Weight

Stand on a bathroom scale and the number staring back at you is not a measure of how much "stuff" you are made of — it is a measure of how hard the Earth is pulling that stuff downward, right now, at that spot. Physics gives this pull a precise name: weight. It sounds like a synonym for mass, and in everyday speech we use the words interchangeably, but in the Gravitation chapter they mean two sharply different things — one a fixed amount of matter, the other a force that changes the moment gravity changes. Get this distinction right and a whole cluster of exam questions (why astronauts float, why a spring balance and a beam balance can disagree, why your weight would be different on the Moon while your mass stays exactly the same) fall into place almost automatically.

The core insight

Weight is the gravitational force $W = mg$ pulling a body toward a planet's centre, a vector measured in newtons that changes with location, unlike mass, the constant scalar amount of matter it is made of.

The core idea

Weight is a force, not an amount of matter. By Newton's second law, a force equals mass times acceleration, and the "acceleration" gravity gives every falling object near a planet's surface is $g$ . So the force of gravity on a body of mass $m$ works out to:

W = mg

where $W$ is the weight (newton, N), $m$ is the mass of the body (kg), and $g$ is the local acceleration due to gravity ( $\approx 9.8\ \text{m/s}^2$ near Earth's surface). A few properties follow directly from weight being a force:

Unit. Weight is measured in newtons (N), the SI unit of force — never in kilograms, which is a unit of mass.
Direction. Weight is a vector: it always points straight down, toward the centre of the attracting body. Mass, by contrast, is a scalar with no direction at all.
Location dependence. Because $W$ depends on $g$ , and $g$ differs from planet to planet (and even shifts slightly across Earth's own surface — a touch weaker at the equator than at the poles, and weaker still with altitude), the weight of the same object changes as it moves, even though its mass never does.
How it is measured. A spring balance measures weight directly: the spring stretches in proportion to the downward pull on it, so its reading depends on $g$ at that location. A beam (or pan) balance instead compares an unknown mass against known masses on the other pan — since both pans feel the same $g$ , it reads mass correctly and gives the same answer anywhere.

Figure

A block hangs from a spring balance on Earth showing a long stretch and full-length weight arrow, while the identical block on the Moon shows a shorter stretch and shorter arrow because g is smaller although the mass label stays the same on both sides

The table below sums up the contrast that Gravitation questions love to test:

	Mass	Weight
What it measures	Amount of matter	Gravitational force on that matter
SI unit	kilogram (kg)	newton (N)
Type of quantity	Scalar	Vector, directed downward
Changes with location?	No	Yes, depends on local $g$
Instrument	Beam balance	Spring balance

An object can never have zero mass, but it can have (approximately) zero weight — an astronaut in continuous free fall in orbit, or an object drifting far from any large mass, feels a vanishingly small net gravitational pull, a state loosely called weightlessness.

Derivation AWeight as the force of gravity, W = mg

The goal is to show, starting only from Newton's second law and the definition of $g$ as the acceleration every freely falling body near a planet's surface shares, exactly why the weight of a body must equal the product of its mass and that local acceleration due to gravity, and why the resulting quantity is a force measured in newtons rather than a second kind of mass.

Step 1. Newton's second law states that any force acting on a body produces an acceleration related by $F = ma$ , where $F$ is the net force, $m$ is the mass, and $a$ is the acceleration produced.

Step 2. Drop a body near Earth's surface (or hold it and imagine releasing it) with no other force acting on it besides gravity. It accelerates downward at a fixed rate that experiment shows is the same for every object regardless of its mass; this rate is called the acceleration due to gravity, $g \approx 9.8\ \text{m/s}^2$ near Earth's surface.

Step 3. Since gravity is the only force producing this acceleration, Newton's second law identifies the force of gravity on the body, which we call its weight $W$ , as $W = ma$ with $a = g$ , giving $W = mg$ .

Step 4. Because $W$ is a force, it inherits force's unit, the newton, and force's vector nature, always directed toward the centre of the attracting body. Since $g$ is fixed by the planet and location, not by the object, doubling $m$ doubles $W$ in direct proportion, while changing location (and hence $g$ ) changes $W$ without touching $m$ at all — the two quantities are linked by $g$ but are not the same thing.

