Newton's law of gravitation

One law for the apple and the Moon

Newton asked whether the force that pulls an apple down could be the same force that holds the Moon on its path. The Moon is $60$ Earth radii away; if the pull falls as $\dfrac{1}{r^2}$, it should be $3600$ times weaker there. The Moon's centripetal acceleration, $\dfrac{v^2}{r}$, is indeed about $\dfrac{9.8\ \text{m/s}^2}{3600}$. One law explains both.

The law of gravitation

Any two bodies attract each other with a force

along the line between them, where $r$ is the distance between their centres and $G = 6.67 \times 10^{-11}\ \text{N m}^2/\text{kg}^2$ is the same everywhere in the universe.

• The two pulls are equal and opposite (a third-law pair), even if one body is far heavier.
• A uniform sphere, or a spherical shell, pulls on outside bodies as if all its mass were at its centre.
• For several bodies, the pulls add as vectors.

Explore the Earth and the Moon.

You should find that doubling the distance quarters the force, while doubling a mass only doubles it.

Watch out. $r$ is measured between centres. For a satellite, it is the Earth's radius plus the height.

A worked example

A $2000\ \text{kg}$ communications satellite is $12\,800\ \text{km}$ from the Earth's centre, twice the Earth's radius.

1. Shortcut. At the surface the pull would be $mg = 19\,600\ \text{N}$.
2. Inverse square. At twice the distance it is a quarter: $4900\ \text{N}$.
3. Check with $G$. With $G = 6.67 \times 10^{-11}\ \text{N m}^2/\text{kg}^2$, $\dfrac{G \times 6.0 \times 10^{24}\ \text{kg} \times 2000\ \text{kg}}{(1.28 \times 10^{7}\ \text{m})^2} = 4890\ \text{N}$.

Think first. Near the Earth, $GM = gR^2$ saves you from using $G$ and $M$ at all.

Practise with new numbers

Try it in Python

The program finds the Earth's pull on a $1000\ \text{kg}$ probe at several distances.

Remember these

• $F = \dfrac{Gm_1m_2}{r^2}$, $G = 6.67 \times 10^{-11}\ \text{N m}^2/\text{kg}^2$, $r$ between centres.
• Equal and opposite pulls on the two bodies.
• Near the Earth, $GM = gR^2$.

Check yourself

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

1. The Sun pulls the Earth with about $3.5 \times 10^{22}\ \text{N}$. How hard does the Earth pull the Sun?

2. Why do you not feel the pull of a friend sitting next to you?

3. If the distance between two bodies halves, what happens to the force?

4. Why does a uniform ball pull like a point mass at its centre?