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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.

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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?

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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?

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[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.
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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} } } {}

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

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(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.

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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} } } {}

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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.

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(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

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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?

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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} } } {}

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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} } } {}

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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} {}

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Questions & Answers

Hey.. I've a question.
Sahim Reply
?
Shii
what would be the correct interrogation "what is time?" or "how much has your watch ticked?"
prakash Reply
a load of 20N on a wire of cross sectional area 8×10^-7m produces an extension of 10.4m. calculate the young modules of the material of the wire is of length 5m
Ebenezer Reply
Young's modulus = stress/strain strain = extension/length (x/l) stress = force/area (F/A) stress/strain is F l/A x
El
so solve it
Ebenezer
please
Ebenezer
two bodies x and y start from rest and move with uniform acceleration of a and 4a respectively. if the bodies cover the same distance in terms of tx and ty what is the ratio of tx to ty
Oluwatola Reply
what is cesium atoms?
prakash Reply
The atoms which form the element Cesium are known as Cesium atoms.
Naman
A material that combines with and removes trace gases from vacuum tubes.
Shankar
what is difference between entropy and heat capacity
Varun
Heat capacity can be defined as the amount of thermal energy required to warm the sample by 1°C. entropy is the disorder of the system. heat capacity is high when the disorder is high.
Chathu
I want learn physics
Vinodhini Reply
sir how to understanding clearly
Vinodhini
try to imagine everything you study in 3d
revolutionary
pls give me one title
Vinodhini
displacement acceleration how understand
Vinodhini
vernier caliper usage practically
Vinodhini
karthik sir is there
Vinodhini
what are the solution to all the exercise..?
What is realm
Vinodhini Reply
The quantum realm, also called the quantum scale, is a term of art inphysics referring to scales where quantum mechanical effects become important when studied as an isolated system. Typically, this means distances of 100 nanometers (10−9meters) or less or at very low temperature.
revolutionary
How to understand physics
Vinodhini Reply
i like physics very much
Vinodhini
i want know physics practically where used in daily life
Vinodhini
I want to teach physics very interesting to studentd
Vinodhini
hy
Aderibigbe
sir
Vinodhini
how can you build interest in physics
Prince
by reading it
Austin
understanding difficult
Vinodhini
vinodhini mam, physics is used in our day to day life in all events..... everything happening around us can be explained in the base of physics..... saying simple stories happening in our daily life and relating it to physics and questioning students about how or why its happening like that can make
revolutionary
your class more interesting
revolutionary
anything send about physics daily life
Vinodhini
How to understand easily
Vinodhini
check out "LMES" youtube channel
revolutionary
even when you see this message in your phone...it works accord to a physics principle. you touch screen works based on physics, your internet works based on physics, etc....... check out google and search for it
revolutionary
hi vinodhini
Karthi
hi
Dada
yea
Solomon
Hi guys
poojaa
hello
Abednego
what is mean by Newtonian principle of Relativity? definition and explanation with example
revolutionary Reply
what is art physics
Akinbulejo Reply
I've been trying to download a good and comprehensive textbook for physics, pls can somebody help me out?
Olanrewaju
try COLLEGE PHYSICS!! I think it will give you an edge.
Lawal
c=1/c1+c2/1+c3 what is the answer
Akinbulejo Reply
got on answers bro
smith
This may seem like a really stupid question, but is mechanical energy the same as potential energy? If not, what is the difference?
Nikki Reply
what is c=1\c1,c=2\c2,c=3\c3
Akinbulejo
mechanical energy is of two types 1: kinetic energy 2: potential energy,so, potential energy is actually the type of mechanical energy ,the mechanical due to position is designated as potential energy
Iram
Thank you!!!!!
Nikki
Can someone possibly walk me through this problem? " A worker drives a 0.500 kg spike into a rail tie with a 2.50 kg sledgehammer. The hammer hits the spike with a speed of 65.0 m/s. If one-third Of the hammer's kinetic energy is converted to the internal energy of rhe hammer and spike.
Nikki
how much does the total internal energy increase
Nikki
you know the mass and the velocity of the hammer. therefore using the equation (mv^2)/2 you can find the kinetic energy. then take one third of this value and that will be your change in internal energy. here, the important thing is that spike is stationary so it does not contribute to initial Energ
Chathu
Thabk you! :)
Nikki
what is the formula for finding the to total capacitance in series arrangement
Austin Reply
Don't know
Ugwu
C = 1/C1+1/C2+1/C3
Nangbun
c=1/c1+1/c2+1/c3
Akinbulejo
what is heat capacity?
smith Reply
Amount of heat that increases the temperature of 1 kg of matter by 1 degree(either celsius or kalvin)
Mateo
it is the ratio between the amount of heat added to an object and the temperature change
Giovanni
what is the different between specific heat capacity and latent heat capacity?
smith
what is sun?
ANYAOGU Reply
in what terms the sun us a star
Jeff
I believe that the sun is a star that went supernova I don't know what terms you're asking about son though
Jeff
sun is the largest, shiniest, brightest star that was formed from a supernova explosion.
Austin
forgot to mention the hottest...😰😰
Austin
it is the largest shiniest brightest star in our Galaxy. There are tons of star much larger than sun.
Claira
sun is a star because of having its own light
Iram
Practice Key Terms 1

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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