the universal gravitational constant, G = 6.67 × 10⁻¹¹ N·m²/kg²

This force is attractive: it acts from each body towards the other, along the straight line connecting them. By the principle of reciprocal actions, body 1 attracts body 2 with a force exactly opposite to that exerted by body 2 on body 1.

Numerical example

Two masses of 1000 kg each, separated by 10 m: F = 6.67 × 10⁻¹¹ × 1000 × 1000 / 10² = 6.67 × 10⁻⁷ N

This value is minuscule: gravitational force is only significant with very large masses, such as those of planets.

Common pitfalls

• d is the distance between the centres of the bodies, not between their surfaces.
• The distance is squared: if d doubles, F is divided by 4, not by 2.
• Do not confuse G (the universal gravitational constant, which is always the same) with g (the acceleration due to gravity, which depends on location).