1. Home
  2. College physics
  3. Fluid statics
  4. Density

11.2 Density  (Page 2/3)

<< Chapter < Page
  College physics     Page 2 / 3
Chapter >> Page >

Calculating the mass of a reservoir from its volume

A reservoir has a surface area of 50 . 0 km 2 size 12{"50" "." 0`"km" rSup { size 8{2} } } {} and an average depth of 40.0 m. What mass of water is held behind the dam? (See [link] 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 size 12{V} {} of the reservoir from its dimensions, and find the density of water ρ size 12{ρ} {} in [link] . Then the mass m size 12{m} {} can be found from the definition of density

ρ = m V . size 12{ρ= { {m} over {V} } } {}

Solution

Solving equation ρ = m / V size 12{ρ= {m} slash {V} } {} for m size 12{m} {} gives m = ρ V size 12{ρ= {m} slash {V} } {} .

The volume V size 12{V} {} of the reservoir is its surface area A size 12{A} {} times its average depth h size 12{h} {} :

V = Ah = 50.0 km 2 40.0 m = 50.0 k m 2 10 3 m 1 km 2 40.0 m = 2 . 00 × 10 9 m 3

The density of water ρ size 12{ρ} {} from [link] is 1 . 000 × 10 3 kg/m 3 size 12{1 "." "000" times "10" rSup { size 8{3} } `"kg/m" rSup { size 8{3} } } {} . Substituting V size 12{V} {} and ρ size 12{ρ} {} into the expression for mass gives

m = 1 . 00 × 10 3 kg/m 3 2 . 00 × 10 9 m 3 = 2.00 × 10 12 kg. alignl { stack { size 12{m= left (1 "." "00" times "10" rSup { size 8{3} } `"kg/m" rSup { size 8{3} } right ) left (2 "." "00" times "10" rSup { size 8{9} } `m rSup { size 8{3} } right )} {} #=2 "." "00" times "10" rSup { size 8{"12"} } `"kg" "." {} } } {}

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 × 10 13 N size 12{ ital "mg"=1 "." "96" times "10" rSup { size 8{"13"} } `N} {} , where g size 12{g} {} is the acceleration due to the Earth’s gravity (about 9 . 80 m/s 2 size 12{9 "." "80"`"m/s" rSup { size 8{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.

Got questions? Get instant answers now!
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)

Section summary

  • Density is the mass per unit volume of a substance or object. In equation form, density is defined as
    ρ = m V . size 12{ρ= { {m} over {V} } } {}
  • The SI unit of density is kg/m 3 size 12{"kg/m" rSup { size 8{3} } } {} .

Conceptual questions

Approximately how does the density of air vary with altitude?

Got questions? Get instant answers now!

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?

Got questions? Get instant answers now!

[link] shows a glass of ice water filled to the brim. Will the water overflow when the ice melts? Explain your answer.

A glass filled to the brim with water and ice cubes.
Got questions? Get instant answers now!

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 size 12{1 "." "610"`"cm" rSup { size 8{3} } } {}

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

(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.

Got questions? Get instant answers now!

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 size 12{"89" "." 0`"cm" rSup { size 8{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 size 12{2 "." "70"`"g/cm" rSup { size 8{3} } } {}

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

(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

Got questions? Get instant answers now!

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?

Got questions? Get instant answers now!

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 × 10 2 kg/m 3 size 12{7 "." 9 times "10" rSup { size 8{2} } `"kg/m" rSup { size 8{3} } } {}

Got questions? Get instant answers now!

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 . 6 g/cm 3 size 12{"15" "." 6`"g/cm" rSup { size 8{3} } } {}

Got questions? Get instant answers now!

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 size 12{"10" rSup { size 8{3} } `"kg/m" rSup { size 8{3} } } {} . The nucleus of an atom has a radius about 10 − 5 size 12{"10" rSup { size 8{ - 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 × 10 8 m size 12{7 times "10" rSup { size 8{8} } `m} {} )?

(a) 10 18 kg/m 3 size 12{"10" rSup { size 8{"18"} } `"kg/m" rSup { size 8{3} } } {}

(b) 2 × 10 4 m size 12{2 times "10" rSup { size 8{4} } `m} {}

Got questions? Get instant answers now!

Questions & Answers

summarize halerambos & holbon
David Reply
the Three stages of Auguste Comte
Clementina Reply
what are agents of socialization
Antonio Reply
sociology of education
Nuhu Reply
definition of sociology of education
Nuhu
what is culture
Abdulrahim Reply
shared beliefs, values, and practices
AI-Robot
What are the two type of scientific method
ogunniran Reply
I'm willing to join you
Aceng Reply
what are the scientific method of sociology
Man
what is socialization
ogunniran Reply
the process wherein people come to understand societal norms and expectations, to accept society's beliefs, and to be aware of societal values
AI-Robot
scientific method in doing research
ogunniran
defimition of sickness in afica
Anita
Cosmology
ogunniran
Hmmm
ogunniran
list and explain the terms that found in society
REMMY Reply
list and explain the terms that found in society
Mukhtar
what are the agents of socialization
Antonio
Family Peer group Institution
Abdulwajud
I mean the definition
Antonio
ways of perceived deviance indifferent society
Naomi Reply
reasons of joining groups
SAM
to bring development to the nation at large
Hyellafiya
entails of consultative and consensus building from others
Gadama
World first Sociologist?
Abu
What is evolutionary model
Muhammad Reply
Evolution models refer to mathematical and computational representations of the processes involved in biological evolution. These models aim to simulate and understand how species change over time through mechanisms such as natural selection, genetic drift, and mutation. Evolutionary models can be u
faruk
what are the modern trends in religious behaviours
Selekeye Reply
what are social norms
Daniel Reply
shared standards of acceptable behavior by the group or appropriate behavior in a particular institution or those behaviors that are acceptable in a society
Lucius
that is how i understood it
Lucius
examples of societal norms
Diamond
Discuss the characteristics of the research located within positivist and the interpretivist paradigm
Tariro Reply
what is Industrialisation
Selekeye Reply
industrialization
Angelo
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 1

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College physics' conversation and receive update notifications?

Ask