Why skis stop you sinking, why divers feel more pressure with depth, and why steel ships float: pressure, hydraulics and Archimedes' principle, explained with diagrams and worked examples.
A person can stand on thin ice if they lie flat, but may fall through if they stand up. A sharp knife cuts bread easily while a blunt one only squashes it. A huge steel ship floats while a small steel nail sinks. All three puzzles are answered by one idea: what matters is not just the size of a force, but how that force is spread out. This article explains pressure in solids, liquids and gases, shows why the sea gets more crushing with depth, and uses Archimedes' principle to explain why things float or sink.
Pressure is the force acting on each unit of area, perpendicular to a surface.
p = F ÷ A (pressure = force ÷ area)
The SI unit is the pascal (Pa): 1 Pa = 1 N/m². Because this is a very small pressure, kilopascals are common: 1 kPa = 1000 Pa. Another unit you may meet is the bar (1 bar = 100 000 Pa).
A student of mass 60 kg stands on both feet; the soles of her shoes touch the ground over a total area of 0.040 m². Find the pressure on the ground. (g = 9.8 N/kg)
Step 1. Weight (the force on the ground) = mg = 60 × 9.8 = 588 N.
Step 2. p = F ÷ A = 588 ÷ 0.040 = 14 700 Pa ≈ 15 kPa.
Check: on one foot (half the area) the pressure doubles, to about 29 kPa.
For a fixed force, a smaller area gives a larger pressure, and a larger area gives a smaller pressure.
You press down on a knife with 20 N. The cutting edge touches the food over an area of 1.0 mm² (1.0 × 10⁻⁶ m²). Find the pressure.
p = F ÷ A = 20 ÷ (1.0 × 10⁻⁶) = 2.0 × 10⁷ Pa (20 MPa).
This is more than a thousand times the pressure of the student standing on the ground, from a force of only 20 N. A blunt edge has a much larger area, so the same force gives far less pressure.
A liquid exerts pressure because of its weight and because its particles collide with surfaces. Several facts hold for a liquid at rest:
p = ρ × g × h
ρ = density of the liquid (kg/m³), g = gravitational field strength (9.8 N/kg), h = depth below the surface (m). This gives the pressure due to the liquid alone (called gauge pressure). Fresh water: ρ ≈ 1000 kg/m³; sea water: ρ ≈ 1025 kg/m³.
(a) Find the pressure due to the water on a swimmer 10 m below the surface of a freshwater lake. (b) Find the pressure on a diver 30 m deep in the sea. (g = 9.8 N/kg)
(a) p = ρgh = 1000 × 9.8 × 10 = 98 000 Pa ≈ 98 kPa. The air above adds about 101 kPa, so the total (absolute) pressure is about 199 kPa, nearly twice normal air pressure.
(b) p = 1025 × 9.8 × 30 = 301 350 Pa ≈ 301 kPa, roughly three atmospheres from the water alone.
As a useful rule, every 10 m of depth in water adds about one atmosphere of pressure. This is why deep divers need special training and equipment, and why submarines have strong hulls.
We live at the bottom of an "ocean" of air. The weight of the air above us creates atmospheric pressure. At sea level the standard value is 101 325 Pa (about 101 kPa, or 1 atmosphere). It does not crush us because the fluids inside our bodies push outwards with the same pressure.
In 1643 the Italian scientist Evangelista Torricelli filled a glass tube with mercury and inverted it in a dish of mercury. The mercury fell to a height of about 760 mm and stayed there, leaving an almost empty space above it. The weight of the column of mercury is balanced by the atmospheric pressure pushing on the surface of the dish. This is the first mercury barometer.
Mercury has a density of about 13 600 kg/m³. Calculate the pressure of a 0.760 m column of mercury.
p = ρgh = 13 600 × 9.8 × 0.760 ≈ 101 300 Pa, matching atmospheric pressure at sea level.
A water barometer would need a tube over 10 m tall, which is why mercury is used.
Atmospheric pressure explains many everyday things. When you drink through a straw, you lower the pressure inside the straw, and the higher pressure of the atmosphere pushes the liquid up. A rubber suction cup sticks because the air is pushed out and the outside air presses it on. In the 1650s Otto von Guericke showed the strength of atmospheric pressure with his famous Magdeburg hemispheres: after the air was pumped out of two joined metal hemispheres, teams of horses could not pull them apart.
Atmospheric pressure decreases with height, because there is less air above. It falls to about half of its sea-level value at around 5.5 km, and to roughly a third at the summit of Mount Everest (8.85 km). This is why mountaineers need extra oxygen at great heights, why aircraft cabins are pressurised, and why your ears "pop" when you climb or descend. Weather forecasters also watch changes in air pressure: falling pressure often brings clouds and rain, while rising pressure often brings fair weather.
