Pascal's Law
When you squeeze an enclosed fluid, the pressure you create pushes out equally in every direction, not just where you pushed it. That means a small force on a small piston can create the same pressure as a much bigger force on a large piston — this is how hydraulic lifts multiply force.
The formula
F₂ / F₁ = A₂ / A₁
- F₁ — input force (N): the force pushed down on the small piston
- A₁ — input area (m²): the area of the small piston
- F₂ — output force (N): the force produced at the large piston
- A₂ — output area (m²): the area of the large piston
Worked example
A hydraulic lift has a small piston with area 0.01 square meters and a large piston with area 0.1 square meters. If you push on the small piston with 10 newtons, how much force appears at the large piston?
- F₁ = 10 N
- A₁ = 0.01 m²
- A₂ = 0.1 m²
- F₂ = F₁ · (A₂ / A₁)
- F₂ = 10 N · (0.1 m² / 0.01 m²)
F₂ = 100 N
Test yourself
Using the same lift, if you push on the small piston with 20 N instead, how much force appears at the large piston?
- Correct answer: 200 N
- 20 N
- 2000 N
Right! The area ratio is still ten, so twenty newtons times ten gives two hundred newtons.
To lift a heavy car using this hydraulic lift, which piston should you push on to use the least force?
- The large piston, because it has 10 times the area
- Correct answer: The small piston, because it needs less input force
- Either piston — it makes no difference
Exactly! Pushing the small piston takes less force, and the pressure transmits to create a much bigger force on the large piston.
Where you see this
A hydraulic car lift at a repair garage is this law with paint on it: the mechanic pushes a small piston with a modest force, the enclosed oil carries that pressure undiminished to every wall of the system, and a large piston under the car turns it into enough force to lift a two-ton vehicle. Hydraulic brakes, dental chairs, and excavator arms all work on the same force-multiplying trick.
Common mistakes
It is easy to imagine the fluid pushing mostly in the direction you squeezed it — an enclosed fluid transmits pressure equally in every direction, so the output piston can point anywhere and still receive the full push. The subtler trap is treating the multiplied force as free energy: when the area ratio is ten, the small piston must travel ten times farther than the large one rises, so work in equals work out and energy conservation still holds.
How it connects
This builds directly on hydrostatic pressure — the same pressure concept, now applied to an enclosed fluid that someone actively squeezes rather than fluid sitting under its own weight. The trade the law makes, force multiplied while distance shrinks, is energy conservation showing up in hydraulics: F₂ / F₁ = A₂ / A₁ is the lever rule restated for fluids. Archimedes' principle, next, returns to fluids at rest and the upward push they give anything placed in them.