Weight, Mass and Gravity

An astronaut who weighs about the same as a large dog on Earth would weigh no more than a small cat on the Moon — yet they are the very same person, made of exactly the same amount of stuff. How can their "weight" change so dramatically when nothing about them has changed? The answer is that weight and mass are two different quantities that everyday speech happily muddles together. Mass measures the amount of matter and never changes; weight is the force of gravity on that mass and depends on where you are. This lesson, part of Topic P2 (Forces) of OCR Gateway Combined Science A, separates mass from weight, works through the equation $W = mg$ , explains why weight changes between planets, and introduces the centre of mass.

By the end of this lesson you should be able to distinguish mass from weight, use and rearrange the equation $W = mg$ with $g = 9.8\text{ N/kg}$ , explain why weight changes with gravitational field strength, describe how weight is measured, and define the centre of mass.

This lesson is AO1 for distinguishing mass from weight and defining centre of mass, strongly AO2 for applying and rearranging $W = mg$ , and AO3 when you analyse why the same object has a different weight on the Moon and explain how that follows from a change in gravitational field strength.

Mass and Weight Are Different

Mass is the amount of matter in an object. It is measured in kilograms (kg) and is a scalar. Crucially, the mass of an object is constant — it does not change if you take the object to the Moon, into space or to the bottom of the sea. A $2 \text{ kg}$ bag of sugar contains the same amount of matter wherever it is.

Weight is the force of gravity acting on an object's mass. It is measured in newtons (N) and is a vector (it always acts downward, toward the centre of the planet). Unlike mass, weight depends on the gravitational field strength where the object is — so the same object weighs less on the Moon than on Earth, because the Moon's gravity is weaker.

	Mass	Weight
What it is	the amount of matter	the force of gravity on the mass
Unit	kilograms (kg)	newtons (N)
Scalar or vector?	scalar	vector (acts downward)
Does it change with location?	no — always the same	yes — depends on gravity
Measured with	a balance (mass balance)	a newtonmeter (force meter)

Exam Tip: A very common misconception is that mass and weight are the same thing. Mass is the amount of matter in kg and never changes; weight is the force of gravity in N and changes with gravitational field strength. Saying you "weigh $70 \text{ kg}$ " is everyday shorthand — strictly, $70 \text{ kg}$ is your mass, and your weight is the force in newtons.

The Weight Equation

The weight of an object depends on its mass and on the gravitational field strength:

$W = mg$

where $W$ is the weight (in $\text{N}$ ), $m$ is the mass (in $\text{kg}$ ) and $g$ is the gravitational field strength (in $\text{N/kg}$ ). On the surface of the Earth:

$g = 9.8\text{ N/kg}$

This means every kilogram of mass is pulled down with a force of $9.8 \text{ N}$ . The equation rearranges to make mass or gravitational field strength the subject:

$W = mg \qquad m = \dfrac{W}{g} \qquad g = \dfrac{W}{m}$

Worked Example 1 — Weight on Earth

Calculate the weight of a $60 \text{ kg}$ student on Earth. ( $g = 9.8\text{ N/kg}$ .)

Step 1 — write the equation: $W = mg$ .

Step 2 — substitute: $W = 60 \times 9.8$ .

Step 3 — calculate: $W = 588 \text{ N}$ .

Answer: the student's weight on Earth is $588 \text{ N}$ .

Worked Example 2 — Weight on the Moon

The same $60 \text{ kg}$ student visits the Moon, where $g = 1.6\text{ N/kg}$ . Calculate their weight there.

Step 1 — write the equation: $W = mg$ .

Step 2 — substitute the Moon's gravitational field strength: $W = 60 \times 1.6$ .

Step 3 — calculate: $W = 96 \text{ N}$ .

Answer: on the Moon the student weighs only $96 \text{ N}$ — far less than the $588 \text{ N}$ on Earth — even though their mass is still $60 \text{ kg}$ , because the Moon's gravity is much weaker.

