11.2 Pressure in a Liquid at Rest

A free GCSE Physics lesson for AQA (8463) from OBA School: how liquid pressure depends on depth and density, and (Higher tier) p = h ρ g. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with every video, practice quizzes and printable worksheets with mark schemes, in our free online course.

What you will learn

• Explain why the pressure in a liquid increases with depth
• Explain why the pressure is the same at all points at the same depth in a liquid at rest
• Explain why a denser liquid gives a greater pressure at the same depth
• (Higher tier) use p = h ρ g to calculate the pressure due to a column of liquid, and rearrange it
• Apply these ideas to dams, divers, water towers and tanks

Key words

• Fluid: A liquid or a gas – a substance that can flow.
• Depth: The vertical distance of a point below the surface of a liquid.
• Column of liquid: The liquid directly above a point. Its weight causes the pressure at that point.
• Density (ρ): Mass per unit volume, in kg/m³. ρ is the Greek letter rho.
• Gravitational field strength (g): The weight of each kilogram: 9.8 N/kg on Earth.
• Liquid pressure: The pressure caused by the weight of liquid above a point. It acts in all directions.

Pressure increases with depth

At any point in a liquid there is a column of liquid above it. The weight of that column pushes down, so it causes a pressure. The deeper the point, the taller the column, the greater its weight and so the greater the pressure.

• Dams are built much thicker at the base, where the water pressure is greatest.
• Divers feel more pressure on their ears as they go deeper. A long snorkel does not work at depth: the water pressure on the chest is too great for the lungs to expand.
• Water towers and tanks are placed high up. Taps on lower floors have a taller column of water above them, so water comes out faster there.

Key fact: Liquid pressure depends only on the depth below the surface and the density of the liquid (and g). It does not depend on the width or shape of the container.

Same depth, same pressure

In a liquid at rest, the pressure is the same at every point on the same horizontal level. If it were not, liquid would flow from the higher-pressure region to the lower-pressure region until the pressures became equal. This is why water “finds its own level”: in connected containers the surface ends up at the same height everywhere.

Liquid pressure also acts in all directions at a point – sideways and upwards as well as downwards. Holes made all around a bottle at the same depth give jets of equal strength.

Denser liquids give greater pressure

A column of a denser liquid has more mass, and so more weight, than the same column of a less dense liquid. So at the same depth, the pressure in a denser liquid is greater. A U-tube with water in one arm and oil in the other shows this: the less dense oil stands higher.

Worked example: (Higher tier) finding the density of an oil with a U-tube

Water column above the interface level = 12.0 cm; oil column = 15.0 cm. Density of water = 1000 kg/m³.
Same pressure at X and Y: hw ρw g = ho ρo g, and g cancels
ρo = 1000 × 12.0 ÷ 15.0 = 800 kg/m³

Answer: density of the oil = 800 kg/m³

Calculating liquid pressure: p = h ρ g (Higher tier)

pressure = height of column × density of liquid × gravitational field strength

p = h ρ g

p in pascals (Pa), h in metres (m), ρ in kilograms per cubic metre (kg/m³), g in newtons per kilogram (N/kg)

Rearranged: h = p ÷ (ρ g) ρ = p ÷ (h g)

Worked example: the bottom of a swimming pool

Depth = 2.5 m, density of water = 1000 kg/m³, g = 9.8 N/kg
p = h ρ g = 2.5 × 1000 × 9.8 = 24 500 Pa

Answer: 24 500 Pa (24.5 kPa)

Worked example: rearranging for depth

A diving suit can safely withstand a pressure due to sea water of 500 kPa. Sea water has a density of 1030 kg/m³.
h = p ÷ (ρ g) = 500 000 ÷ (1030 × 9.8) = 49.5 m

Answer: about 50 m deep

Worked example: force on an aquarium window

The centre of a window is 3.0 m below the water surface. The window has an area of 0.50 m².
p = 3.0 × 1000 × 9.8 = 29 400 Pa
F = p A = 29 400 × 0.50 = 14 700 N

Answer: about 1.5 × 10⁴ N

Exam tip: p = h ρ g gives the pressure caused by the liquid only. The air above the surface adds about 100 kPa, so the total pressure on a diver is (h ρ g + 100 kPa). Always measure h from the surface down to the point, and convert cm to m.

Common mistakes

• Mixing up p (pressure) and ρ (rho, density). They are different letters and different quantities.
• Using the depth in cm. Convert to metres first: 45 cm = 0.45 m.
• Measuring h from the bottom of the container instead of down from the liquid surface.
• Thinking a wider or bigger container gives more pressure at the same depth. Only depth and density matter.
• Forgetting that p = h ρ g leaves out the air pressure on the surface when a question asks for the total pressure.

Revision questions

Try these, then check your answers below each one.

Question 1. Which of these does not affect the pressure at a point in a liquid at rest? (Choose from: the width of the container, the depth below the surface, the density of the liquid, the gravitational field strength.)

Answer: the width of the container

Working: Liquid pressure depends on depth, density and g (p = h ρ g). The width or shape of the container makes no difference.

Question 2. True or false? In a liquid at rest, the pressure is the same at all points on the same horizontal level.

Answer: True

Working: True. If the pressures were different, liquid would flow until they became equal, so in a liquid at rest the pressure is the same along any horizontal level.

Question 3. (Higher tier) Use g = 9.8 N/kg. Density of water = 1000 kg/m³.<br/>Calculate the pressure due to the water at the bottom of a pond 4.0 m deep.

Answer: 39200 Pa

Working: p = h ρ g = 4.0 × 1000 × 9.8 = 39 200 Pa (39.2 kPa).

Related lessons

• Previous lesson: 11.1 Pressure and Surfaces
• Next lesson: 11.3 Atmospheric Pressure
• In this chapter: 11.4 Upthrust and Flotation

Continue in the full course

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