Pressure · Quantitative
Pressure = force ÷ area
A drawing pin is squeezed between one finger and one thumb. The two forces are equal — they have to be. One end goes into the wood and the other does not go into your thumb.
Start here
Press the flat head into your thumb. Now turn it round.
Hold a drawing pin between finger and thumb and squeeze, gently. The head does nothing to your thumb. The point, under the same squeeze, goes straight into the wood.
Your finger and your thumb push on the pin with the same force. So why does only one end go in?
Nothing about the force is different. What is different is the area each end acts on. The head spreads its force across a few square millimetres of thumb; the point concentrates the same force onto a tiny fraction of one. Force divided by the area it is spread over has its own name and its own unit: that is pressure.
Pressure says how concentrated a force is on the surface it meets: the force acting at right angles to that surface, divided by the area it is spread over. It is measured in pascals, Pa, and one pascal is one newton spread over one square metre — 1 Pa = 1 N/m². Pressure acts at right angles to whatever surface it meets, whichever way that surface happens to face.
At the bench · one block, three faces, a tray of sand
Same weight. Different face. Different hole in the sand.
Change a control to begin
The block measures 0.20 m by 0.10 m by 0.05 m, so it has three different faces to stand on. This sand gives way at 6000 Pa. Choose a face. Choose the mass resting on it.
Commit first. You stand the same block on its smallest face instead of its largest one. The weight has not changed. What happens to the pressure under it?
Face on the sand
4 kg
Weight pressing down
—
Area under it
—
Pressure
—
The sand
—
Writing it down · the shape of this relationship
Pressure = force ÷ area
N with m² gives Pa · never N with cm²
The triangle
Cover the one you want
F = P × A
P = F ÷ A
A = F ÷ P
Two things side by side means multiply. One thing over another means divide.
F · force, at right angles to the surface · N
P · pressure · Pa
A · area it acts on · m²
Worked example · one step at a time
A stack of bricks presses down with 24 N on a base of 0.008 m². What is the pressure on the ground?
Step 0 of 5
Convert
24 N stays 24 N · 0.008 m² stays 0.008 m²
The force is already in newtons and the area already in square metres, so there is nothing to convert.
Formula
pressure = force ÷ area
The force is the one at right angles to the ground, which here is the weight.
Insert
pressure = 24 N ÷ 0.008 m²
The area is the part actually touching, not the size of the whole stack.
Fine-tune
24 ÷ 0.008 = 3000
Newtons divided by square metres leaves newtons per square metre.
Answer
pressure = 3000 Pa
Three thousand pascals, because 1 Pa is 1 N spread over 1 m².
Worked example · one step at a time
A brick presses down with 30 N on a face of 200 cm². What is the pressure?
Step 0 of 5
Convert
200 cm² ÷ 10 000 = 0.0200 m²
A pascal is a newton per square metre, and there are 10 000 square centimetres in a square metre.
Formula
pressure = force ÷ area
Force shared out over the area it presses on.
Insert
pressure = 30 N ÷ 0.0200 m²
The converted area goes in. The 200 never does.
Fine-tune
30 ÷ 0.0200 = 1500
Newtons divided by square metres leaves newtons per square metre.
Answer
pressure = 1500 Pa
Insert 200 instead of 0.0200 and the answer comes out 0.15 Pa — ten thousand times too small.
Your turn · the same five steps
Your block: 40 N standing on 0.020 m².
Write all five lines before you check. The numbers are the ones your own bench is showing.
The five lines, marked
Convert
40 N stays 40 N · 0.020 m² stays 0.020 m²
The weight is already in newtons and the face is already in square metres, so there is nothing to convert.
Formula
pressure = force ÷ area
The product force = pressure × area, rearranged for the quantity you want.
Insert
pressure = 40 N ÷ 0.020 m²
The weight is 4 kg × 10 N/kg; the area is the face you chose.
Fine-tune
40 ÷ 0.020 = 2000
Newtons divided by square metres leaves newtons per square metre.
Answer
pressure = 2000 Pa
Under the 6000 Pa this sand gives way at, so the surface holds.
The five lines give 2000 Pa, and the footprint on the bench is drawn 0.20 m × 0.10 m to match.
A tin of paint presses down with 12 N on a base of 50 cm². What is the pressure on the shelf?
This one needs the Convert line to do some work.
The five lines, marked
Convert
50 cm² ÷ 10 000 = 0.0050 m²
A pascal needs square metres, and there are 10 000 square centimetres in one.
Formula
pressure = force ÷ area
Force shared out over the area it presses on.
