Pressure and its variation with depth
What is pressure?
Liquids and gases can flow, so we call them fluids. A fluid has no fixed shape of its own: pour water from a lota into a glass and it takes the shape of the glass.
Press a drawing pin into a board with your thumb. The flat head does not hurt your thumb, but the sharp tip goes straight into the wood. The force is the same at both ends.
What is different is the area. The same force spread over a big area does little; squeezed onto a tiny area it does a lot. That is why knives are sharpened and why school bags have wide straps.
Pressure is the normal force per unit area: P equals F divided by A. Its SI unit is the newton per square metre, called the pascal, Pa.
Now watch a fluid at rest pushing on the wall of its container. The force is always at right angles to the wall. If there were a sideways part, the fluid next to the wall would start to flow.
To find the pressure at a point, imagine a tiny area ΔA around it with a normal force ΔF on it. The pressure is the limit of ΔF over ΔA as the area shrinks to zero.
Pressure has no direction of its own: it is a scalar. At a point in a still fluid it pushes equally on a tiny surface whichever way you turn it.
We also need the density of the fluid, ρ, its mass per unit volume. Water has ρ = 1000 kilograms per cubic metre, and mercury, 13 600: thirteen point six times as much.
Pressure increases with depth
Dive to the bottom of a swimming pool and your ears start to hurt. The deeper you go, the larger the pressure. Let us find out exactly how it grows.
Mark out a vertical cylinder of water inside a tank. Its cross-section is A, its height is h, and its top touches the surface.
Three vertical forces act on it. The air pushes down on its top with P a times A. The water below pushes up on its base with P times A. And its weight, m g, pulls down.
The mass of the cylinder is density times volume: ρ times A h. So its weight is ρ A h g. The sideways forces on its curved wall cancel out.
The water is at rest, so the forces balance: P A minus P a A equals ρ g h A. Cancel the area and we get P equals P a plus ρ g h.
Pressure grows in a straight line with depth. The extra part, ρ g h, is called the gauge pressure: P minus P a. Plot P against h and you get a straight line starting at P a.
Worked example: a swimmer 10 metres below the surface of a lake. ρ g h is 1000 times 9.8 times 10, which is 98 000 pascals. Add the atmosphere, 1.01 times 10⁵, and the total is about 2.0 times 10⁵ pascals: two atmospheres.
Notice what is missing from the formula: the area of the tank and its shape. The pressure depends only on the depth. All points at the same depth in the same still liquid have the same pressure.
Now join three very different vessels at the bottom: a wide jar, a thin tube and a flared funnel. Pour in water. It stands at the same height in all three, and the pressure at the bottom of each is the same.
This surprises people, because the funnel holds far more water. It is called the hydrostatic paradox. The funnel’s sloping walls hold up the extra water; the pressure at the base depends only on h.
Try it: pressure at depth
- Drag the probe (or use the slider). The arrows show pressure pushing equally from every side.
- The amber line is the absolute pressure P = Pₐ + ρgh; the dashed line is the gauge pressure ρgh.
- Switch to mercury: the line is 13.6 times steeper.
- Press Three vessels to see the hydrostatic paradox.
Three vessels
Three vessels with the same base area are filled with water to the same height h. One is a straight jar, one is wide at the top like a funnel, and one is narrow at the top like a bottle. Where is the water pressure at the base the greatest?
- In the funnel, because it holds the most water
- In the bottle, because the water is squeezed into a narrow neck
- It is the same in all three
- In the straight jar, because its walls are vertical
Answer
C. Pressure in a still liquid is P = Pₐ + ρgh. It depends only on the depth h, not on the shape of the vessel or the amount of water. The walls of the funnel hold up the extra water. This is the hydrostatic paradox.
Diving in the sea
A diver is 20 m below the surface of the sea (ρ = 1030 kg/m³). Taking Pₐ = 1.01 × 10⁵ Pa and g = 9.8 m/s², what is the absolute pressure on the diver?
- 2.02 × 10⁵ Pa
- 3.03 × 10⁵ Pa
- 1.22 × 10⁵ Pa
- 2.06 × 10⁴ Pa
Answer
B. Gauge pressure ρgh = 1030 × 9.8 × 20 ≈ 2.02 × 10⁵ Pa. Add the atmosphere above the sea: P = 1.01 × 10⁵ + 2.02 × 10⁵ ≈ 3.03 × 10⁵ Pa, about 3 atm. 2.02 × 10⁵ Pa forgets the atmosphere, and 2.06 × 10⁴ Pa forgets g.
Summary
- Pressure:
P = F / A (unit: pascal, 1 Pa = 1 N/m²) - Density:
ρ = m / V - Pressure at depth h:
P = Pₐ + ρgh - Gauge pressure:
P − Pₐ = ρgh
- A fluid at rest pushes at right angles to any surface it touches. Pressure is a scalar.
- Pressure increases linearly with depth: every 10 m of water adds about 1 atm.
- All points at the same depth in the same still liquid are at the same pressure.
- Hydrostatic paradox: the pressure at the base depends only on the height of the liquid, not on the shape of the vessel or how much liquid it holds.