Pressure, and Pressure in Liquids

What this lesson covers in the specification

Incandio is aligned to this specification. It is not published by, endorsed by or affiliated with Pearson, and it reproduces none of Pearson's wording — the statement numbers are given so you can check every lesson against your own copy.

• 5.5 — Use pressure = force ÷ area
• 5.6 — Pressure at a point in a gas or liquid at rest acts equally in all directions
• 5.7 — Use pressure difference = height × density × gravitational field strength

Specification section 5(b) Density and pressure — also taught in Density, Particles and Changes of State, Specific Heat Capacity.

1 · Understand it

No exam language yet. The only question this section answers is: do I actually understand what is happening?

Press a drawing pin into a board and the same force is applied at both ends — your thumb on the head, the point on the wood. Your thumb is unharmed and the point goes in. What differs is the AREA over which that force is spread, and PRESSURE is the force divided by that area: p = F ÷ A. The unit is the PASCAL, and one pascal is one newton per square metre.

The consequences are everywhere once you look. A sharp knife cuts because a thin blade concentrates the force onto a very small area. Snowshoes and tractor tyres work the other way, spreading a weight over a large area so the pressure on the snow or soil is small enough not to sink in. A camel's foot, an elephant's foot and a stiletto heel are the same physics with three different design goals — and a stiletto heel can exert a greater pressure on a floor than an elephant does, because the area is so much smaller even though the force is far less. Watch the units when you compare them: a square centimetre is a ten-thousandth of a square metre, so an area quoted in square centimetres must be divided by 10 000 before it goes anywhere near the equation.

Pressure, and the area needed to reduce it

(a) A box weighing 600 N rests on a base of area 0.40 m². Find the pressure on the floor. (b) What base area would be needed to reduce the pressure to 500 Pa?

1. (a) p = F ÷ A = 600 ÷ 0.40 = 1500 Pa.
2. (b) Rearrange: A = F ÷ p = 600 ÷ 500 = 1.2 m².
3. Tripling the area divides the pressure by three, because the same weight is spread over three times as much floor.

Answer: 1500 Pa, and 1.2 m² to bring it down to 500 Pa.

Now statement 5.6, which sounds like a footnote and is the most important sentence on the page. At any point in a fluid at rest — a liquid or a gas — the pressure acts EQUALLY IN ALL DIRECTIONS. Not downwards, as intuition insists, but in every direction at once: up, down, sideways and every angle between.

Think of it like being in the middle of a crowd rather than under a pile of books

A stack of books presses down on the one at the bottom and nowhere else — that is a solid, and pressure in a solid does act in the direction of the force. Now imagine standing in the middle of a packed crowd. You are not being pressed only from above; you are squeezed from the front, the back, both sides and every diagonal at once, because the people around you are free to move and are pressing in from wherever there is space. A fluid is that crowd. Its particles move in every direction, so they strike a surface from every direction, and the pressure at any point pushes outward equally on all sides. That is why a submarine is crushed from all around rather than flattened downwards, and why a balloon under water shrinks evenly instead of becoming a pancake.

What follows from pressure acting equally in all directions

1. A hole anywhere in a container leaks — in the side, the bottom or a pipe leading upwards — because the pressure pushes outward on every surface.
2. UPTHRUST exists. The pressure on the bottom of a submerged object is greater than on its top, because the bottom is deeper, and that difference is a net upward force.
3. A HYDRAULIC system works: a pressure applied at one piston is transmitted through the fluid and acts on a second piston of larger area, giving a larger force there.
4. A liquid finds its own level, and pressure at a given depth is the same everywhere in a connected body of fluid regardless of the shape of the container.
5. The pressure of the atmosphere presses on you from all sides, which is why you are not crushed by it — it acts inwards from every direction and is balanced from within.

The pressure in a liquid increases with DEPTH, and the equation for the difference is p = h × ρ × g — the height difference, the density of the liquid, and the gravitational field strength. Three things follow, and all are examined. Deeper means greater pressure, so a dam is built thicker at the bottom. A denser liquid gives a greater pressure at the same depth, which is why mercury barometers are short and water ones would have to be ten metres tall. And the pressure does not depend on the SHAPE or the total amount of liquid at all — only on the depth below the surface.

