Law

Newton's law of gravitation

F_{g} = G \frac{m _{1} m _{2}}{r ^{2}}

Statement

The attractive force between two point masses, proportional to the product of the masses and inversely proportional to the square of their separation.

Symbols and units

$F_{g}$	gravitational force	N
$m_{1}, m_{2}$	masses	kg
$r$	separation	m
$G$	gravitational constant	6.674×10⁻¹¹ N·m²/kg²

When it applies

Point masses, or uniform spheres measured centre to centre.

Derivation

An empirical law, matching the same inverse-square form found for the electric force between charges, but always attractive and vastly weaker:

F_{g} = G \frac{m _{1} m _{2}}{r ^{2}}

Worked examples

Weight from gravitation

Force between Earth (M = 5.97×10²⁴ kg) and a 70 kg person at Earth's surface (r = 6.37×10⁶ m):

F_{g} = \frac{( 6.674 \times 1 0 ^{- 11} ) ( 5.97 \times 1 0 ^{24} ) ( 70 )}{( 6.37 \times 1 0 ^{6} ) ^{2}} \approx 687 N

Matches mg = (70)(9.8) = 686 N — g is this same force law evaluated at Earth's surface.

Watch out for

Valid for point masses, or for uniform spheres measured centre to centre — exactly the same restriction as Coulomb's law.

Connects to

Coulomb's law