How does a hydraulic press work? Pascal's principle and lifting a big load with a small force

In a car repair shop, a mechanic pushes a lever down a few times and the front of a car slowly rises. A digger’s bucket lifts tonnes of soil in one movement. And when your foot presses lightly on the brake, a car weighing more than a tonne comes to a stop. Behind all three is the same idea: pressure applied to a confined liquid is passed on unchanged to every part of the liquid. It is called Pascal’s principle, after the French scientist Blaise Pascal, who stated it clearly in the 17th century. In this post we first define pressure, then take a hydraulic press apart piece by piece.

What is pressure?

A force is a push or a pull. Pressure describes how widely that force is spread over a surface:

p = F / A

Here F is the force at right angles to the surface (in newtons), A is the area of the surface (in square metres) and p is the pressure. Its unit is the pascal: 1 Pa is 1 newton on each square metre. Because this is a very small unit, engineers often use the bar: 1 bar = 100,000 Pa = 100 kPa. Air pressure at sea level is about 1 bar.

The same force on a smaller area gives a larger pressure. The sharp tip of a drawing pin goes into wood easily because the force of your thumb is concentrated on a tiny area. A snowshoe does the opposite: it spreads your weight over a large area and lowers the pressure, so you do not sink into the snow.

In a confined liquid, pressure travels in every direction

Liquids are almost incompressible: even if you raise the pressure on water by 100 bar, its volume shrinks by only about 0.5 per cent. Hydraulic oils behave in a similar way. So when you push on one part of a liquid in a closed container, the liquid does not get “shorter”; your push spreads to every part of the container.

Pascal described this in his treatise on the equilibrium of liquids, published after his death in 1663: imagine a closed vessel full of water with two openings, one a hundred times wider than the other, each fitted with a close-fitting piston. One person pushing the small piston balances a hundred people pushing the large one. Pascal presented the arrangement as a new machine for multiplying forces. The English inventor Joseph Bramah patented the hydraulic press of the kind used in workshops today in 1795.

In a liquid at rest, the pressure is the same at every point at the same depth, and it increases with depth by the weight of the liquid above. In a hydraulic device, the pressure applied by the pistons is usually far greater than this difference, so it is often left out of the calculation.

Two pistons, one pressure

A hydraulic press has two cylinders joined by oil. Let the area of the small piston be A₁ and that of the large piston A₂. A force F₁ on the small piston creates a pressure p = F₁ / A₁ in the liquid. By Pascal’s principle, the same pressure acts under the large piston and pushes it with a force

F₂ = p · A₂ = F₁ · (A₂ / A₁)

So the force grows by the ratio of the areas. Since area grows with the square of the diameter, making the diameter three times larger makes the force nine times larger.

In the figure above, a 100 kg load sits on a large piston 60 mm across, and the small piston is 20 mm across:

• Weight of the load: F₂ = 100 kg × 9.81 m/s² ≈ 981 N
• Area of the large piston: A₂ = π × (0.03 m)² ≈ 0.002827 m², or about 28.3 cm²
• Pressure in the liquid: p = 981 N / 0.002827 m² ≈ 347,000 Pa ≈ 3.47 bar
• Area of the small piston: A₁ = π × (0.01 m)² ≈ 0.000314 m²
• Force needed on the small piston: F₁ = p × A₁ ≈ 109 N

Because the area ratio is (60 / 20)² = 9, a force of 109 N holds up a load of 981 N. The press also pushes the small piston with a lever. If the hand is 5 times as far from the pivot as the piston is, the hand needs only 109 / 5 ≈ 22 N. That is about the force needed to hold a 2.2 kg shopping bag.

The force gain is not free: you pay in distance

No machine creates energy out of nothing. When the small piston moves down by d₁, a volume A₁ · d₁ of oil leaves its cylinder. Since the liquid does not compress, this oil enters the large cylinder and raises the large piston by d₂:

A₁ · d₁ = A₂ · d₂, so d₂ = d₁ · (A₁ / A₂)

If the force is 9 times larger, the distance is 9 times smaller. In our example, one full push of the lever moves the small piston down 30 mm, while the large piston rises only 30 / 9 ≈ 3.3 mm. To lift the load by 6 cm, you have to pump the lever 18 times.

