How To Calculate A Gravitational Force

Newton's law of gravitation says that any two masses attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. Multiply the gravitational constant by both masses, divide by the distance squared, and you have the force in newtons.

The constant is very small, about 6.6743 times 10 to the minus 11, which is why gravity feels weak until enormous amounts of mass are involved. Two people standing a metre apart attract each other, but the force is a few nanonewtons and there is no chance of noticing it. The reason gravity dominates the solar system is that the masses involved are astronomical in a literal sense.

The inverse square law is the part with the widest consequences. Double the distance and the force falls to a quarter. Triple it and the force falls to a ninth. This is the same geometric relationship that governs the falloff of light and sound from a point source, because in each case the effect spreads over a sphere whose area grows with the square of the radius.

Newton's third law means the two forces are always equal and opposite. The Earth pulls on the Moon with exactly the force with which the Moon pulls on the Earth. The difference in what happens next comes entirely from the masses: the Earth's acceleration is about 81 times smaller than the Moon's, which is why the Moon does most of the visible orbiting.

The centre-to-centre distance is a genuine requirement, not a rounding detail. For two spheres the gravitational field outside them is exactly the same as if all the mass sat at the centre, which is why the point-mass formula works for planets. But the distance must be measured between centres, so a satellite's altitude above the surface has to be added to the planet's radius before it goes into the formula.

The reduced mass is a useful by-product when the two bodies are of comparable size. It is the product divided by the sum, and it is the single mass that describes the relative motion of the pair in the two-body problem. For Earth and the Moon it comes out very close to the Moon's own mass, which is the quantitative statement that the Earth barely moves.

The separation energy is the other figure worth keeping. It is the force times the distance, and it tells you the energy you would have to add to pull the pair completely apart. For Earth and the Moon that is about 7.6 times 10 to the 28 joules, which is the number that makes clear how tightly bound even a seemingly empty region of space actually is.

The same equation that lets you weigh a planet also tells you why tides happen. The gravitational field is not uniform across an extended body, so the near side of the Earth is pulled more strongly than the far side, and the difference between the two produces two bulges. That differential force, not the absolute one, is what raises and lowers the oceans.

The same formula that lets you weigh a planet also lets you check the inverse square law. Measuring the field at several altitudes and comparing the values is one of the classic laboratory exercises in physics, because the prediction is exact and the deviation from it is hard to measure with ordinary equipment.