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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.

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?

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?

[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.

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} } } {}

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 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} } } {}

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 × 10 2 kg/m 3 size 12{7 "." 9 times "10" rSup { size 8{2} } `"kg/m" rSup { size 8{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 . 6 g/cm 3 size 12{"15" "." 6`"g/cm" rSup { size 8{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 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} {}

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Physics of the world around us. OpenStax CNX. May 21, 2015 Download for free at http://legacy.cnx.org/content/col11797/1.1
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