What you'll learn
Pressure and hydraulics explains why a sharp knife cuts more easily than a blunt one, why a dam is built thicker at its base, why your ears hurt when diving, and how a car weighing a tonne can be lifted by hand. The unifying idea is that pressure is force spread over an area, and that in a fluid this pressure acts in every direction and is transmitted undiminished throughout. That last property is what makes hydraulic machines possible and gives the topic its practical importance. By the end of this guide you should be able to calculate pressure in solids and in liquids, explain the factors affecting liquid pressure, state and apply Pascal's principle to hydraulic systems, describe atmospheric pressure and how it is measured, apply Archimedes' principle and the law of flotation, and use manometers and barometers.
Key terms and definitions
Pressure — force acting normally per unit area, measured in pascals
Pascal — one newton per square metre, the SI unit of pressure
Fluid — a substance that can flow, meaning a liquid or a gas
Pascal's principle — pressure applied to an enclosed fluid is transmitted equally and undiminished throughout it
Hydraulic system — a machine using an enclosed liquid to transmit and multiply force
Atmospheric pressure — the pressure exerted by the weight of the air above, about 100,000 pascals at sea level
Barometer — an instrument for measuring atmospheric pressure
Manometer — a U-tube containing liquid, used to measure the pressure of a gas supply
Upthrust — the upward force exerted by a fluid on an object immersed in it
Archimedes' principle — the upthrust on an immersed object equals the weight of fluid it displaces
Law of flotation — a floating body displaces its own weight of fluid
Density — mass per unit volume, measured in kilograms per cubic metre
Core concepts
Pressure in solids
Pressure is the force acting normally, meaning perpendicularly, per unit area. It equals force divided by area, with force in newtons and area in square metres, giving pressure in pascals.
The relationship shows that pressure increases when the force increases or when the area decreases. This explains a large number of everyday observations.
A sharp knife has a very small blade area, so even a modest force produces a very large pressure, which cuts. A drawing pin has a broad head to reduce pressure on the thumb and a fine point to concentrate pressure where it enters the wood.
Conversely, a large area reduces pressure. Tractor tyres are wide so the vehicle does not sink into soft ground, camels have broad feet for walking on sand, and the foundations of a building are wider than its walls so that the ground is not crushed.
Unit conversion causes many errors here. If dimensions are given in centimetres, convert to metres before calculating area: one square centimetre is 0.0001 square metres.
Pressure in liquids
Pressure in a liquid has three properties that must be known.
It acts equally in all directions at a given depth, not merely downwards. This is why a hole in the side of a container produces a horizontal jet.
It increases with depth, because a greater weight of liquid lies above. This is why a dam is built thicker at its base and why a hole lower down a container produces a jet travelling further.
It depends on the density of the liquid, so a denser liquid exerts more pressure at the same depth.
It does not depend on the shape or the cross-sectional area of the container. Vessels of very different shapes connected at the base fill to the same level, because the pressure at the base depends only on the vertical depth.
The pressure due to a column of liquid equals the density multiplied by the acceleration due to gravity multiplied by the depth. With density in kilograms per cubic metre, gravitational field strength as 9.8 or 10 newtons per kilogram, and depth in metres, the answer is in pascals.
The total pressure at a depth in an open container is the liquid pressure plus the atmospheric pressure acting on the surface, and questions often specify which is required.
Pascal's principle and hydraulics
Pascal's principle states that pressure applied to an enclosed fluid is transmitted equally and undiminished to all parts of the fluid and to the walls of the container.
This is the basis of the hydraulic press and the hydraulic brake. A small force applied to a small piston creates a pressure, which is transmitted through the liquid to a large piston. Because the pressure is the same at both pistons but the area of the second is much larger, the force produced there is much larger.
The relationship follows directly: since pressure equals force divided by area and the pressure is equal at both pistons, the force on the first divided by its area equals the force on the second divided by its area.
The force multiplication equals the ratio of the areas. If the large piston has ten times the area of the small one, the output force is ten times the input force.
Nothing is gained for nothing: the small piston must move a correspondingly greater distance, because the volume of liquid displaced is the same at both ends. Energy is conserved, and a question asking whether a hydraulic press creates energy should be answered by pointing out the distance trade-off.
Hydraulic systems use liquids rather than gases because liquids are virtually incompressible, so the applied pressure is transmitted immediately and fully rather than being absorbed in compressing the fluid. This is the standard explanation for why brake systems use oil and why air in a brake line makes braking feel spongy.
Atmospheric pressure
The atmosphere exerts pressure because of the weight of the air above. At sea level this is about 100,000 pascals, and it decreases with altitude as less air lies above.
Everyday demonstrations include drinking through a straw, where reducing the pressure inside the straw allows atmospheric pressure to push the liquid up; a rubber suction cup, held by atmospheric pressure after the air beneath is expelled; and a collapsing can, crushed when steam inside condenses and the internal pressure falls.
A simple mercury barometer consists of a tube more than 760 millimetres long, filled with mercury and inverted into a mercury reservoir. The mercury falls until the pressure at the base of the column balances atmospheric pressure, leaving a vacuum above. At standard atmospheric pressure the column stands about 760 millimetres high.
Mercury is used rather than water because of its high density. A water barometer would need a column over ten metres tall to balance the same pressure, which is impractical.
The space above the mercury is a vacuum, so no gas pressure acts downwards on the column, and tilting the tube does not change the vertical height of the column, only its length.
Manometers
A manometer is a U-tube containing a liquid, used to measure the pressure of a gas supply.