Worked Example 3 — Finding mass from weight

An object weighs $49 \text{ N}$ on Earth. Calculate its mass. ( $g = 9.8\text{ N/kg}$ .)

Step 1 — rearrange for mass: $m = \dfrac{W}{g}$ .

Step 2 — substitute: $m = \dfrac{49}{9.8}$ .

Step 3 — calculate: $m = 5 \text{ kg}$ .

Answer: the mass is $5 \text{ kg}$ — and this would be the same on the Moon or anywhere else.

Worked Example 3b — Comparing weight on two planets

A rover has a mass of $150 \text{ kg}$ . Calculate its weight (a) on Earth ( $g = 9.8\text{ N/kg}$ ) and (b) on Mars ( $g = 3.7\text{ N/kg}$ ), and state what fraction of its Earth weight it has on Mars.

Step 1 (a) — write the equation and substitute for Earth: $W = mg = 150 \times 9.8$ .

Step 2 (a) — calculate: $W = 1470 \text{ N}$ on Earth.

Step 3 (b) — substitute the Mars value: $W = mg = 150 \times 3.7$ .

Step 4 (b) — calculate: $W = 555 \text{ N}$ on Mars.

Step 5 — compare: $\dfrac{555}{1470} \approx 0.38$ , so the rover weighs about 0.38 (just over a third) of its Earth weight on Mars.

Answer: the rover weighs $1470 \text{ N}$ on Earth and $555 \text{ N}$ on Mars — about a third as much — even though its mass stays $150 \text{ kg}$ in both places. Note that the ratio of the two weights ( $555 : 1470$ ) is exactly the ratio of the two gravitational field strengths ( $3.7 : 9.8$ ), because the mass cancels: $\dfrac{W_{\text{Mars}}}{W_{\text{Earth}}} = \dfrac{mg_{\text{Mars}}}{mg_{\text{Earth}}} = \dfrac{g_{\text{Mars}}}{g_{\text{Earth}}}$ . This is a neat shortcut — if you know how the gravitational field strengths compare, you know at once how the weights compare.

Exam Tip: Use $g = 9.8\text{ N/kg}$ on Earth unless told otherwise (some questions use $10 \text{ N/kg}$ to simplify). To find mass from weight, rearrange to $m = \dfrac{W}{g}$ . Keep the mass in kg so the weight comes out in newtons. To compare an object's weight on two planets, you can simply compare their gravitational field strengths — the mass is the same in both, so it cancels out.

Why Weight Changes but Mass Does Not

Because weight is the force of gravity on an object ( $W = mg$ ), it depends on the gravitational field strength $g$ at the object's location. Different planets and moons have different gravitational field strengths, so the same object has different weights in different places — while its mass (amount of matter) stays exactly the same.

Location	Gravitational field strength $g$ / $\text{N/kg}$	Weight of a $10 \text{ kg}$ object / $\text{N}$
Earth	$9.8$	$98$
Moon	$1.6$	$16$
Mars	$3.7$	$37$
Jupiter	$24.8$	$248$

The Moon's gravitational field strength is only about a sixth of the Earth's, which is why astronauts on the Moon could leap so high and bound across the surface — their weight was a sixth of its Earth value, while their mass (and so their inertia) was unchanged. On Jupiter, with much stronger gravity, the same object would weigh far more.

Exam Tip: The reason an object's weight differs between Earth and the Moon is the different gravitational field strength: $W = mg$ , and $g$ is smaller on the Moon. The mass does not change — only $g$ (and so the weight) does.

Weight, Mass and Gravity

Weight, Mass and Gravity

Mass and Weight Are Different

The Weight Equation

Worked Example 1 — Weight on Earth

Worked Example 2 — Weight on the Moon

Worked Example 3 — Finding mass from weight

Worked Example 3b — Comparing weight on two planets

Why Weight Changes but Mass Does Not

Weight and Mass Are Directly Proportional

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