Weight on the Moon and other planets: orbits and weightlessness

Weight on the Moon

The Moon's mass and radius give $g = 1.6\ \text{m/s}^2$, about a sixth of the Earth's. So you would weigh a sixth as much there, though your mass is the same. Astronauts bound across the Moon with their heavy suits.

Other worlds

Weight is $mg$ with each world's own $g$: $3.7\ \text{m/s}^2$ on Mars, $24.8\ \text{m/s}^2$ on Jupiter. A $50\ \text{kg}$ person weighs $185\ \text{N}$ on Mars and $1240\ \text{N}$ on Jupiter.

Why does the Moon not fall on the Earth?

It does fall, all the time! Throw a stone sideways from a very high mountain: faster throws land farther away. Throw it fast enough and the ground curves away as fast as the stone falls. It keeps falling round the Earth: it is in orbit.

A satellite near the Earth needs about $7.9\ \text{km/s}$. Higher orbits need less speed, because gravity is weaker there:

You should find that higher orbits are slower and take much longer. At about $36000\ \text{km}$ an orbit takes a whole day, so such a satellite stays above one place: it is used for television.

Weightlessness

Astronauts in an orbiting station float. Gravity has not vanished; up there it is still about $90\%$ of its strength on the ground. They float because they and their station are falling together, so nothing pushes on them. We feel 'weight' from the floor pushing up.

Watch out. 'No gravity in space' is wrong. Without gravity the station would fly off in a straight line.

A worked example

Find the speed and the time per orbit of a satellite $300\ \text{km}$ up, and $2000\ \text{km}$ up.

1. 300 km. $r = 6.7 \times 10^{6}\ \text{m}$: $v = 7.73\ \text{km/s}$, one orbit in about $91$ minutes.
2. 2000 km. $r = 8.4 \times 10^{6}\ \text{m}$: $v = 6.90\ \text{km/s}$, about $127$ minutes.
3. So the higher satellite is slower and takes longer.

Practise with new numbers

Try it in Python

The program works out both orbits.

Remember these

• Weight $= mg$ with the local $g$; mass is the same everywhere.
• An orbit is falling round a planet: the ground curves away as fast as you fall.
• Weightlessness comes from falling freely, not from 'no gravity'.

Check yourself

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

1. What would a $60\ \text{kg}$ astronaut weigh on the Moon?

2. Why does the Moon not crash into the Earth?

3. Is there gravity inside the space station?

4. Which needs the faster speed: a low orbit or a high one?