Pressure, density and Pascal's law

Pressure

Press a drawing pin into a board: the same push that hurts your thumb on the flat end drives the point into wood. What matters is the force on each unit of area:

The unit is the pascal, $1\ \text{Pa} = 1\ \text{N/m}^2$. A fluid (a liquid or a gas) pushes at right angles to every surface it touches, so pressure has no direction of its own.

Density is mass per unit volume, $\rho = \dfrac{m}{V}$: water $1000\ \text{kg/m}^3$, sea water $1030\ \text{kg/m}^3$, mercury $13\,600\ \text{kg/m}^3$.

Pressure grows with depth

A column of liquid of height $h$ presses on its base with its weight, $\rho Ahg$, spread over the area $A$. So below the surface

where $p_0$ is the air pressure on the surface. $\rho gh$ is the gauge pressure, the part above the air's.

You should find that $10\ \text{m}$ of water adds about one atmosphere, so a diver $10\ \text{m}$ down feels about two.

The pressure is the same at every point at one depth in a connected liquid, whatever the shape of the vessel. Water in a narrow pipe and a wide tank joined at the bottom stands at the same level.

Watch out. The pressure depends on the depth, not on how much liquid there is. A tall thin tube of water can press harder on its base than a wide shallow pond.

Pascal's law and the hydraulic lift

Push on a liquid in a closed vessel, and the extra pressure appears everywhere in it, equally. That is Pascal's law.

In a hydraulic lift, a small force $F_1$ on a small piston $A_1$ makes a pressure $\dfrac{F_1}{A_1}$, which acts on the large piston too:

The force is multiplied, but not the energy: the small piston must move much farther than the large one rises.

A worked example

A jack has pistons $4.0\ \text{cm}$ and $36\ \text{cm}$ across. What force on the small piston holds up a $1500\ \text{kg}$ car?

1. Area ratio. $\left(\dfrac{36\ \text{cm}}{4.0\ \text{cm}}\right)^2 = 81$.
2. Force. $\dfrac{1500\ \text{kg} \times 9.8\ \text{m/s}^2}{81} = 182\ \text{N}$.
3. Distance. Pushing the small piston down $40\ \text{cm}$ raises the car $\dfrac{40\ \text{cm}}{81} = 0.49\ \text{cm}$.

Think first. Work in equals work out: $182\ \text{N} \times 0.40\ \text{m} \approx 14\,700\ \text{N} \times 0.0049\ \text{m}$.

Practise with new numbers

Try it in Python

The program works out the jack of the example.

Remember these

• $p = \dfrac{F}{A}$, in pascal; fluids push at right angles to every surface.
• $p = p_0 + \rho gh$; the gauge pressure is $\rho gh$.
• Same depth, same pressure, whatever the shape.
• Pascal's law: $F_2 = F_1\dfrac{A_2}{A_1}$; force is multiplied, energy is not.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. What is the gauge pressure $25\ \text{m}$ down in fresh water?

2. Why do your ears hurt at the bottom of a deep pool, but not at the top?

3. A lift's pistons have areas in the ratio $1 : 40$. What force on the small one holds up $8000\ \text{N}$?

4. Does a hydraulic lift give you energy for free?