Mass and weight: the difference that matters

By the end you can explain the difference between mass and weight, use $W = mg$ with $g = 10\ \text{m s}^{-2}$ , and include weight correctly in force diagrams and $F = ma$ problems.

Mass is the amount of matter in an object. It's a scalar (it has a size but no direction), measured in kilograms (kg), and it does not change when you move the object — a 70 kg person has mass 70 kg on Earth, on the Moon, and in deep space.

Weight is the force with which gravity pulls on that mass. It's a vector (it has a size and a direction), measured in newtons (N), and it does depend on where you are. On Earth it's about 700 N; on the Moon it's about 113 N; in deep space it's nearly zero.

The link is simple:

$W = mg$

Symbol	Meaning
$W$	weight, a force, in newtons (N)
$m$	mass, in kilograms (kg)
$g$	the acceleration of a freely falling object, in $\text{m s}^{-2}$

Where does $W = mg$ come from? Drop an object and ignore air resistance. The only force on it is its weight, and every object falling freely near the Earth speeds up at the same rate, $g$ (about $10\ \text{m s}^{-2}$ : each second it falls, it goes $10\ \text{m s}^{-1}$ faster). Newton's second law says force $=$ mass $\times$ acceleration. With force $W$ and acceleration $g$ , that is exactly $W = mg$ .

It also explains why heavy and light objects fall together (when air resistance is negligible). A 10 kg rock has ten times the weight of a 1 kg rock, but also ten times the mass to get moving, so the acceleration comes out the same.

In 9709, always use $g = 10\ \text{m s}^{-2}$ unless the question explicitly says otherwise.

The weight of a body of mass $m$ is the force $W = mg$ exerted on it by gravity. It acts vertically downwards. In 9709, use $g = 10\ \text{m s}^{-2}$ .

Physics uses $g = 9.81\ \text{m s}^{-2}$ (or 9.8). Mathematics uses $g = 10\ \text{m s}^{-2}$ because it makes the arithmetic clean and the ideas clear. Cambridge explicitly states this for 9709. If you use 9.81 in a 9709 exam, you may not get the exact expected answer and could lose accuracy marks.

Think of it as "Cambridge gravity": a rounded value that everyone taking this exam agrees to use. It is not a mistake; it is the convention.

A crate has mass $40\ \text{kg}$ . Find its weight. Take $g = 10\ \text{m s}^{-2}$ .

A rock weighs $250\ \text{N}$ . Find its mass. Take $g = 10\ \text{m s}^{-2}$ .

A stone of mass $2\ \text{kg}$ falls vertically through air. The air resistance is a constant $3\ \text{N}$ upwards. Find its acceleration. Take $g = 10\ \text{m s}^{-2}$ .

A bag of cement has mass $25\ \text{kg}$ . Calculate its weight. Take $g = 10\ \text{m s}^{-2}$ .

A weight of $340\ \text{N}$ acts on a particle. Find its mass. Take $g = 10\ \text{m s}^{-2}$ .

A ball of mass $0.5\ \text{kg}$ falls vertically. The air resistance is a constant $1.5\ \text{N}$ upwards. Find the acceleration of the ball. Take $g = 10\ \text{m s}^{-2}$ .

Mass vs weight — not the same thing