Pascal's Law & Hydraulic Machines
Squeeze a liquid anywhere and the extra pressure arrives everywhere.
Lesson at a glanceShow the key points
- Pascal's law: a change in pressure at any point of an enclosed liquid is transmitted undiminished to every point of the liquid.
- It follows because pressure differences depend only on height differences (ρgz), which don't change when we push.
- Hydraulic lift: F₁/A₁ = F₂/A₂, so a small force on a small piston supports a large load on a large piston.
- No work is gained: the small piston moves further, A₁d₁ = A₂d₂, so F₁d₁ = F₂d₂.
A mechanic lifts a car with one hand on a lever. The trick is a liquid that passes on every bit of extra pressure, undiminished, to every corner of the machine.
Pascal’s law
We just showed that the pressure difference between two points in a liquid at rest is ρgz, depending only on their height difference z. Suppose we somehow raise the pressure at one point. Since the liquid is incompressible, z doesn’t change, so the pressure difference between that point and any other must stay ρgz. The pressure everywhere else must therefore rise by the same amount.
Fit a piston of area A to a flask full of liquid and push it with force F. The pressure just under the piston rises by F/A, and so does the pressure at every other point in the flask.
The hydraulic lift
Two cylinders of different cross-sections, A₁ (small) and A₂ (large), are joined by a pipe and filled with oil; each has a piston. A valve lets oil pass from the small cylinder into the large one. Push the small piston with F₁: the pressure everywhere rises by F₁/A₁, so the large piston feels an extra upward force
With A₂ much larger than A₁, a small push supports a heavy car. Hydraulic presses, car brakes and excavator arms all work this way.
A hydraulic lift has pistons of area 5.0 cm² and 500 cm². What force on the small piston supports a 1200 kg car? How far does the car rise if the small piston is pushed down 30 cm? (g = 10 m/s²)
- 1Pascal's law: extra pressure applied to an enclosed liquid reaches every point undiminished.
- 2Hydraulic lift: .
- 3Volumes match, , so work in = work out.
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