## Conservation of energy: predicting the forces produced at each piston

Physics Narrative for 11-14

#### How fundamental principles of energy conservation apply to hydraulic systems

We can use the idea of conservation of energy to show how the forces produced at the pistons are related to the distances moved by the pistons. There is no need to bother your pupils with this section of argument; we include it here to demonstrate how the working of a hydraulic machine is underpinned by the fundamental principle of conservation of energy.

For the hydraulic machine, three relationships are true:

energy input is equal to energy output

and energy input = input force × distance moved by input force and energy output = output force × distance moved by output force.

Therefore:

input force × distance input force moves = output force × distance output force moves.

Rearranging the equation reveals that:

input forceoutput force = distance moved by output forcedistance moved by input force .

So we can design systems that either:

Multiply forces: The relatively small input force is applied through a big distance to produce a big output force but acting over a much smaller distance.

Or:

Multiply distances: The relatively large input force is applied through a small distance to produce a small output force but acting over a much bigger distance.

There is always this trade-off between force and distance.