Insert
pressure = 12 N ÷ 0.0050 m²
The converted area goes in. The 50 never does.
Fine-tune
12 ÷ 0.0050 = 2400
Newtons divided by square metres leaves newtons per square metre.
Answer
pressure = 2400 Pa
Insert 50 instead of 0.0050 and the answer comes out 0.24 Pa.
The five lines give 2400 Pa. The whole question turned on the first one.
Key fact
Pressure is the force acting at right angles to a surface divided by the area it acts on, measured in pascals: 1 Pa is 1 N spread over 1 m². Put the same force on a quarter of the area and the pressure is four times as big — which is what a point, a blade and a stiletto heel are all for.
Think again
“A sharp point pushes harder than a blunt one.”
It does not push harder at all. Put a force meter behind a sharp pin and a blunt one and press each into a board until it stops: the meter reads the same. Sharpening something changes no force anywhere — it changes the area that force has to act through, and pressure is force divided by area. That is also why sharpening a knife makes cutting easier without making you stronger, and why a blunt knife needs you to lean on it: you are having to supply extra force to make up for the extra area.
“Pressure pushes downwards.”
Downwards is only where this lesson's examples happen to point, because a weight on sand is the easiest case to draw. Pressure acts at right angles to whatever surface it meets, whichever way that surface faces: a drawing pin pressed sideways into a noticeboard presses sideways, water presses outwards on the walls of a tank as well as down on its base, and the air presses on every side of your body at once. The rule is at right angles to the surface, not towards the floor.
Mastery ladder
Not started yet.
Rungs 3 and 4 you mark yourself.
Rung 1 · Calculate
A crate presses down on the floor with its weight of 600 N. Its base measures 0.30 m². What is the pressure on the floor?
Rung 2 · The one that catches people
Two students both weigh 500 N. One is in flat boots, touching the floor over 0.025 m². The other is in heels, touching it over 0.0005 m². Which statement is right?
Rung 3 · Explain
A drawing pin has a wide flat head and a very sharp point. Explain why it is made that way, using the words force, area and pressure.
Rung 4 · Take it somewhere new
A tracked machine weighs 12 000 N and must not press on soft ground with more than 30 000 Pa. Work out the smallest total track area it can have, and explain why the same machine on wheels touching half that area would sink where the tracks do not.
Key note
Pressure is the force acting at right angles to a surface divided by the area it is spread over: pressure in pascals = force in newtons ÷ area in square metres, and 1 Pa = 1 N/m². The same force on a smaller area gives a higher pressure, and on a larger area a lower one. Pressure is not a force and is not measured in newtons, and it acts at right angles to whichever surface it meets.
Going further
Once you can see pressure as force over area, a lot of design stops looking arbitrary. Anything meant to go into something else concentrates a force into almost no area: nails, needles, studs, chisels, teeth, claws. Anything meant to stay on top of something soft spreads it: snowshoes, tractor tyres, tank tracks, the flat splayed feet of a camel, and the wide concrete footings under a building, which exist only to hand the whole weight of the house to the ground over enough square metres that the ground can take it. Both are the same equation, read in opposite directions.
Liquids give the idea a second life. Because a liquid cannot be squashed much, pressure applied at one place is felt everywhere in it, so a small force on a small piston can hold up a car on a big one — the pressure is the same in both cylinders, and the big piston simply has more square metres for it to act on. That is a hydraulic jack, and it is worth being careful about what it does and does not give you: the big piston pushes with more force, but it moves a much shorter distance, and the two multiply out to the same energy transferred. Force can be multiplied. Energy cannot, which is the same trade you get from a long spanner in the lesson on moments.
Before this lesson
Connects to
At GCSE this becomes
- Pressure in pascals and kilopascals, pressure in fluids and its increase with depth, upthrust and floating, and pressure–volume work on gases.
Where to next
Ask Mr Badmus AI
Got a pressure of your own to work out — a shoe, a tyre, a knife?
Press the pin against your fingertip firmly, but never hard enough to break the skin. You are feeling the difference, not testing how much you can stand.
The sand tray is a teaching model. Its giving-way pressure is fixed at 6000 Pa so that failure is something you can reach; real ground varies with grain size, packing and how wet it is, and gives way gradually rather than at one number. The mass you set is the whole mass resting on the sand, block included, and weight is taken as mass in kilograms × 10 N/kg. The block is drawn to scale in cross-section; the weight arrow uses its own separate scale, and the depth it sinks is drawn as a fixed amount rather than calculated.
Lesson content © MrBadmusAI.