The conservation of energy helps us to understand pressure.

We can also think about pressure as a measure of the compressibility of the fluid. To see what this means read on.

Assume you exert a force on an input piston. For a given volume of fluid being shifted from the input cylinder, the distance moved is calculated by: distance moved by input piston = volume removedarea of piston.

Remembering that energy = input force × distance and Putting these two relationships together, you get:

energy = force × volume removedarea of piston.

Dividing both sides by the volume gives energy inputvolume removed = input forcearea of piston .

The left hand side tells us something about the energy stored for each bit of volume removed, or more formally the energy per cubic metre (measured in joule metre ^{-3}). This measure of the incompressibility of the fluid is called the pressure, and since joule metre ^{-3} is such a mouthful it is usually measured in a special unit, the pascal, abbreviated to Pa, where 1 pascal = 1 joule metre ^{-3}.

Robert Boyle described pressure as the spring of the air

. The following discussion applies to fluids which are quite easy to compress. These are easier to think about, but harder to build reliable engineering structures with. So a way of thinking about hydraulic machines is this: to shift a certain amount of fluid from one cylinder needs an amount of energy for each cubic metre.

As a result of this activity – pushing with a force for a particular distance over a given area – a certain number of joules are shifted. How many joules per metre, the force, depends on both the pressure and the cross-sectional area of the cylinder. To place the fluid in another cylinder needs energy to be transferred. Again the pressure in the fluid and the cross-sectional area of the cylinder fix the force. We are encouraging you to think of this as a two step process – first remove the pressurised fluid from one cylinder – then store it in an interim place, then place it in another cylinder.

This, of course, is not how real hydraulic machines work, but it is an illuminating thought experiment, because it allows you to ask the question: So what is special about the volume of fluid after it has been removed from one cylinder and before it has been placed in the other?

Back to Boyle for the answer: It is just like a compressed spring. There is something special about the state of the fluid. It is squeezed or compressed, just like the compressed spring. As a consequence of this compression it can function as an energy store, and so exert a force when placed in the second cylinder.

Carrying round a lump of compressed fluid is just like carrying round a compressed spring. And compressed fluids do store energy. Think of the energy made available when an air horn sounds, a car air-bag releases or even an inflated, but unsealed, balloon is let go.

Just how are we comparing a spring to the fluid? For a spring it is how far away the spring is from its natural length, the extension, that we use to characterise the compression (or extension). For the fluid it is the excess pressure, the change in pressure from how the fluid was before, that is important in characterising how much energy each cubic metre has available.

The argument above is for compressible fluids. Hydraulic machines mostly use incompressible liquids. But right at the beginning we said that even incompressible liquids compress a bit. Just as very stiff springs are associated with a large store of energy when stretched only a little, so highly incompressible fluids store lots of energy per cubic metre for very little change in volume.