What is Pascal's principle?

Push on a liquid that is shut in, and the extra push doesn't stay where you pressed. It spreads through the whole liquid and presses equally on every wall, in every direction. That is Pascal's principle, written down by Blaise Pascal in the 1650s.

What spreads is pressure: force on each square metre. P = F ÷ A. Its unit, one newton on each square metre, is called a pascal (Pa). The air around you presses at about 101,000 Pa, or 1 bar.

Now the trick. Push a small piston and the same pressure reaches a big piston. The big one has more area, so it gets more force: F₂ = F₁ × A₂ ÷ A₁. A piston 100 times bigger gets 100 times the force. That's a hydraulic lift: your 200 N push, the weight of 20 kg, holds up a 1,500 kg car.

Nothing is free. Liquids barely squash, so the oil you push out of the small cylinder must fit into the big one. The big piston moves 100 times less. Force times distance, the work, is the same on both sides. Energy is conserved (see the conservation of energy box).

Liquids also press because of their own weight. Go deeper and there is more liquid above you, so P = ρ g h: density × gravity × depth. It depends only on the depth, not on the shape of the container. Every 10 m of water adds about one more atmosphere.

Press a car's brake pedal and nothing mechanical joins your foot to the wheels. Your push goes into a small master cylinder full of brake fluid. By Pascal's principle, the same pressure arrives at every wheel through thin steel pipes.

At each wheel, the pressure pushes on the pistons of a calliper, which squeeze pads onto a spinning disc. The calliper's pistons are much bigger than the master's, so the force grows: a car's 57 mm calliper piston has about 6.6 times the area of its 22 mm master piston. With the pedal's lever and the vacuum booster, a firm 150 N press becomes about 12,000 N of squeeze.

A motorcycle's front brake and a mountain bicycle's disc brakes work the same way, with a finger lever instead of a pedal. Because the pressure is the same in every pipe, the brakes on both sides of a car always get the same push. See CarClear, MotorcycleClear and CycleClear.

A bottle jack is a hydraulic lift you pump by hand. Each stroke pushes a little oil from a thin plunger into a fat ram, and one-way valves stop it flowing back. One hand lifts a car, but only about 2 mm per stroke.

Across India, most buildings keep a water tank on the roof. No pump pushes the water down to your tap: its own weight does. The pressure at a tap depends only on the height of the water surface above it, called the head: P = ρ g h.

Each metre of head gives about 0.1 bar. A tap on the ground floor of a four-storey building, 14 m below the tank's water, gets about 1.4 bar and gushes. A tap on the top floor, just 3 m below, gets only 0.3 bar and trickles. That's why top-floor showers are weak, and why people raise the tank on a stand or fit a small pressure pump.

The pipe's shape and length don't change this. The water in it is connected, so Pascal's principle carries the pressure round every bend. (Real pipes do lose a little to friction when water flows.)

A dishwasher uses a pump instead of height. It pushes water into a hollow spray arm. The pressure is the same everywhere inside the arm, so every nozzle shoots its jet out at the same speed, v = √(2P ÷ ρ). The nozzles point a little sideways, so the jets push back and spin the arm. See DishwasherClear.

Your body is mostly water, and water passes pressure on. That is how a blood-pressure cuff works. Pump air into the cuff and its pressure passes straight through the soft flesh of your arm to the artery inside.

When the cuff presses harder than your blood ever does, the artery is squeezed flat and no blood gets through. Let the air out slowly. When the cuff drops just below the peak of each heartbeat, the systolic pressure, blood spurts through with a tap you can hear in a stethoscope. When it drops below the lowest pressure, the diastolic, the blood flows smoothly and the tapping stops. Normal is about 120/80 mmHg.

mmHg means millimetres of mercury: 120 mmHg pushes as hard as a column of mercury 120 mm tall. Blood's own weight matters too. Raise your arm 10 cm above your heart and it reads about 8 mmHg lower. That's why you should rest your arm at heart level. See HeartClear.

Your ears have air on both sides of the eardrum. Behind it, the middle ear is sealed off, except for a thin Eustachian tube to your throat that opens when you swallow or yawn. As a plane comes down, the cabin's pressure rises but the air behind your eardrum stays at the old, lower pressure. The difference pushes on your whole eardrum. Swallow, the tube opens, air flows in, and pop: the pressures match again. See EarClear.

Pascal's barrel. Take a barrel full of water, fix a long thin tube in its lid, and pour water into the tube. Around 7 metres up, the barrel splits, though the tube holds less than a kilogram of water. The pressure under the lid is ρ g h: it depends on the height of water, not how much there is. This famous story is told about Pascal, but it isn't in his own writings, and he may never have done it.

That surprise is the hydrostatic paradox, first explained by Simon Stevin in 1586. A 1 cm tube and a 30 cm pipe, both 10 m tall, push on the lid with exactly the same force, although one holds 0.8 kg of water and the other 700 kg.

Gases squash. Pascal's principle passes pressure through any fluid, but a gas must be squeezed smaller before it pushes back. A bubble of air in a brake pipe soaks up your pedal's movement, so the pedal feels spongy and can sink to the floor before the brakes grip. That's why hydraulics use liquids, and why mechanics bleed air out of brake lines.

Myth-buster: “Hydraulics make free energy.” No. A jack turns 80 N into 10,000 N, but you pump its handle about 50 times to lift a car 10 cm. Force × distance in equals force × distance out, minus a little lost to friction. Hydraulics trade distance for force; they never create energy.