Archimedes' Principle Examples: 45 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Archimedes' Principle.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Physics.

Concept Recap

Archimedes' principle states that the buoyant force on an immersed object equals the weight of the fluid that the object displaces.

A fluid pushes up exactly as much as the displaced fluid would weigh.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Archimedes' Principle asks how mass, volume, pressure, and displacement determine the fluid interaction.

Common stuck point: Students often know a formula related to archimedes' principle but skip the recognition step: Am I reasoning about a fluid or object in a fluid, with volume, area, depth, density, or displaced fluid identified? That leads to a correct-looking substitution attached to the wrong physical model.

Sense of Study hint: Ask: Am I reasoning about a fluid or object in a fluid, with volume, area, depth, density, or displaced fluid identified?

Worked Examples

Example 1

medium

A wooden block of density

600 \text{ kg/m}^3

floats in water. What fraction of its volume is below the surface?

Answer

\frac{V_{s u b}}{V_{b l o c k}} = 0.6

First step

Floating condition: weight = buoyant force, so

\rho_{obj} V g = \rho_f V_{sub} g

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Example 2

medium

2 \text{ kg}

metal object has volume

5 \times 10^{-4} \text{ m}^3

and is fully submerged in water. Find its apparent weight. (

\rho = 1000

g = 10

)

Example 3

medium

A boat of mass

400 \text{ kg}

floats in fresh water. Find the volume of water displaced. (

\rho=1000

g=10

)

Example 4

medium

A cubic block of side

0.1 \text{ m}

and density

1200 \text{ kg/m}^3

is fully submerged in water (

\rho=1000

g=10

). Find the net force and direction.

Example 5

hard

5 \text{ kg}

object hangs from a spring scale. When lowered into water, the scale reads

32 \text{ N}

. Find the object's volume. (

\rho = 1000

g = 10

)

Example 6

hard

A boat of mass

200 \text{ kg}

and base area

5 \text{ m}^2

floats in fresh water. By how much does it sink deeper when a

50 \text{ kg}

passenger steps aboard? (

\rho = 1000

g = 10

)

Example 7

hard

A boat of mass

1000 \text{ kg}

floats in a sealed pool of water sitting on a scale. A

30 \text{ kg}

rock is dropped into the pool from the boat. What does the scale read (just the system) before vs. after? (

g=10

)

Example 8

hard

0.5 \text{ kg}

ball of volume

4 \times 10^{-4} \text{ m}^3

is held submerged at rest. Released, what is its initial acceleration? (

\rho=1000

g=10

)

Example 9

challenge

A crown allegedly of pure gold (

\rho_{Au}=19300

) weighs

19.3 \text{ N}

in air and

18.1 \text{ N}

submerged in water (

\rho_w=1000

g=10

). Is it pure gold?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

An object displaces

0.003 \text{ m}^3

of water (

\rho=1000

g=10

). By Archimedes' principle, find the buoyant force.

Example 2

easy

A submerged object displaces

2 \text{ kg}

of water. Find the buoyant force (

g=10

Example 3

easy

A ship floats by displacing water equal in weight to the ship's

50000 \text{ N}

. What is the buoyant force?

Example 4

easy

An object weighs

40 \text{ N}

in air and

25 \text{ N}

in water. Find the weight of water displaced.

Example 5

easy

Find the volume of fluid displaced if the buoyant force is

50 \text{ N}

in water (

\rho=1000

g=10

Example 6

easy

Only the submerged part of an object displaces fluid. If half is submerged, what fraction of full-submersion buoyancy acts?

Example 7

easy

A rock displaces

0.0008 \text{ m}^3

of water. Find the mass of water displaced (

\rho=1000

Example 8

easy

Two objects displace the same volume in the same fluid. Compare their buoyant forces.

Example 9

medium

0.6 \text{ kg}

object of volume

0.0004 \text{ m}^3

is submerged in water (

\rho=1000

g=10

). Find its apparent weight.

Example 10

medium

A crown of mass

1 \text{ kg}

displaces

0.00006 \text{ m}^3

of water. Find its density and compare to gold (

19300

). (

\rho_w=1000

)

Example 11

medium

A block floats with

0.7

submerged in water (

1000

). Find its density using Archimedes' principle.

Example 12

medium

5 \text{ N}

object is fully submerged and the buoyant force is

3 \text{ N}

. Find the net force and direction.

