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Physics · Forces

Pressure in a column of liquid and upthrust

Use \(p = h\rho g\), explain why pressure increases with depth and density, and explain upthrust, floating and sinking (Higher tier).

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  • 6 key terms
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Teacher resources

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Student handouts

The same files the students see, to print or hand out.

Warm-up

Answer each one, then check.

  1. 1

    What is density?

    Show answerHide answer

    Mass per unit volume

  2. 2

    Write the equation for pressure.

    Show answerHide answer

    \(p = F \div A\)

  3. 3

    What unit is pressure in?

    Show answerHide answer

    Pascal

  4. 4

    What is g?

    Show answerHide answer

    Gravitational field strength, 9.8 N/kg

  5. 5

    What is a fluid?

    Show answerHide answer

    A liquid or gas

Learning Objectives

  1. 1Explain why pressure at a point in a liquid increases with depth and with density.
  2. 2Apply \(p = h\rho g\) and calculate pressure differences at different depths.
  3. 3Explain why a submerged object experiences an upward force called upthrust.
  4. 4Describe the factors that influence floating and sinking.

PRESSURE IN A LIQUID

pressure \(=\) height of column \(\times\) density \(\times\) gravitational field strength \(p = h\rho g\)

The equation is given on the Physics equation sheet. In any calculation g will be given. Pressure is in pascals (Pa), h in metres, ρ in kg/m³ and g in N/kg.

Pressure at a Depth

Calculate the pressure due to the sea water at a depth of 20 m. Density of sea water = 1030 kg/m³; g = 9.8 N/kg.

Show the solutionHide the solution
  1. 1 Write the equation \(p = h\rho g\)
  2. 2 Substitute \(p = 20 \times 1030 \times 9.8\)
  3. 3 Answer \(p = 201\,880\) Pa, about \(2.0 \times 10^5\) Pa

AnswerAbout \(2.0 \times 10^5\) Pa

Pressure Difference

Calculate the difference in pressure between depths of 2.0 m and 5.0 m in fresh water (density 1000 kg/m³). g = 9.8 N/kg.

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  1. 1 Difference in depth \(5.0 - 2.0 = 3.0\) m
  2. 2 Substitute \(p = 3.0 \times 1000 \times 9.8\)
  3. 3 Answer \(29\,400\) Pa

Answer29 400 Pa

Upthrust on a Cube

A cube of side 0.10 m is submerged with its top face at a depth of 0.20 m. Calculate the upthrust. Density of water = 1000 kg/m³; g = 9.8 N/kg.

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  1. 1 Pressure difference between bottom and top \(0.10 \times 1000 \times 9.8 = 980\) Pa
  2. 2 Area of a face \(0.10 \times 0.10 = 0.010\) m²
  3. 3 Force difference \(F = pA = 980 \times 0.010 = 9.8\) N

AnswerUpthrust = 9.8 N, upwards.

Floating and Sinking

Compare upthrust with weight.

  • Float

    Upthrust equals weight when an object floats; its average density is less than the liquid's.

  • Sink

    If the weight is greater than the maximum upthrust, the object sinks.

  • Ships

    A steel ship floats because its shape displaces a lot of water, giving it a low average density.

  • Submarines

    Change their average density by taking in or pushing out water.

Dive Down

A diver is 15 m deep in water (density 1000 kg/m³, g = 9.8 N/kg). Calculate the pressure due to the water. Explain why the pressure is greater than at 5 m.

1. Use p = hρg.

2. Explain with depth.

A good answer shows: p = 15 × 1000 × 9.8 = 147 000 Pa. There is more water above the diver, so its weight produces a greater pressure.

Can I...?

  1. 1State p = hρg.
  2. 2Calculate pressure at a depth.
  3. 3Find a pressure difference.
  4. 4Explain why pressure increases with depth.
  5. 5Explain why pressure increases with density.
  6. 6Explain upthrust.
  7. 7Explain floating.
  8. 8Explain sinking.

