Pressure in Solids and Liquids - Physics for Pilot Assessments

Pressure is what a force becomes once you take account of the area it is spread over, and it is the most useful single idea in this module: it explains why a drawing pin goes in and a thumb does not, why a dam is built thicker at the bottom than at the top, and — in the lessons that follow — why a hydraulic jack works and why anything floats at all. By the end of this lesson you should be able to calculate a pressure from a force and an area and back again, work out the pressure at a given depth in a liquid, and say what that pressure does and does not depend on.

Force spread over an area

Push on something with a given force and what happens depends entirely on how much area you spread that force over. Press on a table with the flat of your hand and nothing happens; press with the same force through a needle and the needle goes in. Nothing about the force changed. What changed is how much material had to resist it: with the flat hand the push is shared out among a great many square millimetres of table, each of which takes a tiny fraction, while under the needle the whole push is concentrated into a fraction of one. Pressure is the name for that share — the force acting perpendicular to a surface, divided by the area it is spread over.

Pressure

p = \frac{F}{A}

$p$: pressure, in pascals (Pa) — one pascal is one newton per square metre
$F$: the force acting perpendicular to the surface, in newtons (N)
$A$: the area the force is spread over, in square metres (m²)

The unit deserves a moment. A pascal is one newton per square metre, and it is a small unit for everyday purposes: a newton is roughly the weight of a small apple, and spreading an apple's weight over a whole square metre is not going to be noticeable. Real pressures are therefore usually large numbers, and the kilopascal — a thousand pascals — is often more convenient. Atmospheric pressure is about 100 000 pascals, or 100 kilopascals, which gives you a yardstick: a few kilopascals is a gentle pressure, and anything in the millions is doing damage. Areas in these items are sometimes quoted in square centimetres, and a square metre is 10 000 square centimetres — a hundred centimetres each way, and a hundred times a hundred is ten thousand — so divide a figure in square centimetres by 10 000 before putting it into the formula.

The two rearrangements come up as often as the original does. Multiply both sides of $\text{pressure} = \frac{\text{force}}{\text{area}}$ by the area: on the right the areas cancel, leaving force = pressure × area. That form is how you get from a pressure back to the total push on a surface, and it is the one hydraulics runs on. Now divide both sides of that new version by the pressure, and the pressures on the right cancel to leave $\text{area} = \frac{\text{force}}{\text{pressure}}$ — the arrangement you want when a question asks how much area is needed to keep a pressure below some limit.

Sharp edges and wide straps

Everything with a working edge is an exercise in making an area small. A knife blade is ground to an edge a fraction of a millimetre wide precisely so that a modest force becomes an enormous pressure along that line, and a blunt knife fails not because you cannot push hard enough but because the same push is now spread over a wider strip. The reverse trick is just as common. A rucksack on thin cords cuts into your shoulders and the same load on wide straps does not; snowshoes stop you sinking into snow your boots go straight through; a tracked vehicle crosses soft ground that would bog a wheeled one; and a bed of nails is survivable only because there are enough of them. In every case the force is fixed and the area is the design variable.

Example 1 — the same force through two areas

You press down with a force of 40 newtons — about the weight of a four-kilogram bag — first through the edge of a knife, whose contact area is 0.000 02 square metres, and then through the flat of your palm, with a contact area of 0.02 square metres. What pressure does each produce?

Pressure is force divided by area, so for the knife the sum is $\frac{40}{0.000\,02}$ .
Clear the decimal by multiplying both numbers by 100 000, which leaves the answer unchanged: $p = \frac{4\,000\,000}{2} = 2\,000\,000\ \text{Pa}$
Now the palm: $p = \frac{40}{0.02}$ Multiply both by 100: $p = \frac{4000}{2} = 2000\ \text{Pa}$
The same 40 newtons produced a thousand times the pressure, because the area was a thousand times smaller. That factor is the entire reason a knife cuts and a palm does not.

Example 2 — working backwards to an area

A machine part must stand on a floor that cannot take more than 50 000 pascals without being damaged. The part weighs 4000 newtons. What is the smallest base area it can have?

