The reason hydraulics are useful is that, for the applications we use them for, hydraulic fluids like water but more frequently oil of some kind can be considered to be incompressible. What’s important about that is it means that if you pressurize a container of hydraulic fluid, the same pressure will be present everywhere in the container. (I am ignoring the effect of gravity because it is irrelevant for hydraulic machines.)
Let’s imagine you have a very thin cylinder and you can apply a force of, say, 100 lb to a piston to drive it down that cylinder. If the area of the head of the piston is one square inch (meaning the radius is about 0.56 inches) that means you’re exerting a pressure of 100 psi. This isn’t very useful, so let’s say the area of the head of the piston is only about 0.01 sq inches (meaning the radius is 0.056 inches or about 1.4 mm). Now you’re exerting a pressure of 10,000 psi.
If you connect the fluid being pressurized by this piston by a pipe that leads to a much larger piston that gets forced out as you press down on your piston, because the pressure is equal, the absolute force can be much larger. Let’s say that you’re a hydraulic press operator and the other end of that press has a piston head that’s one square foot, 144 square inches (with a radius of about 6.8 inches). As you press down on your tiny little piston head, generating 10,000 PSI of pressure, that causes the really big piston head to be driven down with 144 square inches * 10,000 lb / square inch = 1,440,000 lb of force. That’s a tremendous amount of force.
None of this creates an air gap, though. The hydraulic fluid is confined between the two pistons. The only reason the press works at all is that there’s no air gap that ever gets created.
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