Newton's law of gravitation

Every body in the universe pulls every other body. When one of them is the earth we call it gravity; in general it is gravitation. Newton's law of gravitation:

• F = Gm₁m₂/r²: proportional to the product of the masses, and to 1 over the square of the distance between them.
• The force acts along the line joining the two bodies, and it is always attractive.
• G = 6.67 × 10⁻¹¹ N m² kg⁻² is the same everywhere. It was first measured by Cavendish, with a delicate torsion balance.

G is tiny, so everyday masses hardly notice each other: two 100 kg spheres 1 m apart attract with only 6.67 × 10⁻⁷ N. Only when one body is as huge as a planet does the force become large.

The pulls come in equal and opposite pairs (third law). The earth pulls a 100 kg man down with about 1000 N, and he pulls the earth up with the same 1000 N. The earth is so massive that its acceleration is far too small to notice.

Newton tested the law on the moon. The moon is about 60 earth-radii away, so the earth's pull there should be 1/60² = 1/3600 of g. That is exactly the acceleration the moon needs to go round its orbit.

With several masses, find each pull separately, then add them as vectors (the principle of superposition). A third body does not change the pull between the first two. Example: 10 kg and 1000 kg sit 1 m apart. A small mass between them feels no net pull where G × 10/x² = G × 1000/(1 – x)², which gives x = 1/11 m from the 10 kg.

Two useful results for a uniform spherical shell:

• It pulls a mass outside as if all its mass were at its centre. That is why we may measure r from the centres of the earth and the moon.
• It exerts no net force on a mass anywhere inside it.

Gravity cannot be shielded: no screen blocks it. Its tidal effect (the difference in pull across a body) falls off as 1/r³, so the nearby moon raises bigger tides than the far heavier Sun.