Use the arrangement $\text{area} = \frac{\text{force}}{\text{pressure}}$ , which was derived above by dividing both sides of force = pressure × area by the pressure.
So the sum is $\frac{4000}{50\,000}$ . Both numbers divide by 1000, giving $\frac{4}{50}$ .
$\frac{4}{50} = 0.08$ , so the base must be at least 0.08 square metres — 40 centimetres by 20, for instance.
Check by going forwards: $F = 50\,000 \times 0.08 = 4000\ \text{N}$ That is the weight we started with, so the area is right. And note that it is a minimum, not a maximum: a smaller base would concentrate the same weight and push the pressure above the floor's limit.

In a liquid, pressure acts in every direction

A solid transmits a push along the direction you push it. A liquid does not, and that difference is what makes fluids behave so unlike blocks. Because a liquid's particles are free to slide past one another, a liquid cannot hold a sideways shove the way a solid can; what it does instead is press outwards, perpendicular to whatever surface it meets, wherever that surface happens to be. Punch holes around the sides of a can of water and it squirts sideways out of every one of them, not just downwards out of the bottom. That is why pressure at a point in a liquid is a single number with no direction attached to it: it is the same in every direction, and it acts at right angles to any surface you put there — the tank wall, the base, the side of a submerged object, or a diver's eardrum whichever way she turns her head.

The pressure of the fluid above you

So where does that pressure come from? In a liquid open to the air, most of it is simply the weight of the liquid stacked above the point you are interested in. Go deeper and there is more liquid overhead, so more weight to support, so more pressure. That much is intuitive, and it is worth turning into a formula properly rather than leaving it as a slogan, because the derivation is what tells you which quantities matter and which do not.

Imagine a column of liquid standing on a small flat patch of area A at a depth h below the surface. The column's volume is its base area multiplied by its height, so A × h. Its mass is its density multiplied by its volume — that is the arrangement mass = density × volume from the previous lesson — so the mass is ρ × A × h. Its weight is its mass multiplied by g, the gravitational field strength, which is the weight of each kilogram and is about 10 newtons per kilogram on Earth; so the weight pressing down on the patch is ρ × A × h × g. Pressure is that force divided by the area it acts on, so divide by A — and the A cancels, leaving ρgh.

Pressure at a depth in a fluid

p = \rho g h

$p$: the pressure due to the fluid above, in pascals (Pa)
$\rho$: the density of the fluid, in kilograms per cubic metre (kg/m³)
$g$: gravitational field strength, about 10 newtons per kilogram (N/kg) on Earth
$h$: the depth below the surface, in metres (m)

Three things follow from that expression, and all three are examined. It contains the depth, so pressure grows steadily as you go down: twice as deep, twice the pressure from the liquid. It contains the density, so the same depth of mercury produces about 13.6 times the pressure that water does. And the area cancelled out on the way, which means the pressure at a given depth does not depend on how much liquid is sitting there at all.

The deeper the point, the more liquid is stacked above it and the harder the liquid presses. The arrows here show that push acting outwards on the wall; it acts equally in every other direction at the same depth.

Why the container's shape makes no difference

That last consequence is the one people refuse to believe, so it is worth stating bluntly: at the same depth in the same liquid, the pressure is the same in a drinking straw, a bucket, a swimming pool and a lake. A wide container holds far more liquid, but that extra liquid is stacked over a proportionally wider base, and each part of the base only supports the column standing directly above it — which is exactly what the cancelling A was telling you.

Shape does not help either. Whether the container flares outwards, narrows to a neck, or leans over at an angle, the pressure at a point depends only on the vertical distance from the surface down to that point — not on the path the liquid takes to get there, and not on how much of it there is. A tall thin standpipe connected to a wide shallow tank raises the pressure in that tank to whatever its own height dictates, and that is precisely how a header tank in a building supplies pressure to taps several floors below: the pressure at the tap is set by the height of the water above it and by nothing else, which is why the top-floor shower is the weak one.

