Pressure in liquids and in the air

A liquid pushes in every direction

Poke holes at different heights in a bottle of water: water jets out sideways from each one. A liquid pushes on the bottom, on the walls, and up on anything in it. At one depth, it pushes equally in every direction.

The pressure grows with the depth

The deeper you go, the more liquid there is above you, pressing down:

$\rho$ is the density of the liquid and $h$ the depth. In water, every $10\ \text{m}$ adds about $98\ \text{kPa}$.

That is why a dam is thicker at the bottom, and why deep-sea submarines need very strong walls.

The pressure of the air

The air above us has weight, so it presses on everything: about $1.0 \times 10^{5}\ \text{Pa}$ at sea level, the atmospheric pressure. We do not feel it because the fluids in our body push out just as hard. It falls as we climb a mountain.

You should find that the pressure line is straight: twice the depth, twice the pressure. At about $10\ \text{m}$ of water, the water adds as much as the whole atmosphere.

Watch out. The pressure depends on the depth, not on the shape or width of the container.

A worked example

Find the pressure due to the water at depths of $2\ \text{m}$, $10\ \text{m}$ and $25\ \text{m}$, and the total with the air above.

1. 2 m. $1000\ \text{kg/m}^3 \times 9.8\ \text{m/s}^2 \times 2\ \text{m} = 19600\ \text{Pa}$; in all $119600\ \text{Pa}$.
2. 10 m. $98000\ \text{Pa}$; in all $198000\ \text{Pa}$, about twice the air pressure.
3. 25 m. $245000\ \text{Pa}$; in all $345000\ \text{Pa}$.

Practise with new numbers

Try it in Python

The program works out the three depths.

Remember these

• A liquid pushes in every direction; its pressure grows with depth: $P = \rho gh$.
• Air pressure at sea level is about $1.0 \times 10^{5}\ \text{Pa}$.
• Total pressure under water $=$ air pressure $+ \rho gh$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. What is the water pressure $4\ \text{m}$ down in a pool?

2. Why is a dam wall thicker at the bottom?

3. Why do your ears hurt when you dive deep?

4. A tall narrow jar and a wide bowl are filled with water to the same depth. Which has more pressure at the bottom?