Physics · 1 Motion, forces and energy

1.3 Mass and weight

Before you start

A force is a push or a pull, it is measured in newtons (N), and it has a direction.
Acceleration is the change in velocity per unit time, measured in m/s².
Converting grams to kilograms: divide by $1000$ , so $350$ g $= 0.350$ kg.
Rearranging a three-term equation such as $g = W / m$ to make $W$ or $m$ the subject.

Syllabus objectives covered (5)

State that mass is a measure of the quantity of matter in an object at rest relative to the observer
State that weight is a gravitational force on an object that has mass
Define gravitational field strength as force per unit mass; recall and use the equation g = W / m and know that this is equivalent to the acceleration of free fall
Know that weights (and masses) may be compared using a balance
Describe, and use the concept of, weight as the effect of a gravitational field on a mass

Part 1 of 6

Mass is the amount of matter, and it goes where the object goes

Start with the everyday idea of how much stuff there is in something. A full bottle of water has more stuff in it than a half-full one. Mass is the physics word for that amount. Its unit is the kilogram, kg.
When a question says State what mass is, write the syllabus statement in full: mass is a measure of the quantity of matter in an object at rest relative to the observer. 'Quantity of matter' is the part that carries the meaning. Write the whole sentence so nothing is missing.
Mass does not depend on where the object is. A $2.0$ kg bag of sugar taken to the Moon is still $2.0$ kg, because moving it does not add or remove any matter. Hold on to this: the rest of the lesson leans on it.
Mass has a size but no direction, so it is a scalar quantity.
Masses are also given in grams, with $1$ kg $= 1000$ g. The equations in this topic need kilograms, so divide a mass in grams by $1000$ before you use it: $350$ g becomes $0.350$ kg.
Everyday speech blurs mass and weight: 'I weigh $50$ kg' is really a statement about mass. In physics you keep the two words apart, and the next section shows why they cannot be swapped.

Part 2 of 6

Weight is a force, so it is measured in newtons

Weight is a gravitational force on an object that has mass. That sentence is the definition, and it is also the whole idea: an object with mass, in a gravitational field, is pulled by that field. The pull is its weight.
Because weight is a force, its unit is the newton, N, the same unit as any other push or pull. A weight written in kg is wrong, however natural it sounds.
You measure a weight with a newton meter (a spring balance). The object hangs from a spring, the spring stretches until it pulls up as hard as the weight pulls down, and the scale reads that force in newtons.
Weight has a direction as well as a size, so it is a vector quantity. Near the Earth it acts downwards, which more exactly means towards the centre of the Earth.
So mass and weight are different kinds of quantity. Mass is an amount of matter: kg, no direction. Weight is a force: N, with a direction. They are linked, and the link is the next section.

Part 3 of 6

Gravitational field strength: the force on each kilogram

A gravitational field is a region in which a mass experiences a force. The Earth has one around it, and so do the Moon and the other planets.
Some fields pull harder than others. To compare them fairly you need a measure that does not depend on which object you test with, so you ask how much force acts on each kilogram. That is gravitational field strength, defined as force per unit mass. 'Force per unit mass' is the wording to write when asked to Define it.
As an equation, with $g$ the gravitational field strength, $W$ the weight in N and $m$ the mass in kg:
$g = \frac{W}{m}$
The unit comes straight from the equation: newtons divided by kilograms, N/kg.
At the Earth's surface $g$ is close to $9.8$ N/kg, so every kilogram is pulled down with a force of $9.8$ N. The front of Paper 4 states this value: take the weight of $1.0$ kg to be $9.8$ N. Use $9.8$ , the value the paper gives you, not $10$ .
The equation itself is not given. Paper 4 has no formula sheet, so you must recall $g = W / m$ . You will use it in three arrangements. Multiply both sides by $m$ to get the weight, because $m$ is what divides $W$ . Then divide both sides of that by $g$ to get the mass:
$W = m g m = \frac{W}{g}$
The trap here is the mass unit. $g$ is force per kilogram, so the mass must be in kg. Put in $350$ instead of $0.350$ for a $350$ g object and your weight comes out $1000$ times too big.

