FormulaProved

Newton's law of universal gravitation

Statement

$F = G\dfrac{m_1 m_2}{r^2}$ — the attractive force between two point masses $m_1, m_2$ separated by $r$ , with gravitational constant $G \approx 6.674 \times 10^{-11}\ \text{N·m}^2/\text{kg}^2$ .

Why is it true?

Every massive object attracts every other; the force falls off with the square of distance, mathematically the same form as the Coulomb force between two charges.

Two orbiting bodies around their barycenter: change masses and separation to see the 1/r² gravitational force shape elliptical orbits.

Proof sketch

Newton derived this from Kepler's three empirical laws of planetary motion plus his own second law: assuming a circular orbit of radius r and period T, the required centripetal force is $F = m\omega^2 r = m(2\pi/T)^2 r$ . Combined with Kepler's third law ( $T^2 \propto r^3$ ), this gives $F \propto 1/r^2$ .

Stated by

stated 1687Isaac Newton

Topics that use this theorem

Gravity and weight

Related theorems

Newton's second law

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Isaac Newton (1687). Principia Mathematica