A confined fluid can transmit an applied change in pressure from one part of the fluid to another. This is Pascal's principle.

Suppose a fluid is enclosed by movable pistons. If an external force changes the pressure at one piston by

$$\Delta p,$$

then, in an ideal static fluid, that same pressure change is transmitted throughout the connected fluid. Existing pressure differences caused by gravity can still be present; Pascal's principle concerns the added pressure change.

From pressure to force

Pressure is force per area:

$$p=\frac{F}{A}.$$

If a force $F_1$ is applied to a piston of area $A_1$, the pressure increase it produces is

$$\Delta p=\frac{F_1}{A_1}.$$

A second piston of area $A_2$ experiences the same pressure increase, so the resulting force is

$$F_2=\Delta p,A_2.$$

Therefore

$$\boxed{\frac{F_1}{A_1}=\frac{F_2}{A_2}},$$

or

$$\boxed{F_2=F_1\frac{A_2}{A_1}}.$$

A larger output piston can therefore produce a larger force than the applied input force.

Example: a hydraulic lift

An input piston has area

$$A_1=5.0,\mathrm{cm^2}$$

and an output piston has area

$$A_2=200,\mathrm{cm^2}.$$

If the input force is

$$F_1=150,\mathrm N,$$

then

$$F_2=150\frac{200}{5.0}=6000,\mathrm N.$$

The hydraulic system multiplies the force by a factor of $40$.

The area units cancel in the ratio, so converting both areas to square metres is unnecessary as long as the same area unit is used for both.

Force multiplication trades against displacement

Hydraulics does not create motion for free. For an ideal incompressible fluid, the volume pushed into the fluid by one piston must equal the volume displaced at the other:

$$A_1\Delta x_1=A_2\Delta x_2.$$

Thus

$$\boxed{\frac{\Delta x_1}{\Delta x_2}=\frac{A_2}{A_1}}.$$

If the output force is forty times larger, the input piston must move forty times farther than the output piston in the idealized two-piston system.

So a hydraulic device exchanges force for displacement: a large force can be produced over a smaller motion by applying a smaller force over a larger motion.

Idealization and real systems

The simple relation assumes a connected fluid that is effectively static and incompressible, with negligible losses. Real hydraulic systems can have pressure losses, fluid compressibility, seal friction, hose expansion and height differences between pistons.

Pascal's principle does not say that pressure is identical everywhere in a fluid. Hydrostatic pressure can vary with elevation. It says that an externally imposed change in pressure is transmitted through a confined fluid under the ideal static model.