Pascal's Principle and the Hydraulic Lift

A hydraulic lift is two pistons connected by a sealed, incompressible fluid. Push down on the small piston and you raise the pressure of the fluid touching it. Because the fluid is enclosed, that same pressure increase shows up everywhere in the fluid, including under the large piston. Pressure is force divided by area, so the same pressure pressing on a bigger area gives a bigger force. That is the whole trick: you are not multiplying pressure, you are multiplying force by moving to a larger area while pressure stays fixed.

The force ratio exactly follows the area ratio: F1/A1 = F2/A2, so F2 = F1 x (A2/A1). A piston with 25 times the area of the input piston feels 25 times the force. But nothing is free. Fluid is incompressible, so the volume pushed down by the small piston must exactly equal the volume pushed up by the large piston: A1 x d1 = A2 x d2. The small piston has to travel 25 times farther than the large piston rises. Work in, force times distance on the small side, equals work out on the large side. You trade distance for force, never gaining energy.

This connects directly to the pressure-depth relationship from earlier in the unit: both rest on the idea that pressure in a connected fluid depends on depth and applied force, not on the shape or size of the container. It also sets up Bernoulli's equation later, where pressure, speed, and height all trade off while total mechanical energy of the fluid stays constant, the same conservation logic seen here in force and distance.