# 11.2 Density  (Page 2/3)

 Page 2 / 3

## Calculating the mass of a reservoir from its volume

A reservoir has a surface area of $\text{50}\text{.}0\phantom{\rule{0.25em}{0ex}}{\text{km}}^{2}$ 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 $V$ of the reservoir from its dimensions, and find the density of water $\rho$ in [link] . Then the mass $m$ can be found from the definition of density

$\rho =\frac{m}{V}.$

Solution

Solving equation $\rho =m/V$ for $m$ gives $m=\rho V$ .

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

$\begin{array}{lll}V& =& \text{Ah}=\left(\text{50.0}\phantom{\rule{0.25em}{0ex}}{\text{km}}^{2}\right)\left(\text{40.0}\phantom{\rule{0.25em}{0ex}}\text{m}\right)\\ & =& \left[\left(\text{50.0 k}{\text{m}}^{2}\right){\left(\frac{{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{m}}{1\phantom{\rule{0.25em}{0ex}}\text{km}}\right)}^{2}\right]\left(\text{40.0 m}\right)=2\text{.}\text{00}×{\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{3}\end{array}$

The density of water $\rho$ from [link] is $1\text{.}\text{000}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}$ . Substituting $V$ and $\rho$ into the expression for mass gives

$\begin{array}{lll}m& =& \left(1\text{.}\text{00}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}\right)\left(2\text{.}\text{00}×{\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{3}\right)\\ & =& 2.00×{\text{10}}^{\text{12}}\phantom{\rule{0.25em}{0ex}}\text{kg.}\end{array}$

Discussion

A large reservoir contains a very large mass of water. In this example, the weight of the water in the reservoir is $\text{mg}=1\text{.}\text{96}×{\text{10}}^{\text{13}}\phantom{\rule{0.25em}{0ex}}\text{N}$ , where $g$ is the acceleration due to the Earth’s gravity (about $9\text{.}\text{80}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}$ ). 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.

## Section summary

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

## 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?

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

## Problems&Exercises

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

$1\text{.}\text{610}\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{3}$

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 $\text{89}\text{.}0\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{3}$ 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.)

$2\text{.}\text{70}\phantom{\rule{0.25em}{0ex}}{\text{g/cm}}^{3}$

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?

$7\text{.}9×{\text{10}}^{2}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0ex}{0ex}}{\text{kg/m}}^{3}$

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.

$\text{15}\text{.}6\phantom{\rule{0.25em}{0ex}}{\text{g/cm}}^{3}$

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 ${\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}$ . The nucleus of an atom has a radius about ${\text{10}}^{-5}$ 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 $7×{\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{m}$ )?

(a) ${\text{10}}^{\text{18}}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}$

