Pressure and Hydraulics

Why a drawing pin hurts but a flat board doesn't

Press the flat side of a board against your hand and you feel a gentle, harmless push. Press a drawing pin against your hand with the same force and it hurts immediately. The force is identical — so why the difference? The answer is pressure, one of the most useful and surprising ideas in physics. Understanding it explains everything from why snowshoes stop you sinking, to how a tiny push on a brake pedal can stop a two-tonne car.

What is pressure?

Pressure measures how concentrated a force is — how much force is pressing on each unit of area. The formula is:

Pressure = force ÷ area

Or in symbols: P = F ÷ A, where:

• F is the force in newtons (N),
• A is the area in square metres (m²),
• P is the pressure in pascals (Pa), where 1 Pa = 1 N/m².

The key insight is that pressure depends on both the force and the area it is spread over. The same force can produce a tiny pressure or an enormous pressure, just by changing the area.

• Small area → high pressure. A drawing pin, a knife edge, or a stiletto heel concentrate force onto a tiny area, creating huge pressure.
• Large area → low pressure. Snowshoes, tractor tyres, and a camel's wide feet spread weight over a large area, creating low pressure so they don't sink.

This is the same physics behind why pressing on a surface is explored in pressure in liquids and gases.

Worked examples with the pressure formula

Example 1 — basic pressure. A box pushes down with a force of 600 N on the floor. Its base has an area of 2 m². What pressure does it exert?

P = F ÷ A = 600 ÷ 2 = 300 Pa

Example 2 — same force, smaller area. Now the same 600 N box is balanced on legs whose total area is only 0.01 m². What is the pressure now?

P = F ÷ A = 600 ÷ 0.01 = 60 000 Pa

The force never changed, but shrinking the area by 200 times made the pressure 200 times larger! This is exactly why a person standing in stiletto heels can dent a wooden floor that they could walk across safely in flat shoes.

Example 3 — finding force. A hydraulic press produces a pressure of 50 000 Pa on a piston of area 0.4 m². What force does it deliver?

Rearranging: F = P × A = 50 000 × 0.4 = 20 000 N

Pressure in liquids

Pressure behaves in a special way inside liquids. There are two important facts:

1. A liquid cannot be squashed (compressed). Its particles are already close together, so you cannot squeeze it into a smaller space the way you can with a gas.
2. A liquid transmits pressure equally in all directions. Push on a liquid in a sealed container and the pressure increase is felt the same everywhere — sideways, upwards and downwards.

This second fact is so important it has a name: Pascal's principle, after the scientist Blaise Pascal.

Pascal's principle: Pressure applied to an enclosed (trapped) liquid is transmitted equally and undiminished to every part of the liquid and the walls of its container.

It is precisely this principle that makes hydraulics possible.

Hydraulics: multiplying force

A hydraulic system uses a trapped liquid to transmit force from one place to another — and to multiply it. Here is how it works.

Imagine two pistons connected by a pipe full of oil. One piston is small (small area) and one is large (large area).

1. You push the small piston with a modest force. This creates a pressure in the liquid: P = F ÷ A, using the small area.
2. By Pascal's principle, this same pressure is transmitted through the liquid to the large piston.
3. On the large piston, the force is F = P × A, but now using the large area. Because the area is bigger, the force is bigger!

The pressure is identical on both sides, but the larger area produces a larger force. This is force multiplication.

Worked example — a hydraulic lift. You push the small piston (area 0.01 m²) with a force of 100 N.

Pressure created: P = F ÷ A = 100 ÷ 0.01 = 10 000 Pa

This pressure reaches the large piston (area 0.5 m²):

Output force: F = P × A = 10 000 × 0.5 = 5000 N

You pushed with just 100 N, but the large piston pushes back with 5000 N — a 50-fold force multiplication! That is enough to lift a small car. The multiplication factor equals the ratio of the two areas (0.5 ÷ 0.01 = 50).

There's no free lunch — energy is conserved

If you got 50 times the force for free, you might wonder where the extra energy comes from. It doesn't — and this is important. To lift the large piston a small distance, you must push the small piston a much longer distance.

The work done (force × distance) is the same on both sides. You trade distance for force:

• Small piston: small force × large distance.
• Large piston: large force × small distance.

Energy is always conserved. Hydraulics multiply force, never energy. This is the same principle behind levers and simple machines, explored in simple machines.

Hydraulics in the real world

Because trapped liquids transmit force smoothly through flexible pipes and around corners, hydraulics are everywhere:

• Car brakes — a light push on the brake pedal becomes a powerful clamping force at all four wheels.
• Diggers and excavators — hydraulic rams lift heavy buckets of earth.
• Aircraft controls — hydraulics move large wing flaps with precision.
• Garage car lifts and bin lorries — raise huge loads from a small pump.

Liquids are perfect for this because they cannot be compressed, so no force is wasted squashing them.

Try it yourself! 🧪

Build a simple hydraulic machine with syringes.

You need two plastic syringes (one small, one large — no needles), a short length of flexible plastic tubing that fits both, and water.

1. Fill the large syringe and the tubing completely with water, making sure there are no air bubbles (air can be compressed and ruins the effect).
2. Connect the small empty syringe to the other end of the tube, so the system is sealed and full of water.
3. Push the small syringe's plunger. Watch the large syringe's plunger move out — you have transmitted force through the liquid (Pascal's principle in action).
4. Notice that you have to push the small plunger a long way to move the large plunger only a little. That is the force-for-distance trade-off.
5. Try it the other way: push the large plunger and feel how much harder it is to move the small one. The big piston needs a big force to create the same pressure.

You have just built a working hydraulic system and demonstrated both Pascal's principle and force multiplication — the very same physics inside a digger or a car's brakes.