Part 4 of 6

Why g is also the acceleration of free fall

An object is in free fall when its weight is the only force acting on it: no air resistance and nothing holding it up.
Look at the units first. One newton is the force that gives a $1$ kg mass an acceleration of $1$ m/s², so N is the same as kg m/s². Divide by kg and N/kg becomes m/s², the unit of acceleration. Same unit is the first clue that $g$ is an acceleration.
Now the physics. Double the mass of an object and you double its weight. Twice the force acting on twice the mass gives the same acceleration. So, at the same place, every object in free fall speeds up at the same rate, whatever its mass.
That shared acceleration is $g$ . Using $F = m a$ from the forces topic (1.5): the only force is the weight $W = m g$ , so
$a = \frac{W}{m} = \frac{m g}{m} = g$
So the field strength of $9.8$ N/kg at the Earth's surface and the acceleration of free fall of $9.8$ m/s² are the same number, for the same reason. The front of Paper 4 gives both side by side.
Air changes what you see, not the physics. A feather falls slowly through air because the air resistance on it is large compared with its weight, so it is not in free fall. With the air removed, a feather and a coin released together land together.

Part 5 of 6

Weight is what a gravitational field does to a mass

Put the last three sections together. The mass belongs to the object. The field strength belongs to the place. Weight is what you get when the two meet:
$W = m g$
So the same object has different weights in different places, while its mass stays fixed. The Moon's field is weaker than the Earth's: about $1.6$ N/kg at its surface. When a question needs the field strength of the Moon or another planet, use the value the question states.
Work it through for a $60$ kg student. On Earth, $W = 60 \times 9.8 = 588$ N. On the Moon, $W = 60 \times 1.6 = 96$ N. Her mass is $60$ kg in both places. Only the field changed, and so only the weight changed.
The field has a direction, so the weight it produces has one too: towards the centre of the planet. A person in Australia and a person in Canada are both pulled 'down', but those two downs point in different directions in space, because each points at the Earth's centre.
forces diagram — not drawn yet
A circle representing the Earth with a dot at its centre, and four small blocks standing on its surface at the top, bottom, left and right, each with a weight arrow labelled W pointing from the block towards the centre of the circle.
Every weight arrow points at the Earth's centre. 'Down' means towards the centre, wherever you stand.
The Earth's field also gets weaker as you move further from the Earth. So the weight of a space probe falls as it travels away, even though its mass does not change at all.
When a question asks you to Describe weight as the effect of a gravitational field, give the chain in order: the object has mass; it is in a gravitational field; the field exerts a force on the mass; that force is the weight, $W = m g$ , and its size depends on the field strength where the object is.

Part 6 of 6

Comparing weights and masses with a balance

A beam balance is a beam pivoted at its centre, with a pan hanging from each end at the same distance from the pivot. The empty beam is first set so that it is level. Then you put an object in one pan and standard masses in the other.
Each pan is pulled down by the weight of what is in it. The beam is level only when the two weights are equal, because then neither side pulls harder than the other. So the first thing a balance does is compare weights.
apparatus diagram — not drawn yet
An equal-arm beam balance with a central pivot and a level beam; a stone in the left pan and a stack of standard masses labelled 0.24 kg in the right pan, with equal-length downward arrows labelled W under each pan and the two arm lengths marked as equal.
Equal arms and a level beam mean the two weights are equal, and so the two masses are equal.
Both pans are in the same place, so the same $g$ acts on both. Since $W = m g$ , equal weights with the same $g$ means equal masses. That is why the same balance also compares masses: when the beam is level, the mass of the object equals the total of the standard masses.
Now take the level balance to the Moon. Both weights become smaller by the same factor, because the same weaker $g$ acts on both pans. They are still equal, so the beam stays level and the same standard masses still balance the object.
A newton meter behaves differently. It measures one weight against a spring, so on the Moon its reading drops. Keep the difference clear: a newton meter measures a weight; a balance compares two weights, and that comparison tells you the masses are equal wherever you use it.

Worked examples

Example 1. A tin of beans has a mass of $450$ g. Calculate its weight on Earth. Take $g = 9.8$ N/kg.

Show the working

$g$ is force per kilogram, so the mass must be in kg first. Divide by $1000$ .
$m = \frac{450}{1000} = 0.450 kg$
You want the weight, so rearrange $g = W / m$ by multiplying both sides by $m$ .
$W = m g$
Substitute and calculate.
$W = 0.450 \times 9.8 = 4.41 N$
Give the answer to a sensible number of significant figures, here $2$ to match $9.8$ , and write the unit. A weight is a force, so the unit is N, not kg.

Answer: $4.4$ N (from $4.41$ N).

Example 2. On the surface of a planet, a $12.0$ kg object hangs from a newton meter, which reads $44.4$ N. Calculate the gravitational field strength on the planet, and state the acceleration of free fall there.

Show the working

The newton meter reads a force, and the only force it is balancing is the object's weight, so $W = 44.4$ N.
Gravitational field strength is force per unit mass. Divide the weight by the mass.
$g = \frac{W}{m} = \frac{44.4}{12.0} = 3.7 N/kg$
The gravitational field strength and the acceleration of free fall are the same quantity, so the number carries straight over. Only the unit is written differently: N/kg is the same as m/s².

Answer: $g = 3.7$ N/kg; acceleration of free fall $= 3.7$ m/s².

Example 3. An astronaut has a mass of $75$ kg. Calculate her weight on Earth ( $g = 9.8$ N/kg) and on the Moon ( $g = 1.6$ N/kg), and state her mass on the Moon.

Show the working

Weight is the effect of the field on the mass, $W = m g$ . The mass stays at $75$ kg; only $g$ changes between the two places.
On Earth:
$W = 75 \times 9.8 = 735 N$
On the Moon:
$W = 75 \times 1.6 = 120 N$
Her mass on the Moon is still $75$ kg: travelling there does not change the quantity of matter in her.
Sense-check: the Moon's $g$ is about a sixth of the Earth's ( $1.6 / 9.8 \approx 0.16$ ), so her weight on the Moon should be about a sixth of $735$ N. $120$ N is.

Answer: Weight on Earth $= 735$ N; weight on the Moon $= 120$ N; mass on the Moon $= 75$ kg.

Example 4. On Earth, a rock in one pan of a beam balance is balanced by standard masses totalling $0.60$ kg. (i) State the mass of the rock. (ii) Calculate the reading on a newton meter holding the rock on Earth ( $g = 9.8$ N/kg) and on the Moon ( $g = 1.6$ N/kg). (iii) Predict what the beam balance shows on the Moon.

Show the working

(i) The beam is level, so the weights on the two pans are equal. Both pans have the same $g$ , so the masses are equal too. The rock has a mass of $0.60$ kg.
(ii) A newton meter reads the weight, $W = m g$ . On Earth:
$W = 0.60 \times 9.8 = 5.88 N \approx 5.9 N$
On the Moon, same mass, weaker field:
$W = 0.60 \times 1.6 = 0.96 N$
(iii) On the Moon both the rock's weight and the standard masses' weight are reduced by the same factor, because the same $g$ acts on both pans. The weights are still equal, so the beam stays level with the same $0.60$ kg of standard masses.

Answer: (i) $0.60$ kg. (ii) $5.9$ N on Earth (from $5.88$ N); $0.96$ N on the Moon. (iii) The beam stays balanced with the same $0.60$ kg of standard masses.

Where people go wrong

Mistake: Giving a weight in kilograms, for example 'the bag weighs $3$ kg'.

Instead: Weight is a force and is measured in newtons. The $3$ kg is the mass; the weight on Earth is $3 \times 9.8 = 29.4$ N.

Everyday speech uses 'weigh' for mass. In physics, weight is the gravitational force on the mass, so it takes the unit of force.

Mistake: Saying that an object has less mass on the Moon.

Instead: The mass is the same on the Moon. The weight is less.

Mass is the quantity of matter in the object, and moving it does not add or remove matter. Weight depends on the field strength, and the Moon's field is weaker.

Mistake: Putting a mass in grams straight into $W = m g$ , so a $350$ g object comes out with a weight of $3430$ N.

