Density, Fluid Statics | College Physics 2e

Learning objectives

Learning Objectives

By the end of this section, you will be able to:

Define density.
Calculate the mass of a reservoir from its density.
Compare and contrast the densities of various substances.

Which weighs more, a ton of feathers or a ton of bricks? This old riddle plays with the distinction between mass and density. A ton is a ton, of course; but bricks have much greater density than feathers, and so we are tempted to think of them as heavier. (See import-auto-id950378.)

Density, as you will see, is an important characteristic of substances. It is crucial, for example, in determining whether an object sinks or floats in a fluid. Density is the mass per unit volume of a substance or object. In equation form, density is defined as

ρ = \frac{m}{V},

where the Greek letter $ρ$ (rho) is the symbol for density, $m$ is the mass, and $V$ is the volume occupied by the substance.

In the riddle regarding the feathers and bricks, the masses are the same, but the volume occupied by the feathers is much greater, since their density is much lower. The SI unit of density is ${kg/m}^{3}$ , representative values are given in fs-id1769034. The metric system was originally devised so that water would have a density of $1 {g/cm}^{3}$ , equivalent to $10^{3} {kg/m}^{3}$ . Thus the basic mass unit, the kilogram, was first devised to be the mass of 1000 mL of water, which has a volume of 1000 cm³.

Densities of Various Substances
Substance	$ρ (\times 10^{3} {kg/m}^{3}$ or $g/mL)$	Substance	$ρ (10^{3} {kg/m}^{3}$ or $g/mL)$	Substance	$ρ (10^{3} {kg/m}^{3}$ or $g/mL)$
Solids		Liquids		Gases
Aluminum	2.7	Water (4ºC)	1.000	Air	$1 . 29 \times 10^{- 3}$
Brass	8.44	Blood	1.05	Carbon dioxide	$1 . 98 \times 10^{- 3}$
Copper (average)	8.8	Sea water	1.025	Carbon monoxide	$1 . 25 \times 10^{- 3}$
Gold	19.32	Mercury	13.6	Hydrogen	$0 . 090 \times 10^{- 3}$
Iron or steel	7.8	Ethyl alcohol	0.79	Helium	$0 . 18 \times 10^{- 3}$
Lead	11.3	Gasoline	0.68	Methane	$0 . 72 \times 10^{- 3}$
Polystyrene	0.10	Glycerin	1.26	Nitrogen	$1 . 25 \times 10^{- 3}$
Tungsten	19.30	Olive oil	0.92	Nitrous oxide	$1 . 98 \times 10^{- 3}$
Uranium	18.70			Oxygen	$1 . 43 \times 10^{- 3}$
Concrete	2.30–3.0			Steam $(100º C)$	$0 . 60 \times 10^{- 3}$
Cork	0.24
Glass, common (average)	2.6
Granite	2.7
Earth’s crust	3.3
Wood	0.3–0.9
Ice (0°C)	0.917
Bone	1.7–2.0
Silver	10.49

View original figure (external; image not reproduced in this text edition)

Original figure description: A pile of feathers measuring a ton and a ton of bricks are placed on either side of a plank that is balanced on a small support.

A ton of feathers and a ton of bricks have the same mass, but the feathers make a much bigger pile because they have a much lower density.

As you can see by examining fs-id1769034, the density of an object may help identify its composition. The density of gold, for example, is about 2.5 times the density of iron, which is about 2.5 times the density of aluminum. Density also reveals something about the phase of the matter and its substructure. Notice that the densities of liquids and solids are roughly comparable, consistent with the fact that their atoms are in close contact. The densities of gases are much less than those of liquids and solids, because the atoms in gases are separated by large amounts of empty space.

Calculating the Mass of a Reservoir From Its Volume

A reservoir has a surface area of $50.0 {km}^{2}$ and an average depth of 40.0 m. What mass of water is held behind the dam? (See import-auto-id1371721 for a view of a large reservoir—the Three Gorges Dam site on the Yangtze River in central China.)

Strategy

We can calculate the volume $V$ of the reservoir from its dimensions, and find the density of water $ρ$ in fs-id1769034. Then the mass $m$ can be found from the definition of density

ρ = \frac{m}{V} .

Solution

Solving equation $ρ = m / V$ for $m$ gives $m = ρ V$ .

The volume $V$ of the reservoir is its surface area $A$ times its average depth $h$ :

\begin{matrix} V & = & Ah = (50.0 {km}^{2}) (40.0 m) \\ = & ((50.0 k m^{2}) {(\frac{10^{3} m}{1 km})}^{2}) (40.0 m) = 2 . 00 \times 10^{9} m^{3} \end{matrix}

The density of water $ρ$ from fs-id1769034 is $1 . 000 \times 10^{3} {kg/m}^{3}$ . Substituting $V$ and $ρ$ into the expression for mass gives

\begin{matrix} m & = & (1 . 00 \times 10^{3} {kg/m}^{3}) (2 . 00 \times 10^{9} m^{3}) \\ = & 2.00 \times 10^{12} kg. \end{matrix}

Discussion

A large reservoir contains a very large mass of water. In this example, the weight of the water in the reservoir is $mg = 1 . 96 \times 10^{13} N$ , where $g$ is the acceleration due to the Earth’s gravity (about $9.80 {m/s}^{2}$ ). It is reasonable to ask whether the dam must supply a force equal to this tremendous weight. The answer is no. As we shall see in the following sections, the force the dam must supply can be much smaller than the weight of the water it holds back.

View original figure (external; image not reproduced in this text edition)

Original figure description: Photograph of the Three Gorges Dam in central China.

