Weight and Mass

In everyday talk "how heavy is it?" and "how much stuff is in it?" feel like the same question. In physics they are two different quantities, measured in two different units — and telling them apart is one of the biggest ideas in gravity.

Mass is the amount of matter — the amount of stuff — packed into an object. We measure it in kilograms (kg). A 2 kg bag of sugar contains a fixed amount of sugar whether it sits in your kitchen, rides up a mountain, or is flown to the Moon. Mass is the same everywhere.
Weight is the force of gravity pulling on that mass. Like every force it is measured in newtons (N). Weight depends on where you are, because gravity is stronger on some worlds than others. Weight changes.

Same object, two numbers: a fixed mass in kilograms, and a weight in newtons that rises and falls with the strength of gravity around it.

The weight formula

How hard does gravity pull? That depends on two things: how much stuff there is to pull (the mass m), and how strong gravity is where the object sits. The strength of gravity at a place is called the gravitational field strength, written g, and it is measured in newtons per kilogram (N/kg) — the number of newtons of pull on each kilogram of mass. Multiply the two together and you get the weight:

The weight W of an object is its mass m times the gravitational field strength g at its location:

W = m\,g.

W — weight, in newtons (N)
m — mass, in kilograms (kg)
g — gravitational field strength, in newtons per kilogram (N/kg)

The value of g is set by the world you are standing on. On Earth it is about 9.8\ \text{N/kg} (often rounded to 10\ \text{N/kg} for quick sums). Elsewhere it is different:

Place	Gravitational field strength g (N/kg)
The Moon	1.6
Mars	3.7
Earth	9.8
Jupiter	24.8

Because m never changes but g does, the very same object weighs six times less on the Moon than on Earth, and about two and a half times more on Jupiter.

Worked example: weight on Earth

A student has a mass of 50\ \text{kg}. What is their weight on Earth, where g = 9.8\ \text{N/kg}?

Step 1 — write the formula.

W = m\,g.

Step 2 — put the numbers in.

W = 50 \times 9.8.

Step 3 — work it out.

W = 490\ \text{N}.

So a 50 kg student is pulled towards the Earth with a force of 490 newtons. Notice the mass stayed in kilograms and the answer came out in newtons — a weight is always a force.

The same mass, a different world

Take that identical 50 kg student to the Moon, where g = 1.6\ \text{N/kg}. Their mass has not changed — they are still made of exactly the same stuff — but the pull is far weaker:

W_{\text{Moon}} = m\,g = 50 \times 1.6 = 80\ \text{N}.

On Jupiter, with g = 24.8\ \text{N/kg}, the same student would be crushed by a far heavier pull:

W_{\text{Jupiter}} = 50 \times 24.8 = 1240\ \text{N}.

One person, one unchanging mass of 50 kg, three completely different weights: 490 N, 80 N, 1240 N. Only g changed.

Rearranging the formula

W = m\,g ties three quantities together, so knowing any two lets you find the third. Rearrange it just like any equation (this is exactly the substitution and rearranging you meet in algebra):

m = \frac{W}{g}, \qquad\qquad g = \frac{W}{m}.

For example, if a rock weighs 60\ \text{N} on Earth (g = 10\ \text{N/kg}), its mass is m = \dfrac{60}{10} = 6\ \text{kg}. And if a 2\ \text{kg} mass weighs 7.4\ \text{N} on Mars, the field strength there is g = \dfrac{7.4}{2} = 3.7\ \text{N/kg}.

These are the traps that trip up nearly every student — check yourself against all four:

Mass is in kilograms, weight is in newtons. They are not two words for the same thing. If a question gives you kilograms it is talking about mass; newtons means weight (a force). Never write a weight in kg.
Weight changes, mass does not. Fly to the Moon and your weight drops to about a sixth, because g is smaller — but your mass, the amount of stuff in you, is exactly the same as on Earth.
"Weightless" astronauts still have mass. An astronaut drifting in the Space Station feels weightless because they are falling freely, but they are still made of just as much matter. Their mass is unchanged — bump into a wall and it still takes a real shove to get moving.
Don't confuse "heavy" with "big". A huge beach ball has little mass; a small lead ball has lots. Weight follows mass, not size.

Build the weight arrow

In the box below, the square is an object and the red arrow is its weight — the pull of gravity, drawn to a length of m \times g. Slide the mass up and the arrow grows: more stuff, more pull. Then switch the world between Earth, the Moon and Jupiter. The object never changes — same square, same mass in kilograms — but the arrow stretches and shrinks as g changes, and the computed weight in newtons follows W = m\,g.

Why kitchen scales fib about kilograms

Step on a bathroom scale and it proudly reads your "weight" in kilograms. But we just said weight is a force in newtons — so what is really going on?

The scale can only feel the force your body presses down with, which is your true weight in newtons. It then quietly divides by g to turn that force into a mass:

m = \frac{W}{g}.

The scale is built assuming Earth's g \approx 9.8\ \text{N/kg}, so the kilograms it shows are really your mass, worked out from the force it measured. Carry that same scale to the Moon and it would read about a sixth of the number — not because you had lost any stuff, but because it felt a smaller force and still divided by Earth's g. A true reading of mass would need it to divide by the Moon's g instead.

Mass versus weight, side by side

	Mass	Weight
What it is	Amount of matter (stuff)	Force of gravity on that matter
Unit	kilogram (kg)	newton (N)
Type of quantity	Not a force	A force
Changes with location?	No — same everywhere	Yes — depends on g
On the Moon	Unchanged	About 6× smaller
Measured with	A balance (compares masses)	A newtonmeter / force scale
Linked by	W = m\,g

Suppose you have a mass of 60\ \text{kg}. On Earth that is a weight of 60 \times 9.8 = 588\ \text{N} — a comfortable pull your legs handle all day. Now imagine standing on Jupiter, where g = 24.8\ \text{N/kg}:

W = 60 \times 24.8 = 1488\ \text{N},

roughly the weight of two and a half of you pressing down at once — you could barely stand. Hop across to the Moon instead and your weight collapses to 60 \times 1.6 = 96\ \text{N}, light enough to bound along in slow, floating leaps. Same you, same 60 kg of stuff, from featherlight to crushing — just by changing the world beneath your feet.