Newton's Law of Universal Gravitation

Newton's law generalises terrestrial gravity to the cosmos: any two point masses attract along the line joining them with a force proportional to the product of masses and inversely proportional to the square of the distance.

Concept

$\vec F_{12} = -\frac{Gm_1 m_2}{r^2}\hat r_{12}$ where $G = 6.674\times 10^{-11}\,\mathrm{N\,m^2/kg^2}$.

For an extended spherically symmetric body, the field outside equals that of a point mass at the centre (shell theorem). Inside a uniform shell, the field is zero.

Derivation

$G$ was measured by Cavendish (1798) using a torsion balance with two pairs of masses.

Worked Example

Force between Earth ($5.97\times 10^{24}\,\mathrm{kg}$) and Moon ($7.35\times 10^{22}\,\mathrm{kg}$) at $r=3.84\times 10^8\,\mathrm{m}$: $F\approx 1.98\times 10^{20}\,\mathrm{N}$.

Common Confusions

• Force is mutual: same magnitude on each body (Newton's 3rd law).
• $r$ is the separation between centres, not surfaces.
• Superposition: net force = vector sum of pairwise forces.

Key Takeaways

• $F \propto 1/r^2$ — inverse square.
• Universal G is the same constant for all matter.
• Shell theorem simplifies spherical bodies to point masses.