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Why does an astronaut weigh less on the Moon?
Short answer
Because the Moon has far less mass and a smaller radius than Earth: its surface gravity is 1.62 m/s², versus 9.81 m/s² on Earth. An 80 kg astronaut still has 80 kg of mass, but their weight drops from about 785 N to about 130 N, roughly a sixth. Mass doesn't change; what changes is the force with which the ground pulls on them.

Mass and weight aren't the same thing
Mass measures the amount of matter in a body and is the same anywhere in the universe. Weight, on the other hand, is a force: that mass multiplied by the gravitational acceleration of the body one is standing on. On Earth that acceleration is 9.81 m/s²; on the lunar surface, 1.62 m/s². That's why an astronaut with 80 kg of mass weighs about 785 N here and only about 130 N there, roughly a sixth.
A bathroom scale measures force, not matter, which is why it "lies" when you change worlds. A balance scale, which compares masses, would give the same result in both places.
Why the Moon pulls so little
The gravitational pull of a spherical body depends on its mass and the square of the distance to its center. The Moon has about 1.2% of Earth's mass and a radius of about 1,737 km, just over a quarter of Earth's. The smaller radius works in favor of lunar gravity, but it doesn't come close to offsetting the mass deficit, and the net result is that scant sixth.
What that meant for the Apollo program's crews
With pressurized suit and life-support backpack, the combined gear exceeded 80 kg on Earth and was impractical; on the Moon it was equivalent to carrying just over 13 kg. Mission footage shows the consequence: astronauts abandoned normal walking for a bounding lope, because a minimal push was enough to leave the regolith and the fall time was long. Inertia, however, remained that of the full mass, so slowing down or turning took as much effort as on Earth. That split between light weight and unchanged inertia was one of the mobility lessons most repeated in crew reports.
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