P = F/A · P = hρg · and what Pascal noticed
Ten newtons against forty thousand, and the pin wins by a factor of ten thousand. Pressure is never the force on its own — it is the force divided by the patch it arrives through, and once that division is in your hand, dams, brakes, syringes and the weather all start reading as the same idea.
This page covers pressure as force divided by area, why depth and not amount decides the pressure in a liquid, what atmospheric pressure is doing to you right now, and how Pascal's law turns a light push into a heavy lift. By the end you'll be able to:
Almost every machine that turns a small effort into a large one runs on this chapter, and so does every ache you feel going up a mountain:
Searched as: pressure formula, p = f/a, pressure in liquids depends on depth, pascal law hydraulic, atmospheric pressure, gauge pressure meaning.
Predict the pressure on the floor, lock it, then shrink the contact area and watch what a fixed weight can do. 🟢 real P = F/A and P = hρg engine
Guess, then lock — the bench waits until you commit.
Set a weight and a contact area, then shrink the area on its own and watch what a fixed force can do.
Three things can be responsible, and the trick is spotting which one is free to change. Six situations — your answers build the table underneath.
A stiletto heel on a wooden floor. What decides the pressure here?
Ask what is being held fixed. Whatever is left free is what the answer turns on.
Area is the one people leave out, and it is the answer three times in six.
A 700 N person, and a target pressure on the floor. Choose the contact area that delivers it — 3 attempts.
Set the area, then stand. P = F/A.
Each claim breaks something. Tap the broken idea.
P = hρg has no volume in it. A narrow tube and a wide lake filled to the same depth press on their bases with exactly the same pressure — which is the classic connected-vessels result.
Pascal's law multiplies the force and divides the distance by exactly the same factor, so the work at each end matches. A jack is a bargain in effort and never in energy.
You never pull the liquid; the atmosphere pushes it up once you lower the pressure above it. Even a perfect vacuum leaves the air only able to support about 10.3 metres of water.
Thinking beats memorising. Sort every claim about pressure into three boxes.
Two situations, one decision. Force is pitted against area until reading the weight alone stops feeling safe.
Divide the force by the area in each case. The larger answer presses harder.
Pick a job, then choose how much bigger the output piston is. Watch what the extra force costs you in distance.
F₂ = F₁ × A₂/A₁, and the distance goes the other way.
Read the scene. All three are about pressure doing something you had blamed on something else.
You lower the pressure inside the straw. The air outside, still at about 101 kPa, pushes down on the rest of the drink and forces it up the low-pressure path. Nothing is pulled, which is why no straw works beyond about ten metres.
P = hρg, so the pressure at the base of a forty-metre dam is about four hundred kilopascals while the pressure at the top is nearly nothing. The wall is built to match that profile, which is why it tapers.
Atmospheric pressure is the weight of the air above you, so climbing removes some of that column. The air trapped behind your eardrum is still at valley pressure and pushes outwards until a small tube lets it equalise — which is the pop.
In plain terms: a force spread over a wide patch is gentle, and the same force concentrated into a narrow one is violent. Everything else in this chapter is that sentence applied to liquids and gases.
Pressure is force divided by area, and the second number is the one people drop. A 700 newton person lying flat puts under one kilopascal on the floor; the same person on one stiletto heel puts about seven thousand. Nothing about the person changed. This is why knives are sharpened, why snowshoes work, and why a camel has wide soft feet.
P = hρg contains the depth, the density and gravity — and no volume, no width, no shape. A thin tube of water and an entire reservoir filled to the same level press on their bases identically. It also acts equally in every direction, which is why water squirts sideways out of a hole and why a submarine is squeezed from all sides at once.
Press on a trapped liquid and the pressure rises everywhere in it at once, undiminished. Put a small piston at one end and a large one at the other and the same pressure delivers a much larger force at the wide end. That is a jack, a brake and a press — and the price is that the wide end moves correspondingly less, so no energy is created.
A tyre gauge reads gauge pressure, the amount above atmospheric, which is why a flat tyre reads zero rather than 101 kPa. Absolute pressure adds the atmosphere back in, and exam questions swing on which one is wanted. A suction pump cannot lift water past about 10.3 metres, because the atmosphere is doing the pushing and that is all it has. And for a gas rather than a liquid, squeezing the volume raises the pressure — Boyle's law.
Exams reward the method. Work it one step at a time — read the thought, predict the line, then reveal it.
Three small labs, each one native to this idea.
Six wrong ideas, most of them from reading the force and forgetting the area. Tap a card to bust the myth.
A sealed tube of mercury stood upside down in a dish settled at about 76 cm, with an empty space above it. The atmosphere was holding it up, and the first barometer had been built.
Pascal had his brother-in-law carry a barometer up the Puy de Dôme. The mercury column fell as he climbed, proving the air above was doing the pushing.
Pressure applied to a trapped liquid is passed undiminished to every part of it. Every hydraulic machine since rests on that one sentence.
