Mass and Weight

Mass and weight describe different things

A bag of flour labelled 2 kg has a mass of 2 kg. Mass is a measure of the amount of matter in an object. Weight, however, is the force acting on an object due to gravity. Like other forces, weight is measured in newtons (N), not kilograms.

Moving the same bag from Earth to the Moon would not change the amount of matter it contains, so its mass would remain 2 kg. Its weight would become smaller because the Moon's gravitational field is weaker.

Why objects have weight

A gravitational field is a region where a mass experiences a gravitational force. Earth has a gravitational field around it, so an object near Earth experiences a pull towards Earth's centre. This is the object's weight. Gravity acts without Earth needing to touch the object.

Gravitational field strength, $g$, tells us the force acting on each kilogram of mass at a particular location. Its unit is newtons per kilogram (N/kg). For example, a field strength of 10 N/kg means that each kilogram of mass experiences a gravitational force of 10 N.

The same object has a greater weight where the gravitational field is stronger and a smaller weight where it is weaker. Weight therefore depends on both the object's mass and the gravitational field strength at its location.

Calculating weight, mass and field strength

These quantities are linked by:

$$W = mg$$

Here, $W$ is weight in N, $m$ is mass in kg, and $g$ is gravitational field strength in . In calculations, the value of will be given.

Suppose a parcel has a mass of 3 kg and the gravitational field strength is 10 N/kg. Its weight is $W = 3 \times 10 = 30\text{ N}$. This means gravity pulls on the parcel with a force of 30 N.

You can rearrange the equation to find either of the other quantities:

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

For example, an object weighing 20 N in a field of 10 N/kg has a mass of $20 \div 10 = 2\text{ kg}$.

How weight changes with mass

At a constant gravitational field strength, weight is directly proportional to mass:

$$W \propto m$$

The symbol $\propto$ means 'is directly proportional to'. Doubling the mass doubles the weight; tripling the mass triples the weight. The ratio $W/m$ stays constant because it equals $g$.

The graph below shows calculated weights for a field strength of 10 N/kg.

Data for Weight is directly proportional to mass at constant g

A graph of weight against mass is a straight line through the origin. Each additional kilogram adds the same amount of weight. A stronger gravitational field would produce a steeper line because each kilogram would experience a larger force.

Measuring weight with a newtonmeter

Weight is measured using a calibrated spring balance, also called a newtonmeter. An object hanging from its hook pulls on a spring and stretches it. The calibrated scale is marked in newtons, allowing the force to be read directly.

Choose a newtonmeter whose range is suitable for the object. Check that it reads zero with nothing attached, then hold or support it vertically and hang the object securely from the hook. Wait until the object and pointer are still, then read the scale at eye level.

Do not exceed the instrument's maximum force, and keep your feet clear of the hanging object in case it falls. A steady reading gives the object's weight at that location.

Mass, weight and location

• Mass: amount of matter, measured in kg; unchanged by moving an object.
• Weight: gravitational force, measured in N.
• Earth's gravitational field pulls objects towards its centre.
• The same object weighs more in a stronger gravitational field.

Equation and units

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