Pressure at depth

Find the pressure due to the water at a depth of 12 m in a lake. Take the density of water as 1000 kg/m³ and g as 10 N/kg.

1. p = h × ρ × g.
2. = 12 × 1000 × 10.
3. = 120 000 Pa, which is 120 kPa.
4. That is roughly the same as atmospheric pressure at the surface, so a diver 12 m down experiences about twice the pressure they do on land — once from the water and once from the air above it.

Answer: 120 000 Pa from the water alone, and about double atmospheric pressure in total.

That last observation is worth holding on to, because questions often distinguish between the pressure DUE TO THE LIQUID and the TOTAL pressure. The equation p = hρg gives only the pressure produced by the column of liquid. If a question asks for the total pressure on a diver or on a submarine hull, atmospheric pressure — about 100 000 Pa — has to be added, because the air is sitting on top of the water and pressing down on it.

2 · Grade 9 Notes

A different job from the section above. You have already understood it; this is the precise set of things to LEARN — definitions to reproduce word for word, processes in order, equations with units, and the answers that score full marks.

p = F ÷ A

p is pressure in pascals (Pa), F is the force in newtons (N), A is the area in square metres (m²)

Units: Pa, N, m². Rearranged: F = p × A and A = F ÷ p. One pascal is one newton per square metre.

p = h × ρ × g

p is the pressure difference in pascals (Pa), h is the height difference in metres (m), ρ is the density of the liquid in kg/m³, g is gravitational field strength in N/kg

Units: Pa, m, kg/m³, N/kg. This gives the pressure due to the LIQUID only — add atmospheric pressure for the total.

Learn this definition · Pressure

The force acting per unit area, at right angles to a surface. It is measured in pascals, and one pascal is one newton per square metre.

Statement 5.6 — what follows from pressure acting equally in all directions

• A hole ANYWHERE in a container leaks — the side and the bottom alike
• UPTHRUST exists, because the pressure on the bottom of an object is greater than on its top
• HYDRAULIC systems work: pressure applied at one piston acts on another of larger area, giving a larger force
• Pressure at a given depth is the SAME everywhere in a connected liquid, whatever the container's shape
• You are not crushed by the atmosphere because it presses inwards from every direction and is balanced from within

What the pressure in a liquid does and does not depend on

• DEPENDS ON DEPTH — deeper means greater pressure, which is why a dam is thicker at the bottom
• DEPENDS ON DENSITY — a denser liquid gives a greater pressure at the same depth
• DEPENDS ON g — the same depth on the Moon would give a much smaller pressure
• DOES NOT DEPEND on the shape of the container or on how much liquid there is in total
• p = hρg gives the pressure from the LIQUID only; add atmospheric pressure for the total

Pressure in a solid against pressure in a fluid

A. IN A SOLID the force is transmitted in the direction it is applied — a stack of books presses down on the one below and nowhere else.

B. IN A FLUID the pressure at a point acts EQUALLY IN ALL DIRECTIONS, because the particles move in every direction and strike surfaces from every side.

Model answer [3 marks]

A block of weight 480 N rests on a surface of area 0.60 m². Calculate the pressure it exerts. [3]

Using p = F ÷ A, the pressure is 480 ÷ 0.60 = 800 Pa.

Model answer [4 marks]

Explain why a dam holding back a deep lake is built much thicker at the bottom than at the top. [4]

The pressure in a liquid increases with depth, because the pressure due to the liquid is given by p = h × ρ × g and h is the depth below the surface. The water at the bottom of the lake is therefore at a much greater pressure than the water near the surface. Because pressure in a fluid acts equally in all directions, that pressure pushes horizontally against the dam wall as well as downwards, and the horizontal force on the wall is greatest at the bottom. The dam is built thicker there so that it is strong enough to withstand the greater force.

Not this: Pressure in a liquid acts downwards, because that is the direction of the weight of the water above.

This: At any point in a fluid at rest the pressure acts EQUALLY IN ALL DIRECTIONS — up, down and sideways. This is why a hole in the side of a tank leaks, why upthrust exists, and why a dam has to resist a horizontal push.

