The universal law of gravitation

Why does an apple fall?

An apple falls to the ground; the Moon goes round the Earth. Newton saw that one force explains both: the Earth pulls the apple and the Moon. The Moon does not crash because it moves sideways so fast that it keeps falling around the Earth.

Everything pulls everything

Not just the Earth: every object in the universe pulls every other object. You and a friend pull each other too, though far too weakly to notice.

The law

$m_1$ and $m_2$ are the masses, $r$ the distance between their centres and $G = 6.67 \times 10^{-11}\ \text{N m}^2/\text{kg}^2$, the universal gravitational constant. The two forces on the two objects are equal in size and opposite in direction.

The distance is squared

Double the distance and the force falls to a quarter; triple it and it falls to a ninth.

You should find that doubling a mass doubles the force, but doubling the distance quarters it. Gravity is huge only when a mass is huge, like a planet.

Watch out. $r$ is measured between the centres. For something on the Earth's surface, that is the Earth's radius.

A worked example

Find the pull between two students of $55\ \text{kg}$ and $65\ \text{kg}$ standing $1.5\ \text{m}$ apart, and the Earth's pull on a $2\ \text{kg}$ bag.

1. Students. With $G = 6.67 \times 10^{-11}\ \text{N m}^2/\text{kg}^2$, $F = \dfrac{G \times 55\ \text{kg} \times 65\ \text{kg}}{(1.5\ \text{m})^2} = 1.06 \times 10^{-7}\ \text{N}$: far too small to feel.
2. Bag. $F = \dfrac{G \times 6.0 \times 10^{24}\ \text{kg} \times 2\ \text{kg}}{(6.4 \times 10^{6}\ \text{m})^2} = 19.5\ \text{N}$.
3. So the Earth's enormous mass makes all the difference.

Practise with new numbers

Try it in Python

The program works out both forces.

Remember these

• $F = \dfrac{Gm_1m_2}{r^2}$, with $G = 6.67 \times 10^{-11}\ \text{N m}^2/\text{kg}^2$.
• The force grows with each mass and falls with the square of the distance.
• The same law keeps the Moon round the Earth and makes the apple fall.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. The distance between two bodies is halved. What happens to the force?

2. Both masses are doubled and the distance stays the same. What happens to the force?

3. The Earth pulls a stone down. Does the stone pull the Earth?

4. Why is $G$ called universal?