Comparing the work done at the two pistons makes it clear:

• At the small piston: F₁ · d₁ = 109 N × 0.030 m ≈ 3.27 J
• At the large piston: F₂ · d₂ = 981 N × 0.0033 m ≈ 3.27 J

F₁ · d₁ = F₂ · d₂: work is conserved. A hydraulic press multiplies force, not energy. In a real press, friction and leaks mean the work you get out of the large piston is a little less than the work you put into the lever.

Why doesn’t the load drop when the lever rises?

A jack or a hand press pushes the small piston not once but again and again. Two small check valves make this possible. A check valve is a flap that lets liquid pass in one direction only:

1. When you push the lever down, the small piston drives oil forward; the check valve to the large cylinder opens and oil flows into it.
2. When the lever rises, the small piston moves up and draws fresh oil from the tank. Meanwhile the check valve to the large cylinder closes, so the oil in it cannot flow back; the load stays where it is.
3. When the job is done, the release valve is opened. The oil in the large cylinder flows back to the tank and the weight of the load pushes the large piston down.

The pressure gauge in the figure shows the pressure in the oil line. While the large piston rests on its stop at the bottom, the oil carries no load, so the gauge reads 0; from the moment the piston lifts, the pressure is p = F₂ / A₂ and stays there as long as the load is unchanged. It is how heavy the load is, not how fast you pump, that sets the pressure.

Where do we meet hydraulics?

• Brakes: The brake pedal, through a lever, pushes a small piston in the master cylinder. Brake fluid travels through pipes to larger pistons at the wheels, which press the pads against a disc or the shoes against a drum. The pressure reaches every wheel at the same time. If air is left in the lines, the pedal feels “spongy”: the air compresses, and part of your push goes into squeezing it. Brake fluid absorbs moisture over time, which lowers its boiling point, so it is replaced at the intervals the manufacturer recommends.
• Jacks: The bottle jack used to change a tyre or in a garage is the press in the figure standing upright: a lever, a small piston, two check valves, a large piston and a release screw.
• Diggers and other machinery: Hydraulic cylinders move the arm and bucket of an excavator. Here the small piston is driven not by a person but by a pump turned by the engine, and the pressure is often hundreds of bar. The levers the operator moves open and close valves that choose which cylinder the oil goes to.
• Workshop presses, barber’s chairs, car lifts: In each, a small force or a small pump carries a large load through the liquid under a wide piston.

Common misconceptions

• “The pressure is greater at the large piston.” No. The pressure is the same at both pistons; what is greater is the force, because the same pressure acts on a larger area.
• “A hydraulic system produces energy.” No. The force you gain is paid for in distance: F₁ · d₁ = F₂ · d₂.
• “If I pump faster, the pressure rises.” With the same load, the pressure stays roughly the same. Pumping faster lifts the load sooner; it only needs a little extra force to push the oil quickly through narrow passages.
• “Air would work just as well as a liquid.” Air compresses easily. Part of the volume you pump goes into squeezing the air, so the large piston moves late and springily. That is why air is bled out of hydraulic systems.

Try it yourself

Find two needle-free syringes, one small (say 5 mL) and one large (say 20 mL), and a short tube that fits on their tips. Fill the syringes and the tube with water so that no bubbles remain, and connect them. When you press the small syringe’s plunger, you will feel that the large syringe’s plunger moves a shorter distance but is harder to hold back. Then repeat the experiment with a little air left inside: the movement becomes springy and delayed. Do the experiment on a table and do not point the syringe tips at your face.

For pressure in the kitchen, read Why does a pressure cooker cook faster?, and if you wonder why liquids push objects upwards, see What is density?. You can try measuring forces in the Force and the spring scale lesson.