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A piece of hydraulic machinery uses a liquid (usually an oil) under pressure to transmit energy. Lets consider a simple system such as the one that operates the clutch on my car. There are two cylinders with a piston in each and a small tube running between them. When I step on the clutch pedal, I force the piston on one of the cylinders (known as the master cylinder) deeper into the cylinder pushing fluid under pressure into the other cylinder. The fluid exerts pressure on the other cylinder (the slave cylinder) and develops a force which pushes on the clutch. The relationship between force and pressure is:

Force = Area x Pressure

Which can be rearranged with algebra to

Pressure = Force / Area

This, by this is way, is why we measure pressure as in units such as pounds (force) per square inch (area).

If we make the master cylinder smaller than the slave cylinder, say 1 square inch and 2 square inchs respectively, and assume that the pressures are equal, we find that

Master cylinder force = 1/2 Slave cylinder force

So we have to apply half as much force to the master cylinder (at the pedel) as we want from the slave cylinder (the clutch) and I don't have to put as much force on the pedel to operate the clutch. The force amplification is very similar to a lever (such as a pair of plyers) but has the added benifit that the master and slave cylinders can be located anywhere. The tube has many twists and turns, each one of which would require joints if a mechanical system were used. Like a lever, and I must move the master cylinder twice as far at the slave cylinder travels.

Brakes in the car operate similarly, but the tube coming from the master cylinder(s) are branched into several pistons at the wheels. Flexable tubing is used so that the wheels can move freely when the car goes over bumps.

Machines such as backhoes and buldozers use similar priniples. However, in this case, the fluid is kept at constant pressure continually by a pump connected to the engine and sent to the slave cylinders when the operator opens a valve. This allows very large forces to be developed at the slave cylinder.

Pnematic systems are similar to hydrolics, by use gases (typically air) rather than liquid. Since gas can be compressed considerable, a lot of air can be forced into a tank and the energy used later. This make pnematics suitable for systems where large forces and speed are required at the same type such as jackhammers, nailguns, or paintball guns.

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