Define pressure, apply pressure equals force over area, and calculate the pressure due to a liquid column

What this dot point is asking

SEAB wants you to define pressure, to use $p = F/A$ in calculations, and to explain and calculate how the pressure in a liquid increases with depth using $p = h\rho g$. The big idea is that the same force spread over a smaller area gives a larger pressure, and that a tall column of liquid presses harder at the bottom because of the weight of the liquid above.

The answer

What pressure is

Pressure is the force acting per unit area, where the force is perpendicular to the surface:

Force is in newtons and area in square metres, so the unit of pressure is the pascal ($\text{Pa}$), where $1\ \text{Pa} = 1\ \text{N m}^{-2}$. For the same force, a smaller area gives a larger pressure, which is why a sharp knife (small area) cuts more easily than a blunt one.

Pressure in liquids

A liquid exerts pressure on any surface in contact with it. This pressure:

• increases with depth, because there is more liquid pressing down from above,
• acts equally in all directions at a given depth,
• depends on the density of the liquid, not on the shape or width of the container.

The liquid column relationship

The pressure due to a column of liquid of height (depth) $h$ is:

where $\rho$ is the density of the liquid and $g$ the gravitational field strength. This is the extra pressure caused by the liquid; the total pressure also includes the atmospheric pressure pushing down on the surface.

Why depth, not shape, matters

Two containers of different shapes filled to the same depth with the same liquid have the same pressure at the bottom, because $p = h\rho g$ depends only on the depth, the density, and $g$, not on how wide the container is or how much liquid it holds.

Examples in context

Example 1. Dam walls. A dam is built much thicker at the bottom than at the top, because the water pressure increases with depth ($p = h\rho g$). The deepest water presses hardest, so the wall must be strongest there to withstand the larger force.

Example 2. Sharp versus blunt. Pressing a drawing pin pushes the same force onto a tiny point area, giving a huge pressure that pierces the board, while the broad head spreads the force on your thumb over a large area, giving a small, comfortable pressure. Same force, very different pressures, because of the difference in area.

Try this

Q1. A force of $200\ \text{N}$ acts on an area of $0.50\ \text{m}^2$. Calculate the pressure. [2 marks]

• Cue. $p = \dfrac{F}{A} = \dfrac{200}{0.50} = 400\ \text{Pa}$.

Q2. Find the pressure at the bottom of a $5.0\ \text{m}$ deep pool of water ($\rho = 1000\ \text{kg m}^{-3}$, $g = 10\ \text{N kg}^{-1}$). [2 marks]

• Cue. $p = h\rho g = 5.0 \times 1000 \times 10 = 50\,000\ \text{Pa}$.

Q3. Explain why pressure in a liquid increases with depth. [2 marks]

• Cue. The deeper you go, the greater the weight of liquid above pressing down, so the pressure is larger.