Instead: Convert to kg first: $0.350$ kg, giving $W = 0.350 \times 9.8 = 3.43$ N.

$g$ is the force per kilogram, so the mass in the equation must be in kilograms.

Mistake: Thinking a heavier object in free fall has a larger acceleration.

Instead: In free fall every object at the same place has the same acceleration, $g$ .

Twice the mass has twice the weight, so the force per kilogram is unchanged and the acceleration is unchanged. Objects fall at different rates in air only because of air resistance.

Mistake: Predicting that a beam balance needs different standard masses on the Moon.

Instead: It balances with the same standard masses. A newton meter is the instrument whose reading changes.

A balance compares two weights in the same field. A weaker $g$ reduces both weights by the same factor, so they stay equal.

Check yourself

State what mass is a measure of, and give its unit.
Answer
Mass is a measure of the quantity of matter in an object at rest relative to the observer. Its unit is the kilogram, kg.
A friend says, 'My weight is $55$ kg.' What is wrong with this, and what is her weight on Earth?
Answer
Weight is a gravitational force, so it is in newtons; $55$ kg is her mass. Her weight is $W = m g = 55 \times 9.8 = 539$ N (about $540$ N).
Define gravitational field strength, and give its unit.
Answer
Gravitational field strength is the force per unit mass, $g = W / m$ . Its unit is N/kg.
A suitcase has a weight of $196$ N on Earth. Calculate its mass.
Answer
$m = W / g = 196 / 9.8 = 20$ kg.
An apple has a mass of $200$ g. Calculate its weight on Earth.
Answer
$m = 0.200$ kg, so $W = 0.200 \times 9.8 = 1.96$ N (about $2.0$ N).
A stone is dropped near the Earth's surface and air resistance is negligible. State its acceleration, and explain why a stone of twice the mass has the same acceleration.
Answer
$9.8$ m/s². The heavier stone has twice the weight acting on twice the mass, so the force per kilogram, $g$ , is the same, and so is the acceleration.
A $0.50$ kg bag is measured on Earth and on the Moon ( $g = 1.6$ N/kg), once with a beam balance and once with a newton meter. Which instrument gives the same result in both places? What does the newton meter read in each place?
Answer
The beam balance: it balances the bag with $0.50$ kg of standard masses in both places. The newton meter reads $0.50 \times 9.8 = 4.9$ N on Earth and $0.50 \times 1.6 = 0.80$ N on the Moon.
A $30$ kg robot is on a moon where $g = 1.3$ N/kg. State its mass there and calculate its weight.
Answer
Its mass is still $30$ kg. Its weight is $W = m g = 30 \times 1.3 = 39$ N.

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1. A student has a metal block of mass $350$ g.

(a) State what is meant by the mass of an object. 1 mark No calculator

Mark scheme

B1 (a measure of) the quantity / amount of matter in the object (at rest relative to the observer); 'how heavy it is' is insufficient

(b) State what is meant by the weight of an object. 1 mark No calculator

Mark scheme

B1 the gravitational force on the object (that has mass), owtte; 'the mass of the object' is CON

(c) Calculate the weight of the block on Earth. 3 marks

Mark scheme

C1 mass converted to 0.35 kg
C1 W = mg or 0.35 × 9.8 seen
A1 3.4 N (accept 3.43 N); unit required

Question 2. A space probe lands on a planet. On the probe, a test mass of $2.0$ kg hangs from a newton meter. The newton meter reads $7.4$ N.

(a) Define gravitational field strength. 1 mark No calculator

Mark scheme

B1 force per unit mass (accept weight per unit mass, or g = W / m with each symbol identified)

(b) Calculate the gravitational field strength at the surface of the planet. 2 marks

Mark scheme

C1 g = W / m or 7.4 / 2.0 seen
A1 3.7 N/kg (accept 3.7 m/s²); unit required

(c) State the acceleration of a small object falling freely near the surface of the planet. Assume there is no air resistance. 1 mark No calculator

Mark scheme

B1 3.7 m/s² (ecf: the value from (b) with the unit m/s²)

(d) The probe has a mass of $850$ kg. Calculate the weight of the probe on the surface of the planet. 2 marks

