Gravity and weight

A 70 kg person weighs about 687 N on Earth, 260 N on Mars and 113 N on the Moon, with the same mass everywhere.

Mass and weight are different things

In everyday speech "weight" and "mass" are used as if they were the same, but physics keeps them apart. Mass, measured in kilograms, is how much matter an object has and how strongly it resists changes in motion. It is the same on Earth, on the Moon and in space. Weight is the force of gravity on that mass, measured in newtons: W = m g. A 70 kg astronaut has a weight of 70 × 9.81 ≈ 687 N on Earth, 70 × 1.62 ≈ 113 N on the Moon and 70 × 3.71 ≈ 260 N on Mars. Bathroom scales actually measure force and then divide by 9.8 to show kilograms, so on the Moon the same scale would read about 13 kg for an 80 kg person.

Universal gravitation

Newton’s great step in the Principia was to say that the force pulling an apple to the ground is the same force that holds the Moon in orbit. Every mass attracts every other mass with a force F = G m₁ m₂ ÷ r², where r is the distance between their centres and G is a very small constant, about 6.674 × 10⁻¹¹ N·m²/kg² in the NIST list. Because G is so small, the pull between two people is far too weak to notice. It takes a planet-sized mass to produce the weight we feel.

Divide that force by your own mass and you get the surface gravity of a planet: g = G M ÷ R². NASA’s fact sheet gives Earth’s G M as 3.986 × 10¹⁴ m³/s² and its mean radius as 6,371 km, which gives g ≈ 9.82 m/s². The same calculation with the Moon’s numbers gives 1.62 m/s², matching the measured value. Mars comes out between NASA’s equator value of 3.69 m/s² and its pole value of 3.73 m/s²; this course uses 3.71.

Why g changes from place to place

The r² in the formula means gravity weakens with distance from the centre of the planet. Earth is slightly flattened and it spins, so a person at the equator is a little further from the centre and is also being carried around in a big circle. Both effects lower the felt gravity: NASA lists about 9.78 m/s² at the equator and 9.83 m/s² at the poles. For everyday problems we use 9.81 m/s², close to the agreed standard value of 9.80665 m/s².

A few hundred kilometres up, where space stations orbit, gravity is still strong. At 400 km above the surface the formula gives about 8.7 m/s², nearly 90% of the surface value. Astronauts float not because gravity is missing but because they and their station are falling around the Earth together, in permanent free fall.

Weaker gravity changes more than the number on a scale. Suppose you can jump 0.5 m high on Earth. If your legs gave you the same take-off speed on the Moon, the height would grow by the ratio of the two values of g: 0.5 × 9.81 ÷ 1.62 ≈ 3.0 m. On Mars it would be about 1.3 m. Real spacesuits are heavy and stiff, so astronauts never jumped that high, but the calculation shows why everything on the Moon seems to happen in slow motion: falls take longer and throws go further, just as the motion lab shows.