Three Gorges Dam in central China. When completed in 2008, this became the world’s largest hydroelectric plant, generating power equivalent to that generated by 22 average-sized nuclear power plants. The concrete dam is 181 m high and 2.3 km across. The reservoir made by this dam is 660 km long. Over 1 million people were displaced by the creation of the reservoir. (credit: Le Grand Portage)

Test Prep for AP Courses

An under-inflated volleyball is pumped full of air so that its radius increases by 10%. Ignoring the mass of the air inserted into the ball, what will happen to the volleyball's density?

The density of the volleyball will increase by approximately 25%.
The density of the volleyball will increase by approximately 10%.
The density of the volleyball will decrease by approximately 10%.
The density of the volleyball will decrease by approximately 17%.
The density of the volleyball will decrease by approximately 25%.

(e)

A piece of aluminum foil has a known surface density of 15 g/cm². If a 100-gram hollow cube were constructed using this foil, determine the approximate side length of this cube.

1.05 cm
1.10 cm
2.6 cm
6.67 cm
15 cm

A cube of polystyrene measuring 10 cm per side lies partially submerged in a large container of water.

If 90% of the polystyrene floats above the surface of the water, what is the density of the polystyrene? (Note: The density of water is 1000 kg/m³.)
A 0.5 kg mass is placed on the block of polystyrene. What percentage of the block now remains above water?
The water is poured out of the container and replaced with ethyl alcohol (density = 790 kg/m³).
1. Will the block be able to remain partially submerged in this new fluid? Explain.
2. Will the block be able to remain partially submerged in this new fluid with the 0.5 kg mass placed on top? Explain.
Without using a container of water, explain how you could determine the density of the polystyrene mentioned above if the material instead were spherical.

(a) 100 kg/m³ (b) 60% (c) yes; yes (76% will be submerged) (d) answers vary

Four spheres are hung from a variety of different springs. The table below describes the characteristics of both the spheres and the springs from which they are hung. Use this information to rank the density of each sphere from least to greatest. Show work supporting your ranking.


Material Type	Radius of Sphere	Stretch of Spring (from equilibrium)	Spring Constant
A	10 cm	5 cm	2 N/m
B	5 cm	8 cm	8 N/m
C	8 cm	10 cm	6 N/m
D	8 cm	12 cm	10 N/m

Rank the densities of the objects listed above, from greatest to least. Show work supporting your ranking.

Section Summary

Density is the mass per unit volume of a substance or object. In equation form, density is defined as $ρ = \frac{m}{V} .$
The SI unit of density is ${kg/m}^{3}$ .

Conceptual Questions

Approximately how does the density of air vary with altitude?

Give an example in which density is used to identify the substance composing an object. Would information in addition to average density be needed to identify the substances in an object composed of more than one material?

import-auto-id1988132 shows a glass of ice water filled to the brim. Will the water overflow when the ice melts? Explain your answer.

View original figure (external; image not reproduced in this text edition)

Original figure description: A glass filled to the brim with water and ice cubes.

Problems & Exercises

Gold is sold by the troy ounce (31.103 g). What is the volume of 1 troy ounce of pure gold?

$1.610 {cm}^{3}$

Mercury is commonly supplied in flasks containing 34.5 kg (about 76 lb). What is the volume in liters of this much mercury?

(a) What is the mass of a deep breath of air having a volume of 2.00 L? (b) Discuss the effect taking such a breath has on your body’s volume and density.

(a) 2.58 g

(b) The volume of your body increases by the volume of air you inhale. The average density of your body decreases when you take a deep breath, because the density of air is substantially smaller than the average density of the body before you took the deep breath.

A straightforward method of finding the density of an object is to measure its mass and then measure its volume by submerging it in a graduated cylinder. What is the density of a 240-g rock that displaces $89.0 {cm}^{3}$ of water? (Note that the accuracy and practical applications of this technique are more limited than a variety of others that are based on Archimedes’ principle.)

$2.70 {g/cm}^{3}$

Suppose you have a coffee mug with a circular cross section and vertical sides (uniform radius). What is its inside radius if it holds 375 g of coffee when filled to a depth of 7.50 cm? Assume coffee has the same density as water.

(a) A rectangular gasoline tank can hold 50.0 kg of gasoline when full. What is the depth of the tank if it is 0.500-m wide by 0.900-m long? (b) Discuss whether this gas tank has a reasonable volume for a passenger car.

(a) 0.163 m

(b) Equivalent to 19.4 gallons, which is reasonable

A trash compactor can reduce the volume of its contents to 0.350 their original value. Neglecting the mass of air expelled, by what factor is the density of the rubbish increased?

A 2.50-kg steel gasoline can holds 20.0 L of gasoline when full. What is the average density of the full gas can, taking into account the volume occupied by steel as well as by gasoline?

$7 . 9 \times 10^{2} {kg/m}^{3}$

What is the density of 18.0-karat gold that is a mixture of 18 parts gold, 5 parts silver, and 1 part copper? (These values are parts by mass, not volume.) Assume that this is a simple mixture having an average density equal to the weighted densities of its constituents.

$15.8 {g/cm}^{3}$

There is relatively little empty space between atoms in solids and liquids, so that the average density of an atom is about the same as matter on a macroscopic scale—approximately $10^{3} {kg/m}^{3}$ . The nucleus of an atom has a radius about $10^{- 5}$ that of the atom and contains nearly all the mass of the entire atom. (a) What is the approximate density of a nucleus? (b) One remnant of a supernova, called a neutron star, can have the density of a nucleus. What would be the radius of a neutron star with a mass 10 times that of our Sun (the radius of the Sun is $7 \times 10^{8} m$ )?

(a) $10^{18} {kg/m}^{3}$

(b) $2 \times 10^{4} m$

Glossary

density: the mass per unit volume of a substance or object

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