One arm connects to the gas supply and the other is open to the atmosphere. The difference in the liquid levels between the two arms gives the pressure difference between the gas and the atmosphere.
The gas pressure equals atmospheric pressure plus the pressure due to the height difference, which is density times gravitational field strength times the difference in height. If the level is higher on the gas side, the gas pressure is below atmospheric and the height term is subtracted.
Upthrust and Archimedes' principle
Any object immersed in a fluid experiences an upward force called upthrust. It arises because pressure increases with depth, so the pressure on the bottom face of the object is greater than on the top face, giving a net upward force.
Archimedes' principle states that the upthrust on an object immersed in a fluid equals the weight of the fluid displaced by the object.
An object appears lighter in water because the upthrust acts upwards against the weight. The apparent weight equals the true weight minus the upthrust, and the difference gives the upthrust directly — which is how the principle is verified experimentally using a spring balance and a displacement can.
Flotation
The law of flotation states that a floating body displaces its own weight of fluid.
Whether an object floats or sinks depends on the comparison between its weight and the maximum upthrust available. An object floats if its average density is less than that of the fluid, and sinks if it is greater. If the densities are equal, the object remains suspended wherever it is placed.
This explains why a steel ship floats although steel is far denser than water: the hull encloses a large volume of air, so the average density of the ship as a whole is less than that of water. It also explains why a ship floats higher in sea water than in fresh water, since sea water is denser and a smaller volume needs to be displaced to provide the same upthrust.
A submarine controls its depth by taking water into ballast tanks to increase its average density and sink, and expelling it to rise. A hydrometer floats at a depth determined by the density of the liquid and is used to measure that density.
Worked examples
Example 1: Pressure in a solid (3 marks)
A box of weight 480 newtons rests on a floor. Its base measures 40 centimetres by 30 centimetres. Calculate the pressure it exerts.
First convert the dimensions to metres: 0.40 metres by 0.30 metres.
The area is 0.40 × 0.30 = 0.12 square metres.
The pressure is force divided by area, which is 480 ÷ 0.12 = 4,000 pascals. Note that failing to convert from centimetres would give an answer 10,000 times too small.
Example 2: A hydraulic press (4 marks)
In a hydraulic press, the small piston has an area of 0.004 square metres and the large piston an area of 0.10 square metres. A force of 60 newtons is applied to the small piston. Calculate the force produced at the large piston.
The pressure created at the small piston is force divided by area: 60 ÷ 0.004 = 15,000 pascals.
By Pascal's principle this pressure is transmitted undiminished throughout the liquid, so the same pressure of 15,000 pascals acts on the large piston.
The force at the large piston is pressure multiplied by area: 15,000 × 0.10 = 1,500 newtons.
As a check, the area ratio is 0.10 ÷ 0.004 = 25, and 60 × 25 = 1,500 newtons, which agrees.
Example 3: Applying Archimedes' principle (4 marks)
A metal block weighs 8.0 newtons in air and 6.5 newtons when fully immersed in water. Calculate the upthrust and the weight of water displaced.
The upthrust is the difference between the weight in air and the apparent weight in water, which is 8.0 − 6.5 = 1.5 newtons.
By Archimedes' principle, the upthrust equals the weight of the fluid displaced, so the weight of water displaced is also 1.5 newtons.
Since the block sinks, its weight exceeds the maximum upthrust available, which confirms that its average density is greater than that of water.
Common mistakes and how to avoid them
The most frequent error is failing to convert areas from square centimetres to square metres, which makes the calculated pressure wrong by a factor of 10,000.
Students often state that pressure in a liquid depends on the volume of liquid or the shape of the container. It depends only on depth, density and gravitational field strength.
Another common slip is claiming that a hydraulic press creates energy because the output force exceeds the input. Energy is conserved; the small piston moves a proportionally greater distance.
In barometer questions, many candidates say mercury is used because it is a liquid metal. The reason is its high density, which keeps the column to a practical height.
Finally, candidates frequently say a ship floats because it is hollow. The precise reason is that enclosing air lowers the ship's average density below that of water.
Exam technique for "Pressure and Hydraulics"
Convert every length to metres before calculating an area, and write the conversion down so it can be credited.
For hydraulic problems, calculate the pressure at the input piston first and then apply it to the output piston. The two-step method is clearer than using the ratio directly and earns the method marks.
When a question involves liquid pressure, check whether atmospheric pressure should be included. The words total pressure signal that it should.
For upthrust questions, remember that the difference between weight in air and apparent weight in liquid gives the upthrust directly, which by Archimedes' principle is also the weight of liquid displaced.
Explain floating and sinking in terms of average density compared with the fluid, and support it by referring to the weight of fluid displaced.
Quick revision summary
Pressure is force per unit area in pascals, so a small area gives a high pressure, which explains knives and drawing pins, while a large area reduces pressure, which explains wide tyres and foundations. In a liquid, pressure acts equally in all directions, increases with depth and density, and is independent of container shape, with pressure equal to density times gravitational field strength times depth. Pascal's principle states that pressure in an enclosed fluid is transmitted equally throughout, which allows hydraulic systems to multiply force in proportion to the ratio of piston areas, at the cost of the small piston moving further, and liquids are used because they are virtually incompressible. Atmospheric pressure is about 100,000 pascals at sea level, falls with altitude, and is measured by a mercury barometer standing about 760 millimetres high, mercury being chosen for its high density. A manometer measures gas pressure from the difference in liquid levels. Upthrust arises because pressure is greater on the lower face, and by Archimedes' principle it equals the weight of fluid displaced. A floating body displaces its own weight of fluid, and an object floats when its average density is less than that of the fluid.