Example 13

medium

A boat of mass

800 \text{ kg}

floats in water (

\rho=1000

g=10

). Find the volume of water it displaces.

Example 14

medium

An object weighs

60 \text{ N}

in air,

50 \text{ N}

in water (

1000

), and

52 \text{ N}

in oil. Find the oil's density. (

g=10

)

Example 15

medium

A balloon displaces

3 \text{ m}^3

of air (

\rho=1.2

g=10

). Find the buoyant (lift) force.

Example 16

medium

0.9 ext{ kg}

object of volume

0.0001 ext{ m}^3

is submerged in water (

ho=1000

g=10

). Find the buoyant force and apparent weight.

Example 17

medium

A floating object displaces

0.0024 ext{ m}^3

of water (

ho=1000

g=10

). Find the object's weight.

Example 18

challenge

0.5 \text{ kg}

object of volume

0.0003 \text{ m}^3

floats partly in water with a string pulling it down with tension

2 \text{ N}

. Find the submerged volume. (

\rho=1000

g=10

)

Example 19

challenge

A hollow sphere of outer volume

0.002 \text{ m}^3

and mass

1.5 \text{ kg}

is placed in water (

1000

g=10

). Does it float, and if so what fraction is submerged?

Example 20

challenge

An ice cube (

\rho=920

) floats in water (

1000

). When it melts, does the water level rise, fall, or stay the same?

Example 21

easy

A fully submerged object displaces

0.002 \text{ m}^3

of water. Find the buoyant force (

\rho = 1000

g = 10

Example 22

easy

A submerged object displaces a fluid weighing

35 \text{ N}

. What is the buoyant force on it?

Example 23

easy

A swimmer fully submerges and displaces

60 \text{ L}

of water. Find the buoyant force. (

\rho=1000

g=10

;

1 \text{ L} = 10^{-3} \text{ m}^3

)

Example 24

easy

A balloon of volume

0.5 \text{ m}^3

is fully submerged in fresh water. Find the buoyant force. (

\rho=1000

g=10

)

Example 25

easy

An object weighing

50 \text{ N}

in air weighs

30 \text{ N}

when submerged. Find the buoyant force.

Example 26

easy

What volume of water must be displaced for a buoyant force of

80 \text{ N}

? (

\rho = 1000

g = 10

)

Example 27

medium

An iceberg has density

920 \text{ kg/m}^3

and floats in seawater of density

1025 \text{ kg/m}^3

. What fraction is above the surface?

Example 28

medium

An object of volume

0.001 \text{ m}^3

floats half-submerged in oil (

\rho_{oil}=800

). Find its mass. (

g=10

)

Example 29

medium

A balloon filled with helium (

\rho_{He}=0.18 \text{ kg/m}^3

) has volume

2 \text{ m}^3

in air (

\rho_{air}=1.2

). Find the buoyant force on it (

g=10

Example 30

medium

An object of mass

3 \text{ kg}

has buoyant force

40 \text{ N}

when fully submerged in water. What is its volume? (

\rho=1000

g=10

)

Example 31

medium

A floating block displaces

0.30

of its volume in water. Find its density. (

\rho_f = 1000

)

Example 32

hard

A block of density

\rho_{obj}

floats with fraction

f = 0.7

submerged in fluid A (

\rho_A=1000

). In fluid B it floats with fraction

0.5

. Find

\rho_B

Example 33

hard

A hot-air balloon and basket together have mass

400 \text{ kg}

. The balloon's heated air has density

0.9 \text{ kg/m}^3

; surrounding air is

1.2

. Find the minimum balloon volume for liftoff. (

g=10

)

Example 34

hard

2 \text{ kg}

stone is tied to a

0.001 \text{ m}^3

piece of wood (

\rho_{wood} = 600 \text{ kg/m}^3

). The system just barely floats in water (

\rho=1000

g=10

). Find the stone's volume.

Example 35

hard

A submerged

1 \text{ kg}

object of volume

2 \times 10^{-4} \text{ m}^3

is held by a rope from above in water (

\rho=1000

g=10

). Find the rope tension.

Example 36

challenge

An ice cube (

\rho_{ice}=920

) floats in a glass of water (

\rho_w=1000

), with the cube fully fitting inside. The ice melts. Does the water level rise, fall, or stay the same?

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Related Concepts

Buoyancy

Background Knowledge

These ideas may be useful before you work through the harder examples.

buoyancy