Calculating liquid pressure

If the pressure depends on the height of the liquid column alone, then it should be possible to calculate it from that height. Let us find the formula.

How the formula is built

Let the height of the liquid column in a vessel be h and the density of the liquid be ρ.

The volume of liquid standing above unit area of the base is h, so its mass is hρ.

The weight of that column of liquid is therefore hρg.

Since this weight acts on a region of unit area, the pressure at the base is hρg as well.

The formula for liquid pressure

P = h ρ g

h = vertical height of the liquid column (m)

ρ = density of the liquid (kg m⁻³)

g = acceleration due to gravity (m s⁻²)

P = the pressure obtained (N m⁻², that is Pa)

Figure to be added

Figure 15.5 — the pressure at a point at depth h; the column of height h standing above a point inside a liquid of density ρ.

This result is true not only for the pressure at the base of a vessel. If the height of the liquid column above any point inside a liquid is h, the pressure at that point is P = hρg.

Worked example 1

The problem

The depth at a place in a tank is 1.5 m. Find the pressure produced by the water on the bed at that place. (density of water = 1000 kg m⁻³; g = 10 m s⁻²)

The solution

pressure = h ρ g = 1.5 m × 1000 kg m⁻³ × 10 m s⁻²

= 15 000 Pa

Worked example 2

The problem

The depth at a certain place in the sea is 10 m. Find the pressure produced by the sea water on the sea bed at that place. (density of sea water = 1050 kg m⁻³; g = 10 m s⁻²)

The solution

pressure = h ρ g = 10 m × 1050 kg m⁻³ × 10 m s⁻²

= 105 000 Pa

A common mistake

h is not the height of the vessel but the height of the liquid column above the point the question asks about. And it must be substituted in metres, not centimetres.