Density Examples in Physics: 47 Problems with Answers

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

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

Concept Recap

Density is the amount of mass packed into a given volume. In physics, it helps explain why some materials float, sink, or create larger pressure.

Density tells you how tightly matter is packed. Heavy for its size means high density.

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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: Density asks how mass, volume, pressure, and displacement determine the fluid interaction.

Common stuck point: Students often know a formula related to density 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 cylinder has radius

5\text{ cm}

, height

20\text{ cm}

, and mass

2.0\text{ kg}

. Find its density in

\text{kg/m}^3

. (

V = \pi r^2 h

Answer

ρ \approx 1273 {kg/m}^{3}

First step

V = \pi (0.05)^2 (0.20)\approx 1.571\times 10^{-3}\text{ m}^3

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Example 2

medium

Two metals are mixed by volume:

0.0010\text{ m}^3

of metal A (

\rho=8000

) and

0.0010\text{ m}^3

of metal B (

\rho=4000

). Find the average density.

Example 3

hard

An alloy is

40\%

copper by mass (

\rho=8960

) and

60\%

zinc by mass (

\rho=7140

). Find the alloy's density. (Use

1/\rho_\text{alloy} = w_\text{Cu}/\rho_\text{Cu} + w_\text{Zn}/\rho_\text{Zn}

Example 4

hard

Water at

4^\circ

C has density

1000\text{ kg/m}^3

; at

80^\circ

C it has density

972\text{ kg/m}^3

. A

1\text{ m}^3

tank holds water at

4^\circ

C and is heated to

80^\circ

C (sealed off; some leaks out). What mass leaves the tank?

Example 5

challenge

A block of density

\rho_b

floats at the boundary between oil (

\rho_o=750

) on top and water (

\rho_w=1000

) below, with

30\%

of its volume in oil and

70\%

in water. Find

\rho_b

Practice Problems

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

Example 1

easy

A block has mass

60 \text{ kg}

and volume

0.02 \text{ m}^3

. Find its density.

Example 2

easy

Water has density

1000 \text{ kg/m}^3

. What is the mass of

0.5 \text{ m}^3

of water?

Example 3

easy

A metal cube has density

8000 \text{ kg/m}^3

and mass

16 \text{ kg}

. Find its volume.

Example 4

easy

Object A is

2 \text{ kg}

0.001 \text{ m}^3

; object B is

5 \text{ kg}

0.005 \text{ m}^3

. Which is denser?

Example 5

easy

Convert a density of

2 \text{ g/cm}^3

\text{kg/m}^3

Example 6

easy

Ice has density

920 \text{ kg/m}^3

and water

1000 \text{ kg/m}^3

. Will ice float on water?

Example 7

easy

A liquid sample of

250 \text{ g}

occupies

200 \text{ cm}^3

. Find density in

\text{g/cm}^3

Example 8

easy

Two equal-volume blocks: one

3 \text{ kg}

, one

7 \text{ kg}

. Which has greater density?

Example 9

medium

0.3 \text{ m}^3

tank holds oil of density

850 \text{ kg/m}^3

. The empty tank is

20 \text{ kg}

. Find total mass.

Example 10

medium

An alloy mixes

4 \text{ kg}

of metal X (

V=0.0005 \text{ m}^3

) and

6 \text{ kg}

of metal Y (

V=0.001 \text{ m}^3

). Find the alloy's average density.

Example 11

medium

A hollow sphere has outer volume

0.004 \text{ m}^3

and contains

0.001 \text{ m}^3

of air; the shell is

24 \text{ kg}

of metal. Find the average density of the whole sphere.

Example 12

medium

A gas occupies

2 \text{ m}^3

with density

1.2 \text{ kg/m}^3

. It is compressed to

0.5 \text{ m}^3

at constant mass. Find the new density.

Example 13

medium

A wooden plank floats with

80\%

of its volume submerged. Water density is

1000 \text{ kg/m}^3

. Estimate the wood's density.

Example 14

medium

500 \text{ g}

crown has volume

50 \text{ cm}^3

. Pure gold is

19.3 \text{ g/cm}^3

. Is the crown pure gold?

Example 15

medium

A balloon of volume

0.01 \text{ m}^3

is filled with helium (

0.18 \text{ kg/m}^3

). The rubber skin is

0.005 \text{ kg}

. Find the balloon's average density.

Example 16

medium

A river carries silt:

30 \text{ kg}

of sediment (

\rho=2600 \text{ kg/m}^3

) suspended in

1 \text{ m}^3

of water (

\rho=1000

). Find the mixture density.

Example 17

medium

A solid sphere of radius

0.1 \text{ m}

has mass

33.5 \text{ kg}

. Find its density. (

V = \frac{4}{3}\pi r^3 \approx 0.00419 \text{ m}^3

)

Example 18

challenge

A block of density

900 \text{ kg/m}^3

rests at a water (

1000

) and oil (

800

) interface, straddling both. What fraction of its volume is submerged in the water?

