Pressure · printable packet

Pressure is force divided by area; the same force presses harder on a small area and more gently on a large one.

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1. What you will learn

By the end of this lesson you will be able to find a pressure, a force or an area from the other two, and explain why area matters.

2. What you already have

You can find forces in newtons, including weight, and you know how to find the area of a rectangle. This lesson combines force and area into pressure, which explains why the same weight can sink into snow or rest on top of it.

3. Words for this lesson

4. Force spread over area

Rearranged, $F = PA$ finds a force from a pressure, and $A = F/P$ finds the area needed.

Term	What it means
Pressure	Force spread over an area, $P = F/A$.
Pascal	The unit of pressure: one newton per square meter, Pa.
Kilopascal	A thousand pascals, kPa.
Area	The size of a surface, in square meters, m².
Contact area	The part of a surface actually touching another.
Atmospheric pressure	The air's pressure around us, about $101$ kPa at sea level.

Another way: picture

Picture pressing a thumbtack into a bulletin board. Your thumb pushes on the wide, flat head, spreading the force so it does not hurt. The same force comes out at the sharp point, a tiny area, where the pressure is huge and the tack slides into the cork.

Another way: steps

Find the force, often a weight, $mg$.
Find the contact area in square meters.
Divide the force by the area.
Write the unit, pascals.
Check: a smaller area must give a larger pressure.

5. The unit tells you what to do

Newtons per square meter means how many newtons press on each square meter. If a box weighing $500$ N rests on $0.25$ square meters, each square meter carries $500 \div 0.25 = 2000$ newtons, a pressure of $2000$ Pa.

As with density, saying the unit aloud keeps the division the right way around. Newtons come first, so the force goes on top.

6. Small areas, large pressures

A sharp knife has a very thin edge, so a modest push gives an enormous pressure right at the edge, enough to cut through food. A dull knife spreads the same push over a wider edge, and the pressure is too small to cut.

Nails, needles, thumbtacks and ice skates all work the same way. A small contact area turns an ordinary force into a large pressure.

7. Large areas, small pressures

Snowshoes spread a hiker's weight over a big area, so the pressure on the snow is small and the hiker stays on top instead of sinking. Tractors have wide tires for the same reason, so they do not sink into soft fields.

Building foundations spread a building's weight over a large area of ground, and a backpack's wide straps spread its weight over your shoulders so they do not dig in.

8. Finding a force from a pressure

Rearranging gives $F = PA$. If air in a tire pushes with $220$ kPa on a patch of $0.015$ square meters, the force is $220000 \times 0.015 = 3300$ newtons, the share of the car's weight that tire carries.

This is how engineers size hydraulic lifts, dams and aquarium windows: they know the pressure and the area, and multiply to find the force the structure must hold.

9. Converting areas

Areas are often measured in square centimeters, but pascals need square meters. There are ten thousand square centimeters in a square meter, since a meter is a hundred centimeters and area is length times length.

So $50$ cm² is $0.005$ m². Forgetting this conversion gives pressures ten thousand times too small, one of the most common errors in pressure problems.

10. The method, step by step, and how to check it

Checking an answer. A standing person presses on the floor with about ten to twenty kilopascals. A sharp point gives millions of pascals. An answer wildly outside these ranges usually means a unit slip.

11. Why each step is allowed

Pressure is defined as force per unit area, so dividing force by area is simply the definition. When a force is spread evenly, each square meter carries the same share, and that share is the pressure.

Whether something breaks, dents or sinks depends on pressure, not on force alone. That is why the same weight can crack ice under a skate blade yet rest safely on ice under a wide sled.

12. Pressure in the air

The air around us presses on everything with about $101$ kPa, over a hundred thousand newtons on every square meter. We do not feel it because it pushes equally from all sides, and our bodies push back from inside.

Weather forecasts report air pressure because it changes with the weather. Falling pressure often means a storm is coming; rising pressure, fair weather.

13. Pressure in tires and balls

Car tires, bicycle tires and basketballs are filled with air at a pressure above the air outside. In the United States, tire pressure is usually given in pounds per square inch; thirty-two psi is about two hundred twenty kilopascals.

A properly inflated tire keeps a small contact patch on the road. An underinflated tire flattens out, touching more road, which increases friction and wastes fuel.

14. Standing and walking

Standing on both feet, a person spreads their weight over both soles. Standing on one foot halves the area and doubles the pressure. Walking, each step briefly puts your whole weight on one heel, a smaller area still.

That is why high heels can dent soft wooden floors. A narrow heel concentrates the weight onto a tiny area, producing a pressure far greater than a flat shoe does.

15. Common slips

Three errors are common. Multiplying force by area instead of dividing gives a quantity with the wrong unit. Using mass in kilograms instead of weight in newtons leaves out $g$.

The third is forgetting to convert square centimeters to square meters, which makes the pressure far too small. Writing units at every step catches all three.

16. Pressure in liquids

Liquids press on everything in them, and the pressure grows with depth, because deeper water carries the weight of more water above it. Swimmers feel this as a squeeze in their ears at the bottom of a pool.

Divers and submarines must deal with enormous pressures deep in the ocean. Later physics courses show how to calculate pressure at any depth from the weight of the water above.

17. Hydraulic machines

Pressure applied to a trapped liquid spreads through it evenly. A small push on a small piston creates a pressure that acts on a large piston too, where the larger area turns it into a much larger force.

