Hydrostatic Pressure and Pascal’s Principle

In a fluid at rest the pressure increases linearly with depth, p = p₀ + ρgh, the pressure is the same at all points at the same depth, and Pascal’s principle states that a pressure change applied to an enclosed fluid is transmitted undiminished to every point in the fluid and to the walls of the container. In brief: the pressure difference between two points is Δp = ρgΔh, which is the basis of the manometer and the barometer. The force on a vertical wall retaining a fluid of depth H is F = ½ρgH²w for width w, and acts at one-third of the depth above the bottom. The pressure at depth does not depend on the shape of the container.

Dams, submarines and blood pressure gauges rely on hydrostatics. This guide derives the pressure in a static fluid, explains pressure measurement with manometers, calculates the force on dams and walls, compares gauge and absolute pressure in a table, works through numerical examples and corrects common mistakes. A simpler introduction to the topic is given in the basic physics sections; here the emphasis is on quantitative applications.

How does pressure vary with depth?

Consider a small horizontal disc of fluid of area A at a depth h. For the fluid to be in equilibrium, the net vertical force must be zero: the upward force pA at the bottom must equal the downward force p₀A at the top plus the weight of the fluid column ρAhg. Hence p = p₀ + ρgh, where p₀ is the pressure at the free surface (atmospheric pressure for an open container). The same argument for a horizontal layer shows that the pressure is the same at all points at the same depth in a connected fluid: this explains why water in a U-tube or connecting vessels rises to the same level. The pressure acts in all directions, and perpendicular to any surface. If the density varies with depth (a gas, or the deep ocean), integrate: dp = ρg dh. In an incompressible fluid, Pascal’s principle is the statement that a change Δp₀ at the surface adds the same amount at every depth, which is used in hydraulic systems.

How are pressures measured with a manometer?

A U-tube manometer holds a liquid (mercury, water or oil). One arm is connected to the gas at pressure p, and the other is open to the atmosphere (p_atm) or sealed. The difference in the levels of the liquid, h, gives the pressure difference: p − p_atm = ρgh (the gauge pressure). The mercury barometer measures the atmospheric pressure itself with a closed arm (vacuum above the mercury), p_atm = ρgh = 101.3 kPa for h = 760 mm. Blood pressure in medicine is still given in mmHg. The gauge pressure is the pressure above atmospheric; the absolute pressure is the gauge pressure plus atmospheric pressure. Tire pressure gauges show the gauge pressure.

What is the force on a dam or a wall?

Pressure acts on the wall and increases with depth, so the force is not simply the pressure at the bottom times the area. For a vertical rectangular wall of width w retaining a fluid of depth H (with atmospheric pressure on both sides, which cancels), the gauge pressure rises linearly from 0 at the surface to ρgH at the bottom. The average pressure is ½ρgH, so the total force is F = ½ρgH²w. The force acts at a depth of 2H/3 from the surface (H/3 above the bottom), where the “center of pressure” is. The force increases with the square of the depth, which explains why dams are much thicker at the base, and often curved so that the arch carries part of the load. For a submerged flat surface, the force is the pressure at the centroid multiplied by the area.

Gauge and absolute pressure compared

Worked example: manometer, dam and ocean depth

1. Manometer. A gas supply is connected to a U-tube containing mercury (13,600 kg/m³), and the mercury level in the open arm is 0.15 m higher than in the arm connected to the gas. Gauge pressure = ρgh = 13,600 × 9.8 × 0.15 = 20,000 Pa = 20 kPa (150 mmHg). The absolute pressure = 101 + 20 = 121 kPa.
2. Dam. A reservoir of depth H = 20 m behind a vertical wall 100 m wide. Pressure at the bottom: ρgH = 1,000 × 9.8 × 20 = 196 kPa. Force: F = ½ρgH²w = 0.5 × 1,000 × 9.8 × 400 × 100 = 1.96 × 10⁸ N (196 MN, equal to the weight of about 20,000 tonnes). It acts at 20 ÷ 3 = 6.7 m above the bottom.
3. Doubling the depth. At H = 40 m, the force is four times larger: 784 MN.
4. Ocean depth. At the bottom of the Challenger Deep (about 10,900 m), taking an average seawater density of 1,050 kg/m³: p = ρgh = 1,050 × 9.8 × 10,900 = 1.12 × 10⁸ Pa, about 1,100 atmospheres. (Seawater is slightly compressed at this depth, so the real pressure is about 1.1 × 10⁸ Pa.)
5. Connected vessels. A U-tube contains water in one arm and oil (800 kg/m³) in the other. The oil column is 25 cm high above the interface. The water column above the same level is h_w = ρ_oilh_oil ÷ ρ_w = 800 × 0.25 ÷ 1,000 = 0.20 m.

How was hydrostatics developed?

Archimedes laid out the principles of hydrostatics in On Floating Bodies. Simon Stevin (1586) showed the hydrostatic paradox, and Evangelista Torricelli invented the mercury barometer in 1643. Blaise Pascal published the principle named after him in 1663. Daniel Bernoulli’s Hydrodynamica (1738) and Leonhard Euler (1755) gave the equations for fluids at rest and in motion.

Which mistakes are common?

Mistake 1: Using the length of the water column along a slope as h

h is the vertical depth below the free surface.

Mistake 2: Using the bottom pressure times the area for the force on a wall

The pressure varies with depth. Use the average, ½ρgH, for a vertical wall from the surface to the bottom.

Mistake 3: Confusing gauge and absolute pressure

Add atmospheric pressure to gauge pressure when absolute pressure is required.

Mistake 4: Thinking that the shape or amount of liquid matters

At a given depth in a connected liquid, the pressure is the same.