Example 19

challenge

A cube of side

0.1 \text{ m}

and density

2700 \text{ kg/m}^3

is suspended. If

40\%

of its volume is then hollowed out (replaced by vacuum), find the new average density.

Example 20

challenge

Two liquids that do not mix have densities

800

and

1200 \text{ kg/m}^3

. They are combined in equal masses of

6 \text{ kg}

each. Find the average density of the combined (separated) column.

Example 21

easy

A brick has mass

2.4\text{ kg}

and volume

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

. Find its density.

Example 22

easy

Aluminum has density

2700\text{ kg/m}^3

. What is the mass of

0.05\text{ m}^3

of aluminum?

Example 23

easy

Mercury has density

13600\text{ kg/m}^3

and a sample has mass

6.8\text{ kg}

. Find its volume.

Example 24

easy

Convert

7.8\text{ g/cm}^3

\text{kg/m}^3

Example 25

easy

A wooden cube has side

10\text{ cm}

and mass

700\text{ g}

. Find its density in

\text{g/cm}^3

Example 26

easy

Olive oil has density

920\text{ kg/m}^3

. Water is

1000\text{ kg/m}^3

. Will olive oil float on water?

Example 27

medium

A swimming pool holds

50\text{ m}^3

of saltwater with density

1030\text{ kg/m}^3

. Find the total mass of the water.

Example 28

medium

1.5\text{ L}

bottle of cooking oil weighs

1.20\text{ kg}

. Find the oil's density in

\text{kg/m}^3

Example 29

medium

A submarine has total volume

50\text{ m}^3

and mass

48000\text{ kg}

with empty ballast. Water is

1000\text{ kg/m}^3

. Will it float, sink, or be neutrally buoyant?

Example 30

medium

A graduated cylinder reads

40.0\text{ mL}

. A rock dropped in raises it to

58.0\text{ mL}

. The rock weighs

54\text{ g}

. Find its density in

\text{g/cm}^3

Example 31

medium

A hot-air balloon envelope encloses

2000\text{ m}^3

. Hot air inside has density

0.95\text{ kg/m}^3

; outside air is

1.20\text{ kg/m}^3

. What is the net upward buoyant load (in kg-of-air equivalent)?

Example 32

medium

A gas in a sealed piston has density

1.2\text{ kg/m}^3

. The piston is pushed in so the volume drops from

4\text{ m}^3

1\text{ m}^3

(no gas added). Find the new density.

Example 33

medium

An ice cube of density

920\text{ kg/m}^3

floats in seawater of density

1025\text{ kg/m}^3

. What fraction of the cube is submerged?

Example 34

medium

A boat has hull volume

5\text{ m}^3

but the empty boat weighs

2000\text{ kg}

. Water is

1000\text{ kg/m}^3

. What is the maximum cargo mass before it sinks?

Example 35

medium

A steel ball (

\rho = 7850\text{ kg/m}^3

) and an aluminum ball (

\rho = 2700\text{ kg/m}^3

) have the same volume

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

. Find the difference in their masses.

Example 36

hard

A jeweler tests a

200\text{ g}

ring. In air it weighs

200\text{ g}

; submerged in water (which buoys it) it reads

182\text{ g}

on a balance. Find the ring's density. (

\rho_\text{water} = 1.00\text{ g/cm}^3

Example 37

hard

A hollow plastic ball (

\rho_\text{shell}=900\text{ kg/m}^3

, shell volume

0.0006\text{ m}^3

) has an air-filled cavity of

0.0014\text{ m}^3

. Find the average density and decide if it floats on water (

1000\text{ kg/m}^3

Example 38

hard

A swimmer of mass

70\text{ kg}

floats with her body fully submerged but with lungs full of air (

V_\text{lungs}=6\text{ L}

). The rest of her body has average density

1010\text{ kg/m}^3

. Water is

1000\text{ kg/m}^3

. Find her overall density.

Example 39

hard

Equal masses of liquid A (

\rho=600

) and liquid B (

\rho=1200

) — say,

6\text{ kg}

each — are mixed in a container where they stay separated. Find the average density of the combined column.

Example 40

hard

A balloon-and-payload system has mass

250\text{ kg}

and total volume

230\text{ m}^3

. Surrounding air is

1.20\text{ kg/m}^3

. Will it rise, sink, or hover?

Example 41

challenge

A sphere of unknown radius and density

\rho = 2500\text{ kg/m}^3

has mass

1.05\text{ kg}

. Find its radius. (

V = \tfrac{4}{3}\pi r^3

Example 42

challenge

A cylinder of unknown material has mass

1.5\text{ kg}

in air and apparent mass

1.3\text{ kg}

when fully submerged in water (

\rho_w = 1000\text{ kg/m}^3

). Find its density.

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Related Concepts

Mass Pressure Buoyancy

Background Knowledge

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

mass