Define pressure, use pressure = force divided by area, and describe pressure in liquids

What this dot point is asking

SEAB wants you to define pressure, to use the formula pressure $=$ force $\div$ area, to explain everyday effects such as why a sharp knife cuts well, and to describe how pressure in a liquid increases with depth. The big idea is that pressure depends not just on the force but on the area the force is spread over.

The answer

What pressure means

Pressure is the force acting per unit area, pushing at right angles to a surface. The formula is:

In symbols, $P = \dfrac{F}{A}$. The unit is the pascal ($\text{Pa}$), where one pascal is one newton per square metre ($1\ \text{Pa} = 1\ \text{N m}^{-2}$).

Why area matters

For the same force, a smaller area gives a larger pressure, and a larger area gives a smaller pressure. This explains many everyday effects:

• A sharp knife has a tiny edge area, so the same push gives a large pressure and cuts easily.
• A drawing pin has a sharp point (small area) and a wide head (large area): you push the point in with high pressure while your thumb feels low pressure.
• Wide tractor tyres or snowshoes spread weight over a large area, giving a low pressure so they do not sink into soft ground.

Pressure in liquids

A liquid pushes on any surface in contact with it, and in all directions. The pressure in a liquid:

• increases with depth, because deeper down there is more liquid weight pressing from above;
• is greater for a denser liquid;
• acts equally in all directions at a given depth.

This is why your ears hurt at the bottom of a deep pool, why a dam wall is thicker at the bottom, and why water spurts out faster from a hole lower down a tall can.

The depth, density and gravity link

The extra pressure due to a depth $h$ of liquid is given by:

where $\rho$ is the density of the liquid, $g$ is the gravitational field strength, and $h$ is the depth. The formula shows pressure rising in step with depth.

Examples in context

Example 1. Camels and snowshoes. A camel has wide, flat feet that spread its weight over a large area, giving a low pressure so it does not sink into soft sand. Snowshoes work the same way for a person on deep snow. Spreading the same weight over more area lowers the pressure.

Example 2. Water supply towers. Towns store water in tall towers or on hills. The greater the height of water above a tap, the greater the pressure at the tap, because pressure increases with depth below the surface. This is how mains water reaches homes without a pump.

Try this

• Cue. A force of $200\ \text{N}$ acts on an area of $0.50\ \text{m}^2$. Find the pressure. [2 marks] $P = \dfrac{F}{A} = \dfrac{200}{0.50} = 400\ \text{Pa}$.

• Cue. Explain why a person lying on a bed of many nails is not hurt, but standing on one nail would be. [2 marks] Many nails share the weight over a large total area, giving a low pressure; one nail concentrates the force on a tiny area, giving a high pressure.

• Cue. Find the pressure $2.0\ \text{m}$ deep in water of density $1000\ \text{kg m}^{-3}$, with $g = 10\ \text{N kg}^{-1}$. [2 marks] $P = \rho g h = 1000 \times 10 \times 2.0 = 20\,000\ \text{Pa}$.