Pressure as force per unit area

Pressure describes how much force acts on each unit of a surface’s area. It depends on the force acting perpendicular to the surface and the area over which that force is spread.

When the perpendicular force is distributed uniformly over an area, pressure is calculated using:

Here, $$p$$ is pressure, $$F$$ is the perpendicular force and $$A$$ is the area over which the force acts. Pressure is measured in pascals ($$\text{Pa}$$). One pascal is one newton per square metre: $$1\ \text{Pa} = 1\ \text{N/m}^2$$.

How force and area affect pressure

For a fixed area, increasing the force increases the pressure. For a fixed force, decreasing the contact area increases the pressure because the force is concentrated over a smaller area. Increasing the area spreads the same force out and lowers the pressure.

A sharp knife applies force over a small area at its edge, producing a high pressure that helps it cut. Snowshoes spread a person’s weight over a larger area, producing a lower pressure on the snow and reducing how far the person sinks.

Pressure beneath a liquid surface

A liquid exerts pressure beneath its surface. Pressure increases with depth: the deeper a point is, the more liquid there is above it. The weight of this additional liquid produces greater pressure.

Pressure also depends on the liquid’s density, which describes how much mass it has in a given volume. At the same depth, a denser liquid has more mass, and therefore more weight, in the same volume above a point. It produces greater pressure than a less dense liquid at that depth.

Calculating pressure changes in liquids

Pressure increases with depth in a liquid. For a liquid of uniform density, the pressure change between two depths depends on the liquid’s density, the gravitational field strength and the vertical change in depth:

Here, $$\Delta p$$ is the pressure change, $$\rho$$ is the liquid density, $$g$$ is the gravitational field strength and $$\Delta h$$ is the vertical change in depth. The symbol $$\Delta$$ means “change in”: subtract the initial value from the final value.

Use density in $$\mathrm{kg/m^3}$$, gravitational field strength in $$\mathrm{N/kg}$$ and depth change in $$\mathrm{m}$$. The pressure change is then in pascals ($$\mathrm{Pa}$$).

A positive $$\Delta h$$ means the final position is deeper, so $$\Delta p$$ is positive and pressure has increased. Moving to a shallower position gives a negative $$\Delta h$$ and a decrease in pressure.

Calculating a pressure change

In a liquid of density $$1000\ \mathrm{kg/m^3}$$, consider a move $$2\ \mathrm{m}$$ deeper where $$g = 10\ \mathrm{N/kg}$$. The pressure change is:

The pressure at the deeper position is therefore $$20,000\ \mathrm{Pa}$$ greater than at the starting position. This is a pressure difference, not necessarily the total pressure at the deeper position. To find that total pressure, the pressure at the starting position must also be known.

Rearranging the pressure-change equation

If the pressure change and two of the other quantities are known, rearrange the equation to find the remaining quantity: