Newton’s Law of Gravitation

Newton's law of gravitation — every point mass attracts every other point mass with a force proportional to the product of their masses and inversely proportional to the square of their separation.

F = G M m / r²

where G is the gravitational constant, 6.67 × 10⁻¹¹ N m² kg⁻².

What r means — the distance between the centres of the two masses, not the gap between their surfaces. For an object on the Earth's surface, r is the Earth's radius, about 6.37 × 10⁶ m, not zero.

Spherical bodies behave as point masses — a uniform sphere attracts external objects exactly as though all its mass were concentrated at its centre. This is why the law, stated for point masses, can be applied to planets and stars at all.

Always attractive — unlike the electrostatic force, gravity has only one sign. There is no negative mass and no gravitational repulsion.

An inverse square law — tripling the separation reduces the force to a ninth. The force never reaches zero at any finite distance, though it becomes negligibly small.

Example: gravity is extraordinarily weak. The gravitational attraction between two people standing a metre apart is around 10⁻⁷ N — utterly imperceptible. It dominates on astronomical scales only because masses there are enormous and, unlike charge, mass cannot cancel out.

Newton's third law applies — the Earth pulls on you with exactly the force you pull on the Earth. The forces are equal; the accelerations are not, because a = F/m and the masses differ by a factor of about 10²³.

Why G is hard to measure — it is the least precisely known of the fundamental constants, because gravitational forces between laboratory-sized masses are so tiny that they are swamped by almost any other influence. Cavendish first measured it with a torsion balance delicate enough to detect the pull between lead spheres.