Universal law of gravitation

Newton compared the Moon with a falling apple. The Moon (orbit radius 3.84 × 10⁸ m, period 27.3 days) has centripetal acceleration a_m = 4π²R_m/T² ≈ 2.7 × 10⁻³ m/s², far less than g.

If gravity falls off as 1/r², then g/a_m = (R_m/R_E)² ≈ 60² = 3600, and 9.8/0.00272 is indeed about 3600. The same force explains the apple and the Moon.

Law: every pair of bodies pulls on each other. The pull grows with the product of the two masses and falls as 1/(distance)²: F = G m₁ m₂ / r².

In vector form the force on m₂ is F = −G m₁ m₂ r̂ / r², with r̂ pointing from m₁ to m₂; the minus sign means attraction. The force on m₁ is −F, so F₁₂ = −F₂₁ (Newton's third law).

Superposition: the force on one mass from several others is the vector sum of the separate pair forces, each unchanged by the presence of the rest.

Three equal masses at the corners of an equilateral triangle exert zero net force on a mass at the centroid, as symmetry predicts. If the mass at one corner is doubled, the net force points towards that corner.

Shell results: a uniform spherical shell attracts a point outside it as if all its mass sat at its centre; on a point anywhere inside it, the shell's net pull is zero.

The law is for point masses; for an extended body the forces from all its parts are added as vectors. Gravity cannot be shielded: a shell does not block the pull of bodies outside it.