Weight, Mass and Gravity

In everyday speech "weight" and "mass" are used interchangeably, but in physics they are two different quantities with different units.

Mass is a measure of the amount of matter in an object. It is a scalar measured in kilograms (kg).

Weight is the force acting on an object due to gravity. It is a vector measured in newtons (N).

Mass depends only on how much matter an object contains, so it does not change if you move the object somewhere else. A 2 kg bag of flour has a mass of 2 kg on Earth, on the Moon, or floating in deep space.

Weight is different. Weight is a gravitational force pulling the object down, so it depends on the strength of gravity where the object is. The same bag of flour weighs less on the Moon than on Earth because the Moon's gravity is weaker.

Quantity	Symbol	Unit	Type	Changes with location?
Mass	m	kilogram (kg)	scalar	No
Weight	W	newton (N)	vector	Yes

Weight is caused by a gravitational field, the region around a mass (like a planet) where another mass feels a force.

Gravitational field strength (g) is the force of gravity acting on each kilogram of mass. It is measured in newtons per kilogram (N/kg).

The bigger the value of g, the stronger the pull on each kilogram, and so the greater the weight. The value of g depends on the planet or moon and is not something you memorise for the calculation; it will be given in the question.

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

Earth's gravitational field strength is roughly 9.8 N/kg. This means every kilogram of mass on Earth is pulled down with a force of about 9.8 N. Some questions round g to 10 N/kg to keep the arithmetic simple; always use the value stated in the question.

Weight, mass and gravitational field strength are linked by one equation.

$W=mg$

where $W$ is weight in newtons (N), $m$ is mass in kilograms (kg) and $g$ is gravitational field strength in N/kg.

You must recall and apply this equation. The value of $g$ will be given in the question.

Worked example — a schoolbag has a mass of 4.0 kg. Calculate its weight on Earth (g = 9.8 N/kg).

$W=mg=4.0\times 9.8=39.2\text{N}$

The bag weighs 39.2 N.

Worked example — the same 4.0 kg bag is taken to the Moon, where g = 1.6 N/kg. Calculate its weight there.

$W=mg=4.0\times 1.6=6.4\text{N}$

The mass is still 4.0 kg, but the weight has dropped to 6.4 N because the Moon's gravitational field is much weaker.

Because W, m and g are linked, if you know any two you can find the third. Rearranging $W=mg$ for mass gives:

$m=\frac{W}{g}$

Worked example — an object weighs 60 N on Earth (g = 9.8 N/kg). Calculate its mass.

$m=\frac{W}{g}=\frac{60}{9.8}=6.1\text{kg (2 s.f.)}$

The object has a mass of about 6.1 kg.

You can also find g if you measure both weight and mass:

$g=\frac{W}{m}$

Getting the units right is the marker's first check. Weight must be in newtons, mass in kilograms, and g in N/kg. If a mass is given in grams, convert to kilograms first (divide by 1000).

Look again at $W=mg$. At a fixed location, g is constant, so weight depends only on mass.

Weight is directly proportional to mass ($W\propto m$) at a given location. Double the mass and you double the weight.

This is why a heavier object is harder to lift: it has more mass, so gravity pulls on it with a greater force.

Mass (kg)	Weight on Earth (N), g = 9.8
1	9.8
2	19.6
5	49.0
10	98.0

Each time the mass doubles, the weight doubles too. Plotting weight against mass would give a straight line through the origin, the signature of direct proportionality. The gradient of that line is the gravitational field strength, g.

Weight does not act evenly all over an object; for calculations it is treated as acting at one point.

The centre of mass is the single point at which the whole weight of an object can be considered to act.

For a symmetrical object of uniform density, such as a ruler or a ball, the centre of mass is at its geometric centre. On a free body diagram, the weight arrow is always drawn starting from the centre of mass and pointing straight down.

Weight is a force, so it is measured in newtons using a force meter, not a mass balance.

Weight is measured using a calibrated spring-balance called a newtonmeter.

A newtonmeter contains a spring that stretches more as the pulling force increases. Because the stretch is proportional to the force (up to a limit), the scale can be marked directly in newtons. A hanging object stretches the spring by an amount that reads off its weight.

1. Confusing the units of mass and weight

Mass is in kilograms (kg); weight is in newtons (N). Writing a weight in kilograms, or a mass in newtons, loses marks. Check the unit matches the quantity.

2. Saying weight and mass are the same thing

They are different quantities. Mass is the amount of matter (unchanged everywhere); weight is a gravitational force that changes if g changes. An astronaut has the same mass on the Moon but a smaller weight.

3. Forgetting to convert grams to kilograms

$W=mg$ needs mass in kilograms. A mass given as 500 g must become 0.5 kg before you multiply by g, or the answer is out by a factor of 1000.

4. Using the wrong value of g

Use the value of g stated in the question. Do not assume 9.8 N/kg if the question is set on the Moon or Mars, and do not switch between 9.8 and 10 partway through a calculation.

5. Thinking weight acts everywhere on the object

For diagrams and calculations, weight is taken to act at the single point called the centre of mass, and the weight arrow points vertically downwards from there.