Worked example

A 60 kg student stands on a spring balance on Earth's surface, where $g = 9.8\ \text{m/s}^2$ . Find the student's weight. Then, a different spring balance shows a reading of 98 N for a block hanging from it; find the block's mass, and find what the same spring balance would read for that block if it were moved to a spot where $g = 9.75\ \text{m/s}^2$ instead.

Step 1. Weight of the student: apply $W = mg$ directly with $m = 60\ \text{kg}$ and $g = 9.8\ \text{m/s}^2$ :

W = 60 \times 9.8 = 588\ \text{N}

The student's weight is 588 N, while the mass stays 60 kg — the two numbers are different because they represent different physical quantities in different units.

Step 2. Mass of the block: the spring balance reading is the weight, so $W = 98\ \text{N}$ . Rearranging $W = mg$ to solve for $m$ , using $g = 9.8\ \text{m/s}^2$ :

m = \frac{W}{g} = \frac{98}{9.8} = 10\ \text{kg}

Step 3. New reading at $g = 9.75\ \text{m/s}^2$ : the mass of the block does not change — it is still 10 kg — but the weight does, because $g$ changed:

W' = mg' = 10 \times 9.75 = 97.5\ \text{N}

The student weighs 588 N on Earth, the block has a mass of 10 kg, and its weight at the second location is 97.5 N — slightly less than before, purely because $g$ is slightly smaller there, even though not a single gram of the block has changed.

Competitive stretch

If the same 10 kg block were carried to a location where $g = 9.8\ \text{m/s}^2$ exactly (the standard value used to calibrate most school spring balances), it would again read 98 N — showing that a spring balance's reading only matches a beam balance's mass reading when $g$ happens to equal the value it was calibrated against.

Common traps

Calling weight "kilograms." Saying "my weight is 60 kg" mixes up mass and weight; weight must be quoted in newtons, and 60 kg of mass corresponds to a weight of about 588 N on Earth.
Forgetting weight is a vector. Weight always points straight down, toward the planet's centre; this direction matters whenever weight is combined with other forces (like a push or the normal reaction) in a problem.
Assuming a spring balance always reads exactly the mass number. It only coincides with the beam-balance (mass) reading when $g$ equals the value the spring balance was calibrated for; move it to a place with different $g$ , and the reading changes even though the mass has not.
Thinking "weightless" means "no gravity." An astronaut orbiting Earth is still deep inside Earth's gravitational field — that pull is exactly what keeps the spacecraft in orbit. The floating sensation comes from being in continuous free fall together with the spacecraft, not from gravity switching off.

Ask the tutor: Want to see exactly how much your own weight (in newtons) would become on the Moon or another planet, using your unchanging mass, worked out step by step?

If weight changes with location, why does my bathroom scale always show almost the same number every day at home?

Because you are not actually changing location in any way that matters — $g$ at a fixed spot on Earth's surface is essentially constant from one day to the next. The variation of $g$ across Earth (equator to pole, sea level to a mountain) is small, well under half a per cent, so your weight barely shifts even though it is technically not perfectly fixed. To see a genuinely large change in weight, you would need to travel to another world entirely, like the Moon, where $g$ is roughly one-sixth of Earth's value.

Why does a spring balance measure weight while a beam balance measures mass?

A spring balance works by letting gravity stretch a spring; the amount of stretch is proportional to the downward force pulling on the pan, which is weight, so its reading changes if $g$ changes. A beam balance instead puts the unknown object on one pan and known masses on the other, and adjusts until the beam balances. Since gravity pulls equally (same $g$ ) on both pans at once, the balance point depends only on comparing masses, not on the actual value of $g$ — so it reads the same mass anywhere, on Earth, the Moon, or in principle anywhere else.

Does an object in deep space, far from every planet and star, have zero weight and zero mass?

It has essentially zero weight, but its mass stays exactly the same as it always was. Weight is a force that depends on nearby gravity, so far from any large body there is almost nothing left to pull on it, and $W = mg$ gives a value close to zero because $g$ there is close to zero. Mass, on the other hand, is simply the amount of matter in the object, which does not depend on gravity at all — the object still has inertia and would still resist being pushed, exactly as it would on Earth.

Weight

The core idea

Worked example

Competitive stretch

Common traps

Before this

Where this leads