Mechanical · hydraulics

Hydraulic Force

Pressure times area is force — and how a small push lifts a big load.
F = P · A · Pascal
100 bar · Ø50 mm

Cylinder

metric
bar
mm
N

The cylinder

fluid under pressure force Pascal: pressure is equal everywhere in the fluid
Result
Detail
Field notes

Pressure, area, force

How it works

Why hydraulics multiply force

Hydraulics rest on one line: F = P · A. Pressure pushing on an area produces a force. And by Pascal's principle, pressure applied to a confined fluid is the same everywhere in it — so a small piston and a large piston connected by fluid feel the same pressure but produce different forces. The bigger piston wins, in proportion to its area. That's a hydraulic press, a jack, a digger arm.

Worked example

A pump makes 100 bar (10 MPa). Push it against a 50 mm bore piston — area 19.6 cm² — and you get 19.6 kN of force, about 2 tonnes. Feed the same pressure to a 150 mm piston (9× the area) and it pushes harder: ~177 kN.

The catch is stroke: the big piston moves 9× less for the same fluid. Force up, distance down — energy is conserved, exactly like a lever.

Why does area go up with the square of diameter?

A circle's area is π/4·d². Double the bore and you quadruple the area — and the force. It's why a small increase in cylinder diameter buys a big jump in capacity.

bar, psi, MPa — how do they relate?

1 bar ≈ 14.5 psi ≈ 0.1 MPa ≈ 100 kPa. Hydraulic systems often run 100–350 bar (roughly 1,500–5,000 psi). The tool converts as you switch units.

Do I get force for free?

No. The output piston moves less in exact proportion to the force gain, so the work (force × distance) is the same on both sides, minus real-world losses. Hydraulics trade distance for force, like any machine.

What about the rod side?

On a cylinder's retract stroke, the rod takes up part of the piston, so the effective area — and the force — are smaller. This tool models the full-bore (extend) area; subtract the rod area for retract.

Why do both pistons finish their stroke at exactly the same moment?

Because the fluid connecting them is incompressible — whatever volume the input piston displaces has to appear on the output side, instant for instant, not just at the end. Push the input piston at a steady speed and the output piston is forced to move at its own steady (much slower) speed the entire time, which is exactly why the animation has them arrive together rather than the small piston "getting there first."

Ideal, static hydraulics. Assumes an incompressible fluid, no friction or losses, and full-bore (extend) area. Real cylinders lose force to seals, back-pressure and the rod side.