Communicating Vessels and Pressure p = ρgh

Pour oil or kerosene into a water U-tube — or water on top of mercury — and watch the levels settle: columns of different height balance each other, ρ₁h₁ = ρ₂h₂. In vessels of different shapes the water stands at one level, and the pressure on the bottom is the same, although they hold different amounts of water. Drag the pressure probe: it shows the pressure p = ρgh at any point, and the rings mark where the pressure is the same.

◉probe— pressure gauge in the liquid; the rings mark points with the same pressure

The colour of the liquid shows the pressure (scale at the bottom left). Force arrows are not to scale (∝ ∛F); the numbers are exact.

Liquids

h₁ column of the lighter liquid 15 cm

Drag the red probe through the liquid: it shows the pressure p = ρgh, and the rings show where the pressure is the same.

Show pressure as colour

Arrows and labels

Test yourself

Pressure without the atmosphere: p = ρgh, h is the depth below the surface of the same liquid. The arms of the U-tube have equal cross-sections, so the lower liquid drops on the left by as much as it rises on the right. The oscillations after pouring die out; the numbers in Measurements are for equilibrium.

ρ₁

lighter liquid (top left)

h₁

its column above the interface

ρ₂

heavier liquid (below)

h₂ = ρ₁h₁/ρ₂

column on the right above the interface

Δh = h₁ − h₂

difference of the top levels

p = ρ₁gh₁

pressure at the interface level — the same in both arms

p_probe

pressure at the probe (without the atmosphere)

The fountains of Peterhof near St Petersburg have been playing since 1721 without a single pump. Water collects in ponds on the Ropsha heights and runs down a conduit more than 20 kilometres long to the parks, where it rises in the fountains by itself — like in communicating vessels, towards the level it came from. Water towers, the plumbing of tall buildings, locks on rivers and canals that carry ships from one water level to another, gauge glasses on boilers and an ordinary kettle all work the same way: the water in the spout always stands at the same level as in the kettle.

KEY FORMULAS

Hydrostatic pressure

$$p = \rho g h$$

ρ is the density of the liquid, kg/m³; g ≈ 9.8 m/s²; h is the depth below the free surface, m

▶ Detailed Explanation

Each layer of liquid presses with its weight on the layers below, so pressure grows linearly with depth. In water it increases by about 10 kPa for every metre of depth, and in mercury, which is 13.6 times denser, 13.6 times faster.

The pressure on the bottom depends only on the height of the column and the density of the liquid, not on the shape of the vessel or the amount of liquid. This is called the hydrostatic paradox: in vessels of different shapes with equal bottoms, filled to the same level, the pressure and the force on the bottom are the same.

Communicating vessels with different liquids

$$\rho_1 h_1 = \rho_2 h_2 \quad\Rightarrow\quad \frac{h_1}{h_2} = \frac{\rho_2}{\rho_1}$$

h₁, h₂ are the heights of the liquid columns above the interface level, m; ρ₁, ρ₂ are the densities, kg/m³

▶ Detailed Explanation

In communicating vessels with one liquid the levels are the same. If another liquid that does not mix is poured into one arm, the pressures are compared at the level of the interface: in one and the same liquid the pressure is the same at the same height. Hence ρ₁gh₁ = ρ₂gh₂, and the heights of the columns are inversely proportional to the densities.

A 15 cm column of oil (900 kg/m³) balances 13.5 cm of water, and 30 cm of water balances only 2.2 cm of mercury. Liquid manometers work this way: pressure is found from the difference in levels in a U-tube.

Communicating vessels with one liquid

$$\rho g h_1 = \rho g h_2 \quad\Rightarrow\quad h_1 = h_2$$

h₁, h₂ are the heights of the liquid levels in the vessels above the connecting tube, m

▶ Detailed Explanation

The liquid in the connecting tube is at rest only if the pressures on it from the left and from the right are equal. In each vessel the pressure at the level of the tube is ρgh, where h is the height of the column above the tube. For the same liquid, equal pressures mean equal heights: the levels are the same.

The shape of the vessels, their width and the amount of water do not matter: in a wide vessel, a narrow one and a slanted tube the water rises to the same height. That is why the water in a kettle's spout is level with the water in the kettle, and a gauge glass on a boiler shows the water level inside.

Force of liquid pressure on the bottom

$$F = pS = \rho g h S$$

p is the pressure on the bottom, Pa; S is the area of the bottom, m²; h is the height of the liquid level above the bottom, m

▶ Detailed Explanation

The force with which a liquid presses on the bottom equals the pressure times the area of the bottom. It depends on the density, the height of the level and the bottom area, but not on the shape of the vessel or the weight of the liquid poured in. In the simulator three vessels have bottoms of 38.5 cm²; with a water level of 40 cm each bottom feels a force of about 15 N.

The weight of the water in these vessels differs: there is more in the widening one and less in the narrowing one. This is the hydrostatic paradox. The walls take up the difference: in a widening vessel they support part of the water, in a narrowing one they press down on it. The force on the bottom is always equal to the weight of an imaginary column of liquid above the bottom.

Pressure inside a liquid. Each layer of a liquid presses with its weight on the layers below, so pressure grows with depth: p = ρgh, where h is the depth below the free surface. In water the pressure rises by about 10 kPa for every metre: at a depth of 10 m it is 98 kPa — almost one more atmosphere. Mercury is 13.6 times denser than water, and the pressure in it grows 13.6 times faster. At the same depth in the same liquid the pressure is the same in all directions.

Pressure does not depend on the shape of the vessel. The pressure on the bottom is set only by the height of the column and the density of the liquid. In vessels of different shapes with equal bottoms, filled to the same level, the pressure on the bottom is the same, and so is the force F = pS — although the wide vessel holds several times more water than the narrow one. This is called the hydrostatic paradox. The walls of a widening vessel carry the 'extra' water themselves, while in a narrowing vessel the walls press down on the water.

Communicating vessels. These are vessels joined at the bottom. A single liquid settles in them at the same level, whatever their shape. If the levels differed, the pressures on the two sides at the bottom of the connecting tube would be different, and the liquid would flow across until they were equal. In the simulator the probe shows it: in one liquid, points with the same pressure always lie at the same height.

Two different liquids. If oil is poured into one arm of a water U-tube, the pressures are compared at the level of the interface: below it both arms contain the same water, and the pressures are equal. Above that level there is a column of oil h₁ on the left and a column of water h₂ on the right, so ρ₁gh₁ = ρ₂gh₂, or ρ₁h₁ = ρ₂h₂. The heights are inversely proportional to the densities: 15 cm of oil (900 kg/m³) balances 13.5 cm of water, and 30 cm of water balances only 2.2 cm of mercury. A liquid manometer works this way: pressure is measured by the difference of levels in a U-tube.

Try it yourself. Start with an empty left arm (h₁ = 0) — the levels are equal. Add oil and watch the interface go down while the water on the right goes up. Put the probe at the interface — the ring in the right arm is at the same height. Raise the probe into the oil — the point with the same pressure in the water is lower. Choose 'mercury + water': with the same column of water the mercury shifts only a little. In 'Different shapes' add water and compare the volumes in Measurements: the vessels hold different amounts of water, yet the pressure on the bottom is one and the same.

Communicating Vessels and Pressure p = ρgh

See also

KEY FORMULAS

Hydrostatic pressure

Communicating vessels with different liquids

Communicating vessels with one liquid

Force of liquid pressure on the bottom

More on this topic

Other simulators

Problems on this topic