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Pressure and depth

Air and fluids · level 5, lesson 2 of 6

A question to start

A dam wall is much thicker at the bottom than at the top. Divers' ears hurt as they go deeper. Why does the pressure grow with depth?

The idea

Under a liquid, the pressure comes from the weight of the liquid above. Take a column of liquid of height \(h\) over an area \(A\). Its volume is \(Ah\), its mass \(\rho A h\) (where \(\rho\), "rho", is the density), and its weight \(\rho A h g\). The pressure is weight over area:

\[p = \rho g h.\]

This is the extra pressure from the liquid. The total pressure also includes the air pressing on the top:

\[p_\text{total} = p_\text{air} + \rho g h.\]
low pressurehigh pressure:goes farthest

Pressure depends only on the depth, not the shape of the container. A narrow pipe of water 10 m tall pushes on its base with the same pressure as a lake 10 m deep.

Worked examples

1. 10 m of water on Earth. \(p = 1000 \times 9.81 \times 10 = 98{,}100\) Pa, nearly as much again as the whole air above.

2. The total. \(101{,}400 + 98{,}100 = 199{,}500\) Pa, about twice sea-level pressure. Every 10 m down adds about one more "air's worth".

3. A water tank. A tank 2 m deep: \(1000 \times 9.81 \times 2 = 19{,}600\) Pa at the bottom from the water.

Try it

  1. Why does pressure grow with depth?
  2. Find the water pressure (not counting the air) 25 m down in the sea. Take seawater as 1,025 kg/m³ and \(g = 9.81\) N/kg.
  3. Why is a dam thicker at the bottom?
  4. A thin tube and a wide tank both hold water 3 m deep. Which has more pressure at the bottom?
Answers
  1. There is more liquid above, pressing down.
  2. \(1025 \times 9.81 \times 25 \approx 251{,}000\) Pa.
  3. The water pressure is greatest at the bottom.
  4. Neither: the pressure is the same, because the depth is the same.

Common mistake

"More water means more pressure." It is the depth that counts, not the amount. A huge shallow lake pushes less on its floor than a narrow deep well.

Where it leads