💦 Pressure due to the liquid
If the question asks for the pressure difference, or the pressure 'due to the water', use p = h × ρ × g and stop there.
Density and Pressure » Pressure in Liquids and Gases
What you'll learn this session
Study time: 30 minutes
Edexcel spec: 5.6, 5.7
- Why the pressure at a point in a liquid or gas at rest acts equally in all directions
- How to use pressure difference = height × density × gravitational field strength
- Why pressure gets bigger as you go deeper, and bigger again in a denser liquid
- When to add atmospheric pressure to find the total pressure
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Unlock This CoursePressure in every direction
A solid block only presses down on the surface it rests on. Liquids and gases are different. They can flow, so they press on everything they touch: the bottom of a container, its sides, and any object placed inside them.
The key idea is this: the pressure at a point in a gas or liquid at rest acts equally in all directions. It pushes up, down and sideways with the same size. 'At rest' simply means the liquid or gas is still, not flowing along.
Remember, a fluid is a liquid or a gas. Everything in this lesson is true for both.
Three things you can picture show that pressure acts in all directions.
💧 Holes in a bottle
Make a small hole in the side of a plastic bottle full of water. The water spurts out sideways, at right angles to the wall. Water has weight that pulls it down, yet it pushes sideways too.
🎈 A round balloon
The air inside a balloon pushes outwards on every part of the rubber with the same pressure. It does not just push on the bottom. That is why the balloon is stretched evenly and stays round.
🌊 A diver
A diver under water feels the water pressing on the chest, the back, the sides and the ears all at once. Turning around makes no difference, because the pressure is the same in every direction.
Deeper means more pressure
Key terms:
- Depth: the vertical distance below the surface of a liquid.
Think about a point under water. All the water above that point has weight, and that weight presses down. Go deeper and there is more water above you, so the pressure is bigger.
You can see this with the bottle. To draw it, sketch a tall bottle full of water with three small holes in its side: one near the top, one in the middle and one near the bottom. Draw a jet of water leaving each hole. The jet from the top hole is weak. The jet from the bottom hole comes out fastest, because the pressure is greatest at the bottom.
The same idea explains the shape of a dam. A dam holds back deep water. The water pressure on the wall is small near the surface and very large at the bottom, so the wall is built much thicker at the bottom than at the top.
The pressure difference equation
pressure difference = height × density × gravitational field strength
p = h × ρ × g
This equation is not given in the exam. You must learn it.
| Quantity | Symbol | Unit |
|---|---|---|
| pressure difference | p | pascal (Pa) |
| height | h | metre (m) |
| density | ρ | kilogram per metre cubed (kg/m3) |
| gravitational field strength | g | newton per kilogram (N/kg) |
The equation gives the difference in pressure between two points in a fluid, where h is the vertical height between them. If one point is the surface of a liquid, then h is the depth, and p is the pressure caused by the liquid alone.
Use g = 10 N/kg unless the question gives another value. Remember, the density of water is 1000 kg/m3.
The equation tells you three things:
- Double the depth and the pressure from the liquid doubles.
- A denser liquid gives more pressure at the same depth. Sea water (about 1030 kg/m3) gives a little more pressure than fresh water.
- Only h, ρ and g matter. The width and shape of the container do not appear in the equation. The pressure 2 m down in a narrow pipe is the same as the pressure 2 m down in a wide lake of the same water.
Worked example 1: a lake
Calculate the pressure difference between the surface of a freshwater lake and a point 12 m below the surface.
p = h × ρ × g
p = 12 × 1000 × 10
p = 120 000 Pa
Worked example 2: height in centimetres
A vase holds water to a depth of 15 cm. Calculate the pressure of the water on the bottom of the vase.
h = 15 ÷ 100 = 0.15 m
p = h × ρ × g = 0.15 × 1000 × 10
p = 1500 Pa
Total pressure: adding the atmosphere
The air around us is a fluid too, and it has weight. The air above us presses on everything at the Earth's surface. This is called atmospheric pressure, and at sea level it is about 100 000 Pa. A question will normally tell you the value to use.
The atmosphere presses on the surface of a lake or the sea, so a diver has both the water and the air pressing down from above.
➕ Total pressure
If the question asks for the total pressure, work out h × ρ × g and then add atmospheric pressure.
Worked example 3: total pressure on a diver
A diver is 25 m below the surface of the sea. The density of sea water is 1030 kg/m3. Atmospheric pressure is 100 000 Pa. Calculate the total pressure on the diver.
pressure due to the water: p = h × ρ × g = 25 × 1030 × 10 = 257 500 Pa
total pressure = 257 500 + 100 000 = 357 500 Pa
Rearranging the equation
You can rearrange the equation to find a height or a density:
- height = pressure difference ÷ (density × gravitational field strength), h = p ÷ (ρ × g)
- density = pressure difference ÷ (height × gravitational field strength), ρ = p ÷ (h × g)
Worked example 4: finding the depth
The oil in a storage tank causes a pressure of 16 000 Pa at the bottom of the tank. The density of the oil is 800 kg/m3. Calculate the depth of the oil.
h = p ÷ (ρ × g)
h = 16 000 ÷ (800 × 10) = 16 000 ÷ 8000
h = 2.0 m
The equation works for gases too
Air is much less dense than water, about 1.2 kg/m3 near the ground, so the pressure changes far more slowly with height. Go up and there is less air above you, so the pressure falls. This is why atmospheric pressure is lower at the top of a mountain than at sea level.
Worked example 5: up a tall building
A lift rises 50 m inside a building. The density of air is 1.2 kg/m3. Calculate the change in air pressure.
p = h × ρ × g = 50 × 1.2 × 10
p = 600 Pa, so the air pressure at the top is 600 Pa lower than at the bottom.
Common mistakes
- Leaving the height in centimetres: change it to metres first by dividing by 100.
- Using a density in g/cm3: it must be in kg/m3 to get an answer in pascals.
- Adding atmospheric pressure when the question only asks for the pressure difference, or forgetting it when the question asks for the total pressure.
- Saying that water pressure only pushes downwards: it acts equally in all directions.
- Thinking a wider tank has more pressure at the bottom: width does not matter, only depth and density do.
Exam-style question
A research submarine is at rest 150 m below the surface of the sea.
(a) State the equation linking pressure difference, height, density and gravitational field strength. [1 mark]
(b) The density of sea water is 1030 kg/m3. Calculate the pressure difference between the surface of the sea and the submarine. Use g = 10 N/kg. [2 marks]
(c) Atmospheric pressure is 100 000 Pa. Calculate the total pressure on the submarine. [1 mark]
(d) The hull of the submarine is made equally strong on its top, bottom and sides. Explain why. [2 marks]
(e) The submarine moves to the same depth in a freshwater lake. Explain how the pressure due to the water changes. [2 marks]
Model answer
(a) pressure difference = height × density × gravitational field strength, or p = h × ρ × g (1).
(b) p = 150 × 1030 × 10 (1) = 1 545 000 Pa (1).
(c) total pressure = 1 545 000 + 100 000 = 1 645 000 Pa (1).
(d) The pressure at a point in a liquid acts equally in all directions (1), so the water pushes just as hard on the sides and bottom of the hull as on the top (1).
(e) Fresh water has a lower density than sea water (1). The height and g are the same, so the pressure due to the water is smaller (1).
Exam tip
Read the wording of the pressure question twice. 'Pressure difference' or 'pressure due to the water' means h × ρ × g only. 'Total pressure' means you must add the atmospheric pressure given in the question. Write the equation in words or symbols first, because it often earns a mark of its own.