Summary & Exam Focus

  • \(p = h\rho g\).
  • Pressure increases with depth and density.
  • Upthrust = difference in force on bottom and top.
  • Floats if upthrust equals weight.

Exam focus

Calculate the pressure due to the sea at a depth of 20 m. Density = 1030 kg/m³; g = 9.8 N/kg. (2 marks) (2 marks)

Substitute into p = hρg.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Upthrust
The resultant upward force on a submerged object due to pressure.
Column of liquid
The liquid above a point.
Density
Mass per unit volume.
Submerged
Completely under the surface.
Displaced
Pushed out of the way.
Average density
Total mass divided by total volume.

Questions and answers

10 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Calculate 3 marks Easier

A submarine is at a depth of 20 m in sea water of density 1030 kg/m³. Gravitational field strength = 9.8 N/kg. Calculate the pressure on the submarine due to the water.

Mark scheme — 3 marks available

  • Correct equation — 1 mark
  • Correct substitution — 1 mark
  • 201 880 Pa — 1 mark

Model answer

\(p = 20 \times 1030 \times 9.8 = 201\,880\) Pa (about \(2.0 \times 10^5\) Pa)

2. Exam question Explain 3 marks Easier

Explain why the pressure at the bottom of a swimming pool is greater than the pressure near the surface.

Mark scheme — 3 marks available

  • Pressure due to the weight of water above — 1 mark
  • More water above at the bottom — 1 mark
  • So greater pressure — 1 mark

Model answer

The pressure is caused by the weight of the water above. At the bottom there is more water above, so the weight and therefore the pressure is greater.

3. Exam question Calculate 3 marks Easier

Calculate the difference in pressure between depths of 2.0 m and 5.0 m in fresh water. Density of water = 1000 kg/m³; g = 9.8 N/kg.

Mark scheme — 3 marks available

  • Height difference 3.0 m — 1 mark
  • Correct substitution — 1 mark
  • 29 400 Pa — 1 mark

Model answer

\(p = 3.0 \times 1000 \times 9.8 = 29\,400\) Pa

4. Exam question Explain 3 marks Easier

A brick is completely submerged in water. Explain why it experiences an upward force called upthrust.

Mark scheme — 3 marks available

  • Pressure greater on the bottom — 1 mark
  • Force greater on the bottom surface — 1 mark
  • Resultant upward force — 1 mark

Model answer

The pressure increases with depth, so the pressure on the bottom surface is greater than on the top surface. The force on the bottom is larger, giving a resultant upward force.

5. Exam question Explain 4 marks Easier

A steel ship floats but a steel ball of the same mass sinks. Explain why.

Mark scheme — 4 marks available

  • Ship displaces a large volume — 1 mark
  • Upthrust equals weight for the ship — 1 mark
  • Ball: upthrust less than weight — 1 mark
  • Comparison of average density — 1 mark

Model answer

The ship's shape displaces a large volume of water, so the upthrust equals the ship's weight and its average density is less than water. The ball has a small volume so the upthrust is less than its weight and it sinks.

6. Multiple choice 1 mark Easier

Pressure in a liquid depends on...

  1. A depth and density Correct
  2. B colour only
  3. C volume of container only
  4. D shape only

Why: p = hρg.

7. Multiple choice 1 mark Core

Pressure increases as depth...

  1. A decreases
  2. B increases Correct
  3. C stays the same
  4. D is zero

Why: More liquid above.

8. Multiple choice 1 mark Core

Upthrust acts...

  1. A downwards
  2. B sideways
  3. C upwards Correct
  4. D in circles

Why: It opposes the weight.

9. Multiple choice 1 mark Core

An object floats when upthrust is...

  1. A smaller than its weight
  2. B zero
  3. C double its weight
  4. D equal to its weight Correct

Why: Balanced forces.

10. Multiple choice 1 mark Stretch

A denser liquid gives a...

  1. A greater pressure at the same depth Correct
  2. B smaller pressure
  3. C no pressure
  4. D lower upthrust only

Why: ρ is in the equation.