17.5 Stationary liquids

Before you read this, I suggest you read post 16.44.

In post 16.37 we thought about liquids as fluids – things that can flow. But if we have liquid in a container, it doesn’t usually flow.

Suppose the container has vertical sides and a cross-sectional area of A. If the column of liquid in the container has a height of h, the volume of liquid is hA. Then the mass of liquid is hAρ, where ρ is its density (post 16.44) and its weight (post 16.17) is hAρg where g is the modulus of the gravitational field (post 16.16).

This weight is a force that pushes down on the base of the container; the force acting on a unit area is called the pressure and is given by

p = hAρg/A = hρg.

Strictly, pressure is the component of a force (post 16.50) acting on a unit area of surface perpendicular to the force. Since pressure is defined in terms of a component of a force, it is a scalar and not a vector (post 17.2).

The previous paragraphs assume that ρ doesn’t change when the height, h, of the liquid column increases. You might expect that as h increases, the pressure p will increase and that the liquid molecules would be squashed closer together (post 16.37), making it more dense. And you would be right! But the effect is slight. We might need to consider it in the deepest oceans and would certainly need to consider this effect in gases (where the molecules are further apart – post 16.37). It is because liquid molecules cannot be squashed very much closer together (liquids are almost incompressible) that we use a liquid (brake fluid) in the braking system of a car to transmit pressure.

The units in which pressure is measured are the units of force (post 16.13) divided by the units of area N/m2 or N.m-2. However, this unit is given a special name – the pascal (abbreviated to Pa).

At a depth h’, which is less than h, the pressure acting on the liquid below is h’ρg. This pressure pushed down on the liquid below it – but the liquid below doesn’t move. Why not?

The only possible explanation is that equal and opposite pressures exist at any point in a liquid – acting from side-to-side as well as from top-to-bottom.


In conclusion, the pressure in a liquid increases with depth and acts equally in all directions. That is why dams are built much wider at the bottom than at the top. The base of the dam has to withstand a much higher pressure, from the water in a reservoir, than the top. And the pressure pushes sideways, against the wall of the dam, as well as downwards.


Related post

16.44 Density
16.37 Solids, liquids and gases

Follow-up posts

20.25 Pressure distribution in fluids

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