A force spread over an area
A push or a pull is called a force, and forces are measured in newtons, written N. When a force presses on a surface, it does not press on a single point: it is spread over the whole area it touches.
Suppose a block presses down on a table with a force of 12 N, and the bottom of the block covers 4 cm² of the table. The 12 N is shared equally over the 4 square centimeters, so each square centimeter takes 12 ÷ 4 = 3 newtons. The force on each square centimeter is called the pressure. Here the pressure is 3 N/cm², read "3 newtons per square centimeter".
12 N spread over 4 cm²: 3 N on each square centimeter.
A smaller area means a bigger pressure
Now turn the block so that the same 12 N presses on only 2 cm². The force has not changed, but there are only 2 square centimeters to share it, so each one takes 12 ÷ 2 = 6 newtons. Half the area gives twice the pressure.
This is why a sharp knife cuts: its edge is very thin, so the force of your hand presses on a tiny area and the pressure is large.
The same 12 N on 2 cm²: 6 N on each square centimeter, twice as much as before.
It works the other way too. Spread the same 12 N over 6 cm² and each square centimeter takes only 12 ÷ 6 = 2 newtons. A bigger area gives a smaller pressure, which is why wide tires and snowshoes stop things from sinking into soft ground.
The same 12 N over 6 cm²: 2 N on each square centimeter.
Pressure = force ÷ area
To find a pressure, divide the force by the area: pressure = force ÷ area. This is the same kind of calculation as density, which is mass ÷ volume. Both share an amount out over each single unit, and the answer tells you how much lands on one of them.
The unit of a pressure is made from the two units in the calculation. A force in newtons over an area in square centimeters gives newtons per square centimeter, N/cm². A force in newtons over an area in square meters gives newtons per square meter, .
Finding the force
If you know the pressure and the area, multiply to find the whole force: force = pressure × area. A pressure of 3 N/cm² means 3 N on each square centimeter, so on 4 cm² the force is 3 × 4 = 12 N in all, the force we started with.
And if you know the force and the pressure, divide to find the area: area = force ÷ pressure. 12 N at 3 N/cm² presses on 12 ÷ 3 = 4 cm².
The usual mistakes
Multiplying the force by the area to find the pressure. Pressure shares the force out over the area, so it divides: for 12 N on 4 cm², 12 ÷ 4 = 3, so the pressure is 3 N/cm², not 12 × 4 = 48.
Adding the pressure to the area to find the force. Each of the 4 cm² takes 3 N, so the force is 4 × 3 = 12 N, not 4 + 3 = 7 N.
Using the wrong area. A person standing on two feet spreads their weight over both feet, so the area is twice the area of one foot.
Worked example: A Hiker on Snow in Boots and in Snowshoes: The Pressure Each Way, and the Heaviest Pack
Question A hiker weighs 720 N and stands on snow with his weight spread evenly over both feet. Each of his boots has a sole of area 0.03 m2, and each of his snowshoes has an area of 0.12 m2. (a) What pressure does he put on the snow in boots, and what pressure in snowshoes, in N/m2? (b) The snow holds any pressure up to and including 3500 N/m2 without a foot sinking in. What is the heaviest pack, in newtons, that he can carry on snowshoes without sinking? Ignore the weight of the snowshoes.
1.In boots, the hiker's 720 N presses on 2 × 0.03 = 0.06 m2 of snow.
In boots, 720 N presses on 2 × 0.03 = 0.06 m2. 2.The pressure is 720 ÷ 0.06 = 12000 N/m2.
The pressure in boots: 720 ÷ 0.06 = 12000 N/m2. 3.In snowshoes, 720 N presses on 2 × 0.12 = 0.24 m2, so the pressure is 720 ÷ 0.24 = 3000 N/m2.
In snowshoes the area is 0.24 m2: 720 ÷ 0.24 = 3000 N/m2. 4.(a) 12000 N/m2 in boots and 3000 N/m2 in snowshoes: four times the area gives a quarter of the pressure.
(a) 12000 N/m2 in boots and 3000 N/m2 in snowshoes. 5.At 3500 N/m2, the 0.24 m2 under the snowshoes holds up at most 3500 × 0.24 = 840 N.
At the limit, the snow under the snowshoes holds up 3500 × 0.24 = 840 N. 6.The hiker is 720 N of that, so 840 − 720 = 120 N is left for the pack.
The hiker is 720 N of that: 840 − 720 = 120 N remain. 7.(b) The heaviest pack is 120 N. Check: (720 + 120) ÷ 0.24 = 3500 N/m2, which is the limit exactly.
(b) The heaviest pack is 120 N.
Answer: (a) 12000 N/m2 in boots and 3000 N/m2 in snowshoes; (b) 120 N
Common mistakes
- Using the area of one boot or one snowshoe: he stands on both feet, so his weight is spread over twice that area.
- Giving 840 N as the answer: that is the most the snow holds up in total, and the hiker's own 720 N is part of it.