Mark-losing trap. Pressure in a fluid acts EQUALLY IN ALL DIRECTIONS. This single sentence explains upthrust and hydraulics.

Mark-losing trap. p = hρg depends on DEPTH, not on the shape of the container or the total amount of liquid.

Mark-losing trap. p = hρg gives the pressure from the LIQUID only. Add atmospheric pressure if the total is wanted.

Mark-losing trap. Area must be in SQUARE METRES. A cm² is a ten-thousandth of a m², not a hundredth.

3 · Prove it — the five questions

The five questions climb Grade 6 → Grade 7 → Grade 8 → Grade 9 → Grade 9 challenge, and are marked inside Incandio on your own device, by rule, with an authored diagnosis of the mistake you actually made. The mark schemes stay in the app so that the practice is worth doing; the questions themselves are here.

1. Grade 6 · Calculate [2 marks] — A force of 200 N acts on an area of 0.50 m². Calculate the pressure.
2. Grade 7 · State [2 marks] — At a point in a liquid at rest, in which direction does the pressure act?
3. Grade 8 · Calculate [3 marks] — Calculate the pressure due to the water at a depth of 25 m. The density of water is 1000 kg/m³ and g is 10 N/kg.
4. Grade 9 · Explain [5 marks] — Select every statement that belongs in a full-mark explanation of why a dam holding back a deep reservoir is built much thicker at the bottom than at the top.
5. 9+ · Analyse [6 marks] — A student says: 'A stiletto heel cannot possibly exert more pressure on a floor than an elephant's foot — an elephant weighs thousands of times more than a person.' Select every statement that belongs in a full-mark analysis.

The people behind this science

Two ways into the same idea — the one who worked out why anything floats, and the one who found the limit that revealed the atmosphere has weight. Inside Incandio each of them answers knowing exactly which lesson you have just finished.

Archimedes — the one who worked out why anything floats

Upthrust is the consequence of everything on this page, and On Floating Bodies is where it was first established: a body immersed in a fluid is pushed up by a force equal to the weight of the fluid it displaces. That follows directly from pressure increasing with depth and acting in all directions — the pressure on the bottom of an object exceeds the pressure on its top, and the difference is the upthrust. He proved it from stated assumptions rather than noticing it, which is why the result has survived unchanged for over two thousand years.

• “Why does a heavy ship float when a small stone sinks?”
• “What exactly is pushing an immersed object upwards?”
• “How did you prove the floating result rather than just observing it?”
• “What happens to the water when an object is put into it?”
• “Does the shape of an object matter, or only how much it displaces?”

Galileo Galilei — the one who found the limit that revealed the atmosphere has weight

Florentine well-diggers had found that no suction pump, however good, could raise water more than about ten metres, and Galileo took the problem seriously enough to write about it near the end of his life. The explanation is on this page: a pump does not pull water up, it removes the air above so that atmospheric pressure pushes the water up — and atmospheric pressure can only support a column of about ten metres of water. Torricelli, his student, followed the reasoning through and built the first barometer, using mercury precisely because its far greater density makes the column short enough to fit in a room.

• “Why can no pump lift water higher than about ten metres?”
• “What is actually pushing the water up a pipe?”
• “How would you show that the air itself has weight?”
• “Why would mercury give a much shorter column than water?”
• “What did the well-diggers of Florence teach you?”

Carry on through the course

• Previous lesson: Density
• Next lesson: Particles and Changes of State
• Density — Not how heavy something is, but how much of it is packed into the space it occupies.
• Particles and Changes of State — What the particles are doing in each state — and why a boiling pan stays at 100 °C however hard you heat it.
• Specific Heat Capacity — Why the sand burns your feet and the sea is freezing on the same afternoon.
• Gas Pressure and Absolute Zero — Where the pressure of a gas actually comes from, and why the temperature scale has to start somewhere else.
• The Kinetic Theory of Gases — One picture — fast random molecules bouncing off walls — that predicts everything a gas does.
• All of Physics · Physics key terms · required practicals · equations · Incandio Science · The 4PH1 specification