The Law of Gravitation: Quick Revision

A brief note on The Law of Gravitation. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• Every object in the universe pulls every other object. The pull acts along the line joining their centres.
• The law: $F = \dfrac{Gm_1m_2}{r^2}$, with $G = 6.67 \times 10^{-11}\ \text{N m}^2/\text{kg}^2$, the same everywhere.
• The pull grows with each mass and falls with the square of the distance between the centres.
• The pulls on the two objects are equal and opposite, even when one is the Earth and the other an apple.
• Near a planet every object falls with the same acceleration, $g = \dfrac{GM}{R^2}$. On the Earth $g = 9.8\ \text{m/s}^2$.
• $g$ gets smaller as you go higher, because you move away from the centre.

The facts at a glance

A worked example

Two lead spheres of $40\ \text{kg}$ and $25\ \text{kg}$ have their centres $0.5\ \text{m}$ apart. Find the pull between them.

1. The law. $F = \dfrac{G \times 40\ \text{kg} \times 25\ \text{kg}}{(0.5\ \text{m})^2}$, with $G = 6.67 \times 10^{-11}\ \text{N m}^2/\text{kg}^2$.
2. Work it out. The top is $6.67 \times 10^{-8}\ \text{N m}^2$; divide by $0.25\ \text{m}^2$.
3. So $F = 2.67 \times 10^{-7}\ \text{N}$, far too small to notice. Gravity is only strong when one of the masses is huge.

Practice with new numbers

A program to try

The program works out the pull between the spheres, then $g$ on the Earth.

Mistakes to avoid

• Forgetting to square the distance. Twice as far gives a quarter of the pull, not a half.
• Measuring $r$ from the surface. It runs from centre to centre: for the Earth, add its radius.
• 'The Earth pulls the apple harder than the apple pulls the Earth.' The pulls are equal; the apple moves more because its mass is tiny.
• 'Heavy objects have a bigger $g$.' $g$ depends on the planet, not on the falling object.

Check yourself

1. Two bodies attract with $35\ \text{N}$. The distance between them is halved. What is the new force?

2. Why do we not feel the pull of the person sitting next to us?

3. A planet has the Earth's radius but four times its mass. What is $g$ there?

4. Where is $g$ bigger, at sea level or on top of a mountain?

5. The Earth pulls a $0.15\ \text{kg}$ apple with $1.47\ \text{N}$. How hard does the apple pull the Earth?