Car lifts, dentist chairs and car brakes all use this trick. A modest push on a brake pedal becomes a force large enough to stop a moving car.

18. Estimating pressures

A few benchmarks help. A book lying on a table presses with a few hundred pascals. A person standing presses with about fifteen thousand. A car tire holds about two hundred thousand. The point of a thumbtack pressed by a thumb reaches hundreds of millions.

Comparing an answer with these benchmarks shows quickly whether the area was converted correctly.

19. Pressure and damage

Whether a surface dents or breaks depends on pressure. Wooden floors, ice and soft ground can each withstand only so much before giving way. Engineers design floors, roads and bridges to handle the largest pressures expected.

Heavy trucks have many wheels to spread their weight, keeping the pressure on the road low enough to avoid damage. Highway rules limit the weight on each axle for exactly this reason.

20. Pounds per square inch

In the United States, many gauges read pressure in pounds per square inch, written psi. A bicycle tire might be pumped to sixty psi, a car tire to thirty-two, and a basketball to about eight. One psi is about six thousand nine hundred pascals, or almost seven kilopascals.

To convert a tire gauge reading to kilopascals, multiply the psi value by about 6.9. A car tire at thirty-two psi holds about two hundred twenty kilopascals more than the air outside. Scientists use pascals because they fit neatly with newtons and meters, while psi remains common on gas station air pumps.

21. Why we do not feel the air

The air presses on your hand with about a hundred thousand newtons on every square meter, yet your hand does not feel crushed. The reason is that the air presses equally on every side, top and bottom, left and right, and the fluids inside your body push outward just as hard.

You notice air pressure only when it changes quickly, such as when your ears pop in an airplane or an elevator in a tall building. The pressure outside changes faster than the air trapped inside your ears can adjust, and the difference pushes on your eardrums.

22. Designing with pressure

Engineers design with pressure all the time. The legs of a heavy piano sit on wide cups so they do not dent the floor. A crane spreads its weight on large pads before lifting. Tent stakes are sharpened so they slide easily into the ground.

23. In the world: skating at Rockefeller Center

Each winter, skaters glide on the famous rink at Rockefeller Center in New York City. A skate blade touches the ice along a thin edge, an area of only a few square centimeters, so even a light skater presses on the ice with well over a million pascals.

That concentrated pressure lets the blade bite into the ice, giving the grip skaters need to push off and turn. A skater standing in ordinary boots, touching the ice over a much larger area, presses with a pressure thousands of times smaller and slips instead of gliding. Blades are sharpened regularly to keep the edge thin and the pressure high.

24. In the world: snowshoes in Yellowstone

In winter, deep, soft snow covers much of Yellowstone National Park. A hiker in ordinary boots sinks up to the knees with each step, because their weight on the small area of the boot soles makes a pressure the fluffy snow cannot hold.

Snowshoes spread the same weight over an area about ten times larger, cutting the pressure to about a tenth. The snow can support that, so the hiker stays near the surface. Wildlife uses the same trick: the snowshoe hare has broad, furry feet, while heavier animals like moose, with narrow hooves, struggle in deep snow.

25. A bigger force does not always mean more pressure

It is natural to think a heavier object always presses harder. But pressure depends on area as well as force. A heavy elephant with wide feet can press on the ground with less pressure than a light person in high heels.

A related error is to think a sharp knife cuts because you push harder. You push with the same force; the thin edge makes the area tiny, so the pressure becomes large enough to cut.

26. A box on the floor

27. A thumbtack

28. Crossing a frozen pond

29. Your turn: a $600$ N person stands on an area of $0.04$ m². What pressure do they exert?

30. Guided practice

A force of $1500$ N presses evenly on an area of $0.5$ m². What is the pressure, in Pa?

31. Guided practice

Complete the worked solution: a $45$ kg hiker stands on snow. Both boots together touch an area of $0.03$ m², and a pair of snowshoes covers $0.3$ m². With $g = 9.8$ N/kg, find the pressure in Pa standing on both boots, on one boot, and on the snowshoes.

32. Guided practice

33. Practice

A brick weighing $24$ N can rest on a large face of area $0.024$ m² or a small end of area $0.006$ m². Fill in the pressure on the ground in Pa lying flat, the pressure standing on its end in Pa, and how many times larger the second pressure is.

34. Practice

A person weighing $1200$ N stands on a surface, touching it over an area $A$, in square meters. Write the pressure on the surface, in Pa, as a function of $A$.

35. Practice

A car tire is inflated to $200$ kPa above the air outside, and the patch touching the road has an area of $0.018$ m². About how much force, in N, does that tire carry?

36. Somewhere new

Picture a hiker in boots on the Appalachian Trail, with a mass of $80$ kg resting on a contact area of about $0.04$ m². With $g = 9.8$ N/kg, what pressure does the person put on the surface, in kPa?

37. Lesson test

Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.

38. Test question

A person weighing $900$ N stands on a surface, touching it over an area $A$, in square meters. Write the pressure on the surface, in Pa, as a function of $A$.

39. What you can do now

You can use pressure. Explain to someone why snowshoes keep a hiker on top of deep snow.

Working for the steps left to you

	value
pressure lying flat (Pa)
pressure on its end (Pa)
times larger

29. Your turn: a $600$ N person stands on an area of $0.04$ m². What pressure do they exert?, step 3

	force divided by the area it acts on	one newton on each square meter	a tiny area, so a large pressure	a large area, so a small pressure
pressure
one pascal
a sharp knife edge
snowshoes