Pressure in Liquids and Liquids at Rest

1Deriving P = hρg

Picture a column of liquid of height h and base area A above a point. Its volume is A × h, its mass is ρ × A × h (where ρ is the density), and its weight is ρAhg. Dividing that weight by the area A gives:

P = hρg

Here h is the depth in m, ρ is the density in kg/m³ and g is the gravitational field strength (9.8 N/kg). This is the pressure due to the liquid alone. At an open surface the air presses down too, so the total pressure = atmospheric pressure + hρg.

Worked example 1: diving in Lake Kivu

What is the pressure due to the water alone at a depth of 10 m in Lake Kivu? Take ρ = 1000 kg/m³ and g = 9.8 m/s².

P = hρg = 10 × 1000 × 9.8 = 98 000 Pa (98 kPa). That is more than the whole atmosphere presses at the lake surface: Lake Kivu lies about 1460 m above sea level, where air pressure is only about 85 kPa. A diver at 10 m therefore feels a total pressure of about 85 + 98 = 183 kPa, more than double the pressure at the surface.

Worked example 2: a water tank for a school

A water tank stands so that its water surface is 5.0 m above a tap. What is the water pressure at the tap (above atmospheric)? Take g = 9.8 m/s².

P = 5.0 × 1000 × 9.8 = 49 000 Pa. The higher the tank, the stronger the flow from the tap.

2Properties of pressure in a liquid at rest

1. It increases with depth (P ∝ h), so dam walls are thicker at the bottom.
2. It increases with the density of the liquid: salty sea water presses harder than fresh water at the same depth.
3. At a given point it acts equally in all directions: up, down and sideways.
4. It is the same at all points at the same depth in the same liquid.
5. It does not depend on the shape or width of the container. Only the vertical depth matters.

3Equilibrium of a liquid at rest

Because points at the same level must have the same pressure, the free surface of a liquid at rest is always horizontal. If one part of the surface were higher, the extra pressure beneath it would push liquid sideways until the levels were equal again.

Open the Under Pressure simulation. Drag the pressure gauge down into the pool and watch the reading rise steadily with depth. Then try the trapezium-shaped pool: at the same depth, the reading is the same whatever the shape.

Practice