Liquids are almost incompressible. Pascal's principle says that a pressure applied to a confined fluid is transmitted undiminished to every part of the fluid and to the walls of its container. A hydraulic system uses this to multiply force. A small force on a small piston creates a pressure; the same pressure acts on a large piston and gives a large force.
F₁ ÷ A₁ = F₂ ÷ A₂ (same pressure in both cylinders)
A force of 50 N pushes a small piston of area 4.0 cm². The large piston has an area of 200 cm². What force acts on the large piston, and how far does it move when the small piston moves 25 cm?
Step 1. F₂ = F₁ × (A₂ ÷ A₁) = 50 × (200 ÷ 4.0) = 2500 N.
Step 2. The liquid volume moved is the same on both sides, so the large piston moves 25 cm ÷ 50 = 0.50 cm.
The force is multiplied by 50, but the distance is divided by 50. No energy is created: work in equals work out (50 N × 0.25 m = 12.5 J; 2500 N × 0.0050 m = 12.5 J). Car brakes, garage jacks, diggers and aircraft control systems all use hydraulics.
When an object is put in a fluid, the fluid pushes it upwards with a force called upthrust (or buoyant force). It exists because pressure increases with depth: the bottom of the object is deeper than the top, so the upward push on the bottom is bigger than the downward push on the top.
Archimedes' principle states that the upthrust on an object in a fluid equals the weight of the fluid it displaces. The ancient Greek scientist Archimedes of Syracuse (about 287–212 BCE) studied this idea. A popular story, recorded centuries later by the Roman writer Vitruvius, says he discovered it in a bath while checking whether a crown was pure gold. Historians treat the tale with caution, but the principle itself is correct and is used every day.
upthrust = ρfluid × g × Vdisplaced
Vdisplaced is the volume of fluid pushed out of the way, which equals the volume of the object that is under the surface.
A block measures 0.20 m × 0.10 m × 0.050 m and is fully under water. Find the upthrust. If the block has a mass of 0.50 kg, will it float? (ρwater = 1000 kg/m³)
Step 1. Volume = 0.20 × 0.10 × 0.050 = 1.0 × 10⁻³ m³.
Step 2. Upthrust = 1000 × 9.8 × 1.0 × 10⁻³ = 9.8 N.
Step 3. Weight = mg = 0.50 × 9.8 = 4.9 N. The upthrust (9.8 N) is greater than the weight (4.9 N), so the block floats.
Step 4. Its density is 0.50 ÷ 1.0 × 10⁻³ = 500 kg/m³, which is half the density of water, so it floats with half of its volume under the surface.
Compare the weight W of the object with the largest upthrust U it could get (when fully under the surface):
This explains why a steel ship floats: its hull is hollow, so it displaces a huge volume of water. The ship's average density, counting all the air inside, is less than that of water. A solid steel nail, with an average density about eight times that of water, sinks.
A ship has a total mass of 5.0 × 10⁶ kg. What volume of water must it displace to float in (a) fresh water (1000 kg/m³) and (b) sea water (1025 kg/m³)?
A floating ship's weight equals the weight of the water it displaces, so the displaced water has mass 5.0 × 10⁶ kg.
(a) V = m ÷ ρ = 5.0 × 10⁶ ÷ 1000 = 5000 m³.
(b) V = 5.0 × 10⁶ ÷ 1025 ≈ 4880 m³.
The ship needs to displace less in the denser sea water, so it floats slightly higher there.
Try each question before opening the answer.
p = F ÷ A = 500 ÷ 0.25 = 2000 Pa.
Skis have a large area, so the skier's weight is spread out and the pressure on the snow is small.
p = ρgh = 1000 × 9.8 × 5.0 = 49 000 Pa.
Pressure in water increases with depth, so the water pushes harder on the lower part of the wall, which needs to be stronger.
F₂ = 30 × (100 ÷ 2.0) = 1500 N.
U = ρgV = 1000 × 9.8 × 0.0030 = 29.4 N.
The ship's hull encloses a large volume of air, so it displaces a large volume of water and its average density is below that of water. The nail is solid steel with a density far greater than water.
There is less air above, so atmospheric pressure is lower and supports a shorter column of mercury.
Safe classroom activities: press a pencil against the skin with the point and the blunt end to feel the difference in pressure; use a plastic bottle with three holes at different heights (cover them first, then uncover) to show the water jets leaving with different strengths; use a syringe to show how hydraulic systems transmit a push; weigh an object in air and then in water with a newton meter and compare the loss with the weight of displaced water in a eureka can; test floating with a lump of plasticine formed into a ball and into a boat. Keep water spills away from electrical sockets. Numerical values are rounded; please check that they match the data booklet used by your examination board.
This article is original EDUSAMBAM writing. Definitions, constants and historical facts were checked against the sources below. Diagrams are original and were not copied from any source.
Note on the numbers: densities and pressure values are rounded standard figures. Please check them against your own syllabus or exam board specification before relying on them.
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