An elephant and a woman in high heels: who sinks deeper into soft ground? It may be the woman. The elephant weighs fifty times more, but its weight spreads over four broad feet. Hers is concentrated on the thin tip of a heel.

What sinks into the ground is not force but pressure: the force divided by the area it rests on.

$$p=\frac FA.$$

The unit of pressure is the pascal: $1\ \text{Pa}=1\ \text{N/m}^2$, a force of one newton spread over one square metre. It is a tiny pressure; at sea level the atmosphere exerts about $100{,}000$ Pa.

The treatise on liquids

Around 1653 Pascal wrote the Treatise on the equilibrium of liquids, published in 1663, after his death. In it is the rule we now call Pascal’s principle:

An increase in pressure applied at one point of a still, enclosed liquid is transmitted equally to every point of the liquid and to the walls of the container.

Squeeze a tube of toothpaste in the middle and the paste comes out of the nozzle: the pressure of your finger got there. Fill a balloon with water, prick it in several places and squeeze: water spurts out of every hole, in every direction.

The barrel that burst

The story goes that Pascal gave a famous demonstration. He took a barrel full of water, tightly closed, and fitted a thin, very tall tube into its lid, about 10 metres high. He climbed onto a balcony and poured water into the tube. With little more than a litre, the barrel burst.

We do not know whether the scene happened exactly like that, but the numbers show that it could have. The pressure at the bottom of a column of liquid of height $h$ is

$$p=\rho gh,$$

where $\rho$ is the density of the liquid and $g\approx9.8\ \text{m/s}^2$. For 10 metres of water, with $\rho=1000\ \text{kg/m}^3$,

$$p=1000\cdot9.8\cdot10=98{,}000\ \text{Pa}.$$

What matters is the height of the column, not the amount of water. If the tube has a cross-section of $1\ \text{cm}^2$, 10 metres of water is only $1$ litre. But the $98{,}000$ Pa is transmitted to the whole lid of the barrel. If the lid has an area of $0.2\ \text{m}^2$, the force on it is

$$F=pA=98{,}000\cdot0.2=19{,}600\ \text{N},$$

the weight of two tonnes. One litre of water in a thin tube pushes on the lid like a car standing on it.

The hydraulic press

Pascal realized he could use the principle to multiply forces. Imagine two cylinders full of water, joined at the bottom, each closed by a piston. One is narrow, the other wide. Pushing the narrow piston with a force $F_1$ increases the pressure by $F_1/A_1$, and that same increase reaches the wide piston:

$$\frac{F_1}{A_1}=\frac{F_2}{A_2}\quad\Longrightarrow\quad F_2=F_1\cdot\frac{A_2}{A_1}.$$

A man pushing the small piston will match the force of a hundred men pushing the piston a hundred times wider.

— Blaise Pascal, Treatise on the equilibrium of liquids

If the small area is $2\ \text{cm}^2$ and the large one is $8\ \text{cm}^2$, the ratio is $4$. A force of $50$ N on the small piston becomes $200$ N on the large one. This is how the hydraulic jack that lifts a car works, as do the brakes that stop a car with the pressure of one foot, and excavators.

Ideal model: pistons at the same height, small area of 2 cm² and an incompressible fluid.

Change the force and the ratio between the areas. The force on the large piston increases, but its travel shrinks in the same proportion.

There is no free lunch

If the press multiplies the force by 4, is it creating energy? No. When the small piston goes down a distance $d_1$, it pushes a volume $A_1d_1$ of water. That same volume goes into the wide cylinder, which rises only $d_2$:

$$A_1d_1=A_2d_2\quad\Longrightarrow\quad d_2=d_1\cdot\frac{A_1}{A_2}.$$

The force is 4 times larger, but the large piston moves 4 times less. The work, force times distance, is the same on both sides:

$$F_1d_1=F_2d_2.$$

To lift a car 10 cm with a jack, you pump many times, pushing lightly, over a long distance. Pascal had already noticed this: the press is a lever made of water.

Exercise: the area ratio is 5 and the small piston goes down 10 cm. How far does the large one rise?

The volumes are equal, so $d_2=\dfrac{10}{5}=2$ cm. In exchange, the force on the large piston is 5 times greater.

Exercise: what is the water pressure 3 metres down in a swimming pool?

The increase in pressure caused by the water is $\rho gh=1000\cdot9.8\cdot3=29{,}400$ Pa, about 30% of atmospheric pressure. Adding the atmosphere, the total pressure is close to $130{,}000$ Pa. That is what you feel in your ears when you dive.

Sources: Pascal biography, MacTutor · Traités de l’équilibre des liqueurs, on Wikisource (French).

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