F = Gm₁m₂/r² — Newton's Law of Universal Gravitation

Newton's law of universal gravitation: the attractive force between two masses — with a worked Earth-surface example.

F = G\frac{m_1 m_2}{r^2}

Gravitational force equals G times the product of the masses divided by the distance squared.

What it means

Every mass attracts every other mass, with a force proportional to both masses and falling off as the square of the distance. Doubling the distance quarters the force — this inverse-square law is why planets orbit in ellipses and why gravity is negligible between everyday objects.

Variables

Symbol	Name	Unit	Meaning
$F$	Gravitational force	N	Attractive force between the masses
$G$	Gravitational constant	N·m²/kg²	Universal gravitational constant
$m₁, m₂$	Masses	kg	The two attracting masses
$r$	Separation	m	Distance between centers of mass

Assumptions

Both bodies treated as point masses
Only gravity acts (no other forces)
r measured between centers of mass

Worked example

Problem

What gravitational force does Earth exert on a 70 kg person standing at the surface?

Given:

G = 6.67430×10⁻¹¹ N·m²/kg²
M = 5.97237×10²⁴ kg (Earth)
m = 70 kg (person)
r = 6.371×10⁶ m (Earth's radius)

Write the law: $F = G\frac{Mm}{r^2}$
Substitute: $F = 6.67430\times10^{-11} \times 5.97237\times10^{24} \times 70 / (6.371\times10^6)^2$
Numerator: $3.986004\times10^{14} \times 70 = 2.7902\times10^{16}$
Denominator: $(6.371\times10^6)^2 = 4.0590\times10^{13}$
Divide: $F = 2.7902\times10^{16} / 4.0590\times10^{13} \approx 687\ \mathrm{N}$

≈ 687 N

The 687 N result is the person's weight — mass times g (70 × 9.81 = 686.7 N). The universal law and the familiar W = mg are the same physics.