Pascal’s Principle and Hydraulic Machines
Pascal’s principle states that a pressure change applied to an enclosed incompressible fluid is transmitted undiminished to every part of the fluid and to the walls of its container, and hydraulic machines use it to multiply force: F2 ÷ A2 = F1 ÷ A1. In brief: a small force on a small piston creates a large force on a large piston, in proportion to the ratio of the areas. The distances move in the opposite ratio, so the work is conserved: F1d1 = F2d2 in the ideal case.
Car jacks, brakes, excavators and aircraft control surfaces use hydraulic systems. They rely on liquids being almost incompressible. This guide states the principle, explains how a hydraulic press works, compares the pistons in a table, works through a force multiplication calculation including the distances moved, and corrects common mistakes.
What is Pascal’s principle?
Liquids are practically incompressible, so when an additional pressure is applied to any part of a confined liquid, it appears everywhere in the liquid at the same time, with the same value, and acts perpendicular to every surface in contact with the liquid. (Gravity also makes the pressure vary with depth, but this effect is added to the change which is transmitted.) The principle is named after Blaise Pascal, who published it in 1653.
How does a hydraulic machine work?
A hydraulic system has two cylinders, connected by a tube and filled with oil, each with a piston. A force F1 on the small piston of area A1 creates a pressure p = F1 ÷ A1. By Pascal’s principle, the same pressure acts on the large piston of area A2, so the force on it is
F2 = p × A2 = F1 × (A2 ÷ A1)
The force is multiplied by the ratio of the areas. Energy is conserved, though: when the small piston moves a distance d1, it displaces a volume A1d1 of liquid, and the large piston moves a distance d2 = d1 × (A1 ÷ A2), which is much smaller. The work done on the system equals the work done by it, F1d1 = F2d2, minus the losses through friction and leaks. Therefore a hydraulic machine is a force multiplier, not an energy multiplier. The liquid must have no air bubbles (they are compressible).
What are hydraulic machines used for?
- Car brakes: pressing the pedal creates pressure in the brake fluid, which pushes the brake pads at all wheels equally.
- Hydraulic jacks and lifts in garages, and hydraulic presses in factories.
- Excavators, cranes and dump trucks, in which hydraulic cylinders move arms.
- Dentist’s chairs and aircraft landing gear and flaps.
Small and large pistons compared
| Quantity | Small piston | Large piston |
|---|---|---|
| Area | A1 (small) | A2 (large) |
| Pressure | p | p (equal) |
| Force | F1 (small) | F2 = F1A2/A1 (large) |
| Distance moved | d1 (large) | d2 = d1A1/A2 (small) |
| Work | F1d1 | F2d2 (equal in the ideal case) |
Worked example: a hydraulic jack
The small piston has an area of 2.0 cm2 = 2.0 × 10−4 m2, and the large piston has an area of 80 cm2 = 8.0 × 10−3 m2. A force of 100 N is applied to the small piston, which moves 0.40 m.
- Pressure: p = 100 ÷ 2.0 × 10−4 = 5.0 × 105 Pa.
- Area ratio: 80 ÷ 2.0 = 40.
- Output force: F2 = pA2 = 5.0 × 105 × 8.0 × 10−3 = 4,000 N (equal to 100 × 40). This can lift a mass of 4,000 ÷ 9.8 = 408 kg.
- Distance moved by the large piston: d2 = 0.40 ÷ 40 = 0.010 m = 1.0 cm.
- Work check: input = 100 × 0.40 = 40 J; output = 4,000 × 0.010 = 40 J. In real jacks the output is smaller because of friction.
How was the principle discovered?
Blaise Pascal formulated the principle in his Treatise on the Equilibrium of Liquids (written 1653, published 1663). Joseph Bramah patented the hydraulic press in 1795. During the Industrial Revolution, hydraulic power was widely used in docks, cranes and machine tools, and today it is found in vehicles and aircraft.
Which mistakes are common?
Mistake 1: Thinking that hydraulics give free energy
Force is multiplied, but the distance is reduced by the same factor. Work is not increased.
Mistake 2: Saying that force is transmitted equally
It is the pressure that is transmitted equally, not the force.
Mistake 3: Using diameters instead of areas
The force ratio equals the ratio of areas, which is the square of the ratio of diameters.
Mistake 4: Ignoring air in the system
Air bubbles compress, and the pedal feels soft. Brake systems are bled to remove air.