Otto von Guericke pumped the air from two metal hemispheres and two teams of horses failed to pull them apart. Nothing held them but the atmosphere.
One pascal is one newton per square metre — a tiny amount, which is why almost every real pressure is quoted in kilopascals or bars.
Built to a profile that matches hρg — almost nothing at the top, hundreds of kilopascals at the base, and the thickness follows.
A light push on the pedal becomes a heavy clamp at four wheels, because the brake fluid passes the pressure on undiminished.
Every ten metres adds another atmosphere, which is why depth limits exist and why ascending too fast is dangerous.
One sentence each: what is pressing, and on what.
Drag the slider. Six pressures, from a hundredth of a pascal to three hundred and sixty gigapascals.
Sixteen orders of magnitude, and the same division underneath every one of them.
Each one is a build and a measurement — the measuring is what turns a demo into a project.
Build: a tall plastic bottle with holes punched at three heights, filled with water over a tray.
Measure: how far each jet lands from the base. The lowest hole throws furthest, and the ratio follows the depth.
Build: stand on squared paper and draw round both shoes, then count the squares.
Measure: your weight in newtons divided by that area. Repeat on one foot and on tiptoe.
Build: a little water boiled in an empty drinks can, then inverted into cold water.
Measure: the can's dimensions, and work out the force the atmosphere applied to collapse it.
Build: a thin and a thick syringe joined by tubing, filled with water and no air bubbles.
Measure: the force and the distance at each end. Multiply them and compare the two products.
Build: a long clear tube of water inverted into a bowl, raised as high as you safely can.
Measure: the column height and predict it from Pₐₜₘ/ρg. Then work out why mercury needs only 76 cm.
Build: a sealed syringe with known volumes, loaded with measured masses on the plunger.
Measure: volume against total pressure, then plot P against 1/V. A straight line, and its gradient gives you the atmosphere.
Straight answers, each one complete on its own:
Because pressure is force divided by area, and the pin's point is about forty million times smaller than the elephant's feet. Ten newtons through a hundredth of a square millimetre works out at roughly a gigapascal, while forty thousand newtons through four thousand square centimetres is about a hundred kilopascals. The pin wins by a factor of around ten thousand.
No. Pressure in a liquid is hρg, which contains the depth, the density and gravity and nothing else. A narrow tube and a wide reservoir filled to the same level press on their bases with exactly the same pressure, which is why water finds the same level in every arm of a set of connected vessels whatever their shapes.
Because you never pull the liquid at all. Sucking lowers the pressure inside the straw, and the atmosphere pushes down on the rest of the drink to force it up. The atmosphere has only about 101 kPa to give, which will support roughly 10.3 metres of water and no more, however perfect the vacuum you create.
Because it pushes on every side of you at once, and the fluids inside your body push back at the same pressure, so nothing is being squeezed in any particular direction. You notice it only when the balance breaks — climbing a mountain, taking off in an aircraft, or watching a can collapse when the air is pumped out of it.
Gauge pressure is how much a pressure exceeds the atmospheric pressure around it, and it is what a tyre gauge or a blood pressure cuff reads. Absolute pressure adds the atmosphere back in. A tyre reading 220 kPa on the gauge is really at about 321 kPa absolute, and exam questions often turn on which of the two is being asked for.
No. It multiplies the force by the ratio of the piston areas and divides the distance moved by exactly the same ratio, so the work done at each end is identical apart from friction. Pressing the small piston through twenty centimetres to raise the large one by two millimetres is the honest trade, and there is no way around it.
Because mercury is about 13.6 times denser than water, and the height a column reaches is P/ρg. The same atmospheric pressure that supports 10.3 metres of water supports only 0.76 metres of mercury. That is the whole reason Torricelli used mercury: a water barometer would have to be taller than a house.
No. At any given depth it acts equally in every direction — down, sideways and up. That is why water squirts horizontally from a hole in the side of a bucket, why a submarine is squeezed from all sides, and why an object submerged in water feels a net upward push from the pressure being larger on its underside than on its top.
Spot these before the exam does.
Seven question formats, the way LLOS.ai serves them — multiple choice, multiple-correct, fill-in-the-blank, match, sequence, read-think-connect, and write-your-own. Every question has layered hints: a quick nudge, the reasoning, then a deeper connection — so a wrong answer opens a door, never a dead end. 🟢 received from a board-tagged question bank · seed toward 2,000
Pick your board — the set re-tunes to its wording and emphasis. Competitive draws the JEE / NEET / Olympiad lane.
Four things, from a pin to an elephant. Tap them in order, hardest press first. Then read the arithmetic — and see what fooled you.
Four tonnes standing on four broad feet, about four thousand square centimetres in all.
A sixty-kilogram person with their weight momentarily on one stiletto heel.
An eighty-kilogram person in walking boots, both feet flat on the ground.
A thumb pushing a drawing pin into a board with about ten newtons.
Tap the four in the order you believe, hardest press on the ground first.
Ten newtons and forty thousand, and the ten wins. Once you are dividing by the area without being told to, the rest of fluids follows almost by itself.