The atmosphere, and how it is measured

You are at the bottom of an ocean of air, and it presses on you exactly as water would. The same reasoning applies — the weight of the air above you, spread over the area you present to it — but the calculation is not as clean as ρgh, because air is a gas and therefore compressible: it is denser near the ground where it is squeezed by everything above it, and it thins out gradually with height, so there is no sharp surface and no single density to put into the formula. What can be said is the answer. At sea level the atmosphere presses at about 101 000 pascals, near enough 100 kilopascals, in all directions; on a single square metre that is a force of about 100 000 newtons. You do not notice it because it presses equally on every side of you, inside and out.

The instrument that measures it is a barometer, and the simplest version is a glass tube rather more than a metre long, sealed at one end, filled with mercury, and stood upside down with its open end under the surface of a dish of mercury. The mercury runs down the tube a little way and then stops, leaving a vacuum above it and a column standing about 760 millimetres high. Nothing is holding that column up from above — there is nothing above it. What holds it up is atmospheric pressure pushing down on the mercury in the dish and therefore up into the tube, and the column stops falling at the height where its own ρgh exactly matches that atmospheric pressure. So the height of the column is a direct measurement of the pressure of the atmosphere, and when the atmosphere presses harder the column rises.

Nothing but vacuum sits above the column, so nothing is pressing it down from there. The atmosphere pressing on the dish's open surface is what holds it up, and the height at which the column's own ρgh balances that push — 760 millimetres on a standard day — is a direct reading of atmospheric pressure.

Example 5 — checking the barometer against the formula

Confirm that a 760-millimetre column of mercury really does correspond to about 100 kilopascals. Mercury's density is 13 600 kilograms per cubic metre; take g as 10 newtons per kilogram.

The formula needs the height in metres, so convert first: 760 millimetres is $\frac{760}{1000} = 0.76$ metres.
Pressure = density × g × height = 13 600 × 10 × 0.76.
Take it in two steps. 13 600 × 10 = 136 000. Then 136 000 × 0.76 = (136 000 × 0.75) + (136 000 × 0.01) = 102 000 + 1360 = 103 360 pascals.
That is about 103 kilopascals, and the small excess over 100 comes from using 10 for g rather than the true 9.81 — put 9.81 in and it lands almost exactly on 101 kilopascals.
It also shows why mercury is used rather than water: water is 13.6 times less dense, so by the same formula a water barometer would need a column over ten metres tall.

One piece of vocabulary before leaving this. A pressure quoted as an absolute pressure includes the atmosphere; a pressure quoted as a gauge pressure is the amount by which it exceeds atmospheric, and that is what most instruments actually read, because the instrument is itself surrounded by the atmosphere and responds only to the difference. A tyre inflated to 200 kilopascals gauge is at about 300 kilopascals absolute. So when a question says a gauge reads zero, that means atmospheric pressure, not a vacuum.

Why it matters in aviation

Force divided by area is why a large aircraft rolls on so many wheels. Its weight has to reach the ground without the contact patch of any single tyre pressing harder than the pavement can bear, so the load is shared between several wheels on each leg, and the tyres are inflated to high pressures to keep those patches small and firm. That is also why tyre pressures are checked closely rather than casually: an underinflated tyre changes the size of its own contact patch and shifts load onto its neighbours. Aerodrome pavements are themselves rated for the loads they will accept, so an aircraft can be limited by the surface rather than by its own structure.

Pressure differences also do the measuring. The pitot-static system works by comparing the pressure of air brought to rest with the pressure of the undisturbed air alongside the fuselage, and the instruments respond only to that difference. A pressurised cabin is likewise held at a higher pressure than the air outside, and the difference presses outwards on every square metre of skin, so the fuselage carries it as a genuine structural load.

What the test asks

Two traps, and both are about ρgh. The first is the container: an item will offer you two vessels of different widths, or one with an odd tapering shape, and invite you to say that the wider or the fuller one has the greater pressure at the same depth. It does not — only depth and density appear in the formula, and the area cancelled during the derivation. The second is confusing the total force on a surface with the pressure on it. A wide tank base really does carry a far greater total force than a narrow one at the same depth, because force = pressure × area, and an item will use that true statement to make the false claim about pressure look reasonable. Read carefully which of the two the question wants. Expect also a plain p = F/A item where the area is given in square centimetres and the whole trap is failing to convert it.