Liquid Pressure and Depth
The pressure in a liquid at rest increases with depth and depends on the density of the liquid: p = ρgh, where ρ is the density, g the gravitational field strength and h the depth below the surface. In brief: the pressure acts in all directions at a given depth, and does not depend on the shape of the container or the amount of liquid. Total pressure at depth h is p = patm + ρgh. Water pressure increases by about 100 kPa (1 atmosphere) for every 10 m of depth.
Divers feel pressure on their ears, dams are thicker at the bottom, and water towers are built high. All are consequences of liquid pressure. This guide explains where liquid pressure comes from and derives the formula, discusses the properties of liquid pressure, compares depths in a table, works through calculations for a diver and a dam, and corrects common mistakes.
Why does pressure increase with depth?
At any point in a liquid, the liquid above is pushing down with its weight. The deeper the point, the more liquid is above it, so the greater the weight per unit area. Particles of the liquid also collide with surfaces and with each other, so the pressure acts in all directions: sideways and upward as well as downward, and always at right angles to a surface in contact with the liquid.
What is the formula for liquid pressure?
Consider a column of liquid of height h and cross-section A. Its volume is Ah, its mass is ρAh, and its weight is ρAhg. The pressure at the bottom caused by this weight is the weight divided by the area:
p = ρgh
This is the gauge pressure (due to the liquid only). The absolute pressure includes the pressure of the air above the surface: p = patm + ρgh, with patm = 101 kPa. For fresh water ρ = 1,000 kg/m3, and for sea water about 1,025 kg/m3. The pressure difference between two points in the same liquid is ρgΔh.
What are the properties of liquid pressure?
- It increases linearly with depth.
- It is the same at all points at the same depth in a connected liquid at rest, in all directions.
- It does not depend on the shape of the vessel or the width of the container. This is known as the hydrostatic paradox: a thin tube of water 10 m high gives the same pressure at the bottom as a wide lake of the same depth (Pascal’s barrel experiment, 1646).
- It depends on the density: a denser liquid gives a higher pressure at the same depth (mercury gives 13.6 times the pressure of water).
- The free surfaces of one liquid in connected vessels are at the same level.
Pressure at different depths in water
| Depth (m) | Gauge pressure ρgh (kPa) | Absolute pressure (kPa) |
|---|---|---|
| 0 | 0 | 101 |
| 1 | 9.8 | 111 |
| 10 | 98 | 199 |
| 30 | 294 | 395 |
| 100 | 980 | 1,081 |
Worked example: a diver and a dam
- Diver at 10 m. Gauge pressure = ρgh = 1,000 × 9.8 × 10 = 98,000 Pa = 98 kPa. Absolute pressure = 101 + 98 = 199 kPa, about twice the pressure at the surface.
- Force on a window. A submarine window of area 0.50 m2 at a depth of 10 m (assuming the pressure inside balances the atmosphere): F = pA = 98,000 × 0.50 = 49,000 N, which is about the weight of a 5-tonne mass.
- Dam. The water behind a dam is 30 m deep. The pressure at the base = 1,000 × 9.8 × 30 = 294,000 Pa = 294 kPa, which is three times as much as at 10 m, and so the dam wall is thicker at the base.
- Different liquids. At 0.76 m depth, mercury (13,600 kg/m3) gives p = 13,600 × 9.8 × 0.76 = 101,300 Pa, equal to atmospheric pressure. Water would need a depth of 10.3 m for the same pressure.
How was liquid pressure studied?
Simon Stevin showed in 1586 that the pressure of a liquid depends on the depth and not on the shape of the vessel. Blaise Pascal repeated the argument in his barrel experiment in 1646, in which he burst a barrel by pouring water into a long thin tube attached to it. The formula p = ρgh is the basis of the design of dams, water supply systems and submarines.
Which mistakes are common?
Mistake 1: Using the length of the container instead of the depth
h is the vertical depth below the surface of the liquid.
Mistake 2: Thinking that pressure depends on the amount of liquid
A small lake and a large ocean give the same pressure at the same depth.
Mistake 3: Forgetting atmospheric pressure when absolute pressure is needed
Add patm if the surface is open to the air and the absolute value is required.
Mistake 4: Saying that pressure acts only downward
It acts in all directions, perpendicular to any surface.