Mark scheme

C1 W = mg or 850 × 3.7 seen (ecf from (b))
A1 3100 N (accept 3145 N or 3.1 × 10³ N); unit required

(e) Explain why the mass of the test mass on the planet is the same as on Earth, but its weight is different. 2 marks No calculator

Mark scheme

B1 mass is the quantity of matter, which does not change when the object is moved / does not depend on location, owtte
B1 weight is the effect of the gravitational field on the mass / W = mg, and g on the planet is different from (smaller than) g on Earth

Question 3. An equal-arm beam balance is used on Earth. A stone is placed in the left pan. The beam is level when standard masses with a total mass of $0.24$ kg are in the right pan.

(a) State the mass of the stone. 1 mark No calculator

Mark scheme

B1 0.24 kg; unit required

(b) Explain why a level beam shows that the stone and the standard masses have equal masses. 2 marks No calculator

Mark scheme

B1 the beam is level when the weights on the two pans are equal / the downward forces on the two sides are equal
B1 the same gravitational field strength acts on both pans, so equal weights mean equal masses (W = mg), owtte

(c) The balance, the stone and the standard masses are taken to the Moon. Predict whether the beam is still level. Give a reason for your answer. 2 marks No calculator

Mark scheme

B1 the beam is still level / still balances
B1 both weights decrease by the same factor / the same (smaller) g acts on both pans, so the weights are still equal; a statement that the mass of the stone changes is CON

(d) The gravitational field strength on the Moon is $1.6$ N/kg. Calculate the weight of the stone on the Moon. 2 marks

Mark scheme

C1 W = mg or 0.24 × 1.6 seen (ecf from (a))
A1 0.38 N (accept 0.384 N); unit required

How this is examined

Working out a weight from a mass with $W = m g$ : a two- or three-mark Calculate that gives a mass, possibly in grams, and asks for the weight The mass converted to kg; $W = m g$ or the substitution (e.g. $0.350 \times 9.8$ ) written down; the answer with the unit N. The conversion and the equation are C marks, so they can still score when the final answer is wrong.
Finding $g$ on another planet, then reading off the free-fall acceleration: a Calculate from a mass and a newton meter reading, followed by a short State of the acceleration of free fall $g = W / m$ with the numbers in; the answer in N/kg; then the same number given as the acceleration with the unit m/s². A wrong $g$ carried forward correctly still earns the acceleration mark through ecf.
Why mass stays the same but weight changes in a different place: a two-mark Explain set on the Moon or another planet One point about mass: it is the quantity of matter and does not depend on where the object is. One point about weight: it is the effect of the gravitational field on the mass, $W = m g$ , and $g$ is different there (smaller on the Moon).
Why a balance compares masses, and why it still balances on the Moon: a two-mark Explain or Predict about a beam balance moved to a place with a different $g$ 'The beam is level when the weights on the two pans are equal'; 'the same $g$ acts on both pans, so equal weights mean equal masses'; for the Moon, 'both weights decrease by the same factor, so it still balances'.
The three one-line statements: mass, weight and gravitational field strength: one-mark State or Define parts, at the start of a longer question Mass: 'quantity of matter in an object'. Weight: 'gravitational force on an object'. Gravitational field strength: 'force per unit mass'. 'How heavy something is' does not earn the mark for either mass or weight.

Words to know

Mass: A measure of the quantity of matter in an object at rest relative to the observer. Measured in kg; it does not depend on where the object is.
Weight: The gravitational force on an object that has mass. Measured in N, and directed towards the centre of the planet.
Gravitational field: A region in which a mass experiences a force.
Gravitational field strength: The force per unit mass, $g = W / m$ . Measured in N/kg; close to $9.8$ N/kg at the Earth's surface.
Acceleration of free fall: The acceleration of an object when its weight is the only force acting on it. Equal to $g$ , so $9.8$ m/s² near the Earth's surface.
Newton meter: A spring balance that measures a force, such as a weight, in newtons.
Beam balance: A beam pivoted at its centre with a pan at each end. It is level when the weights on the two pans are equal, so it compares weights and, in the same field, masses.

All Physics lessons Next: 1.4 Density