Universal Gravitation

Scalar form

Vector form

Newton's law of universal gravitation describes gravity as a force by stating that every particle attracts every other particle in the universe with a force that is proportional to their masses and inversely proportional to the square of the distance between their centers of mass.

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Physics Classical mechanics Gravity Inverse-square law

\htmlData{term=force}{\mathbf{F}_{21}} = - \htmlData{term=gravitational-constant}{G} \frac{ \htmlData{term=mass-one}{m_1} \htmlData{term=mass-two}{m_2} }{ |\htmlData{term=displacement}{\mathbf{r}_{21}}|^{2} } \htmlData{term=unit-vector}{\hat{\mathbf{r}}_{21}}

Notes

The scalar form gives the magnitude of the attraction. The vector form adds direction: the force on one mass points along the line to the other and is always attractive.

Newton published the law in the Principia (1687). It is an excellent approximation whenever masses move slowly compared with light and spacetime curvature is weak. The constant $G$ was measured later; Cavendish's 1798 torsion-balance experiment gave the first laboratory value.

Sources

Philosophiæ Naturalis Principia Mathematica — Isaac Newton (1687)
Original statement of the law.
Newton's law of universal gravitation (2026)
Modern scalar and vector notation.