(b) $2×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{m}$

hi guys can you teach me how to solve a logarithm?
how about a conceptual framework can you simplify for me? needed please
Villaflor
Hello what happens when electrone stops its rotation around its nucleus if it possible how
Afzal
I think they are constantly moving
Villaflor
yep what is problem you are stuck into context?
S.M
not possible to fix electron position in space,
S.M
Physics
Beatriz
yes of course Villa flor
David
equations of kinematics for constant acceleration
A bottle full of water weighs 45g when full of mercury,it weighs 360g.if the empty bottle weighs 20g.calculate the relative density of mercury and the density of mercury....pls I need help
well You know the density of water is 1000kg/m^3.And formula for density is density=mass/volume Then we must calculate volume of bottle and mass of mercury: Volume of bottle is (45-20)/1000000=1/40000 mass of mercury is:(360-20)/1000 kg density of mercury:(340/1000):1/50000=(340•40000):1000=13600
Sobirjon
the latter is true
Sobirjon
100g of water is mixed with 60g of a liquid of relative density 1.2.assuming no changes in volume occurred,find the average relative density of the mixture...take density of water as 1g/cm3 and density of liquid 1.2g/cm3
Lila
plz hu can explain Heisenberg's uncertainty principle
who can help me with my problem about acceleration?
ok
Nicholas
how to solve this... a car is heading north then smoothly made a westward turn during the travel the speed of the car remains constant at 1.5km/h what is the acceleration of the car? the total travel time of the car as it smoothly changed its direction is 15 minutes
Vann
i think the acceleration is 0 since the car does not change its speed unless there are other conditions
Ben
yes I have to agree, the key phrase is, "the speed of the car remains constant...," all other information is not needed to conclude that acceleration remains at 0 during the entire time
Luis
who can help me with a relative density question
Lila
1cm3 sample of tin lead alloy has mass 8.5g.the relative density of tin is 7.3 and that of lead is 11.3.calculate the percentage by weight of tin in the alloy. assuming that there is no change of volume when the metals formed the alloy
Lila
morning, what will happen to the volume of an ice block when heat is added from -200°c to 0°c... Will it volume increase or decrease?
no
Emmanuel
hi what is physical education?
Kate
BPED..is my course.
Kate
No
Emmanuel
I think it is neither decreases nor increases ,it remains in the same volume because of its crystal structure
Sobirjon
100g of water is mixed with 60g of a liquid of relative density 1.2.assuming no changes in volume occurred,find the average relative density of the mixture. take density of water as 1g/cm3 and density of liquid as 1.2g/cm3
Lila
Sorry what does it means"no changes in volume occured"?
Sobirjon
volume can be the amount of space occupied by an object. But when an object does not change in shape it will still occupy the same space. Thats why the volume will still remain the same
Ben
Most soilds expand when heated but if it changes state at 0C it will have less volume. Ice floats because it is less dense ie a larger mass per unit volume.
Richard
how to calculate velocity
v=d/t
Emeka
Villaflor
Villaflor
v=d/t
Nisha
hello bro hw is life with you
Mine is good. How about you?
Chase
Hi room of engineers
yes,hi sir
Okwethu
hello
akinmeji
Hello
Mishael
hello
Jerry
hi
Sakhi
hi
H.C
so, what is going on here
akinmeji
Ajayi
good morning ppl
ABDUL
If someone has not studied Mathematics enough yet, should theu study it first then study Phusics or Study Basics of Physics whilst srudying Math as well?
whether u studied maths or not, it is advisable to start from d basics cuz it is essential to know dem
Nuru
yea you are right
wow, you got this w/o knowing math
Thomas
I guess that's it
Thomas
later people
Thomas
mathematics is everywhere
Anand
thanks but dat doesn't mean it is good without maths @Riaz....... Maths is essential in sciences particularly wen it comes to PHYSICS but PHYSICS must be started from the basic which may also help in ur mathematical ability
Nuru
A hydrometer of mass 0.15kg and uniform cross sectional area of 0.0025m2 displaced in water of density 1000kg/m3.what depth will the hydrometer sink
Lila
16.66 meters?
Darshik
16.71m2
aways
,i have a question of let me give answer
aways
the mass is stretched a distance of 8cm and held what is the potential energy? quick answer
aways
oscillation is a to and fro movement, it can also be referred to as vibration. e.g loaded string, loaded test tube or an hinged door
what property makes the magnet to break?
difference between charge and force
charges: these are groups of ions which can be positive or negative. while force is said to be push or pull of an object. Generally, charge deals with ions while force deals with particles.
akinmeji
define oscillation
Munachukwu
what is up thrust
upward force exerted by water on a body.
Matthew
A hydrometer of mass 0.15kg and of uniform cross sectional area 0.0025m2 displaced in water of density 1000kg/m3.what depth will the hydrometer sink
Lila
archimedes principle
Nicholas
mg=p(area x height)g
Nicholas
solve for height
Nicholas
what is matter
Anything that has mass. And it occupies space. So it has volume.
wenhe
stages of matter.
daniel
States of matter are solid liquid gas and plasma
wenhe
what are Newtown's law
Lila
An object as rest will remain at rest or in motion will remain at motion. Force is mass times acceleration. And a force will have equal and opposite reaction.
wenhe
do we have any thing like plasma as a states of matter
daniel
no
Nicholas
Plasma is a state of matter. It's much later on. The sun for example is not a gas ball. Its plasma.
wenhe
why is plasma a state of matter
Lila
In lower grades you are taught that there are 3 states of matter. But later on, you'll learn that if you eat gas enough. It turns to plasma
wenhe
Heat*
wenhe
yes we have plasma as a state of matter
akinmeji
it is anything that occupie space and has mass
Anand
I never head of that before
daniel
Anything that occupies space and has mass is matter.
wenhe
what is the difference between vapor and gas
Lila
Plasma is best described as an ionized gas because it is made up of positively and negatively charged particles.
akinmeji
Vapour is gas.
wenhe
Aside plasma, we also have Bose- Einstein. this is also another state of matter
akinmeji
was bose Einstein discovered by Albert Einstein
Lila
We have like 6 states or more. But those are newly discovered.
wenhe
OK, thanks at list I have experience about that one now.
daniel
Einstein​ predicted it
wenhe
With another person
wenhe
what are the other states
Lila
Fermionic condensate
wenhe
OK what is Albert Einstein
daniel
And maybe more. I just rmb those. Tbh for school you only need to know 4.
wenhe
Albert Einstein​is the name of a scientist
wenhe
OK
daniel
what is the physical state of water and salt
Lila
can I ask off topic question?
Kristine
Both are matter. So can theoretically exist in all those states. But in room temperature, water is liquid and salt is solid.
wenhe
Yes
wenhe
Kristine
I'm not very good at maths. But if I can answer... Lol
wenhe
The reflector of a radiotelescope is in the shape of a parabola revolve its axis, if the diameter of the reflector is 400ft. above the vertex of the parabola, what should be the depth of reflector?
Kristine
Are there any other factors given?
wenhe
Or just that?
wenhe
matter is merely energy condensed to a slow vibration
Vashist
Cuz with those factors, I can't solve it.
wenhe
none
Kristine
Well. There must be another factor given or something. Like an equation of the parabola. Or else I can't solve it
wenhe
explain why water and salt are compounds
Lila
That's chem. But both water and salt are made of more than 1 element. So they are compounds.
wenhe
Compound meaning 2 or more things added together.
wenhe
salt chemically is combination of sodium n chloride whereas water is oxygen n hydrogen
Friday
deficiency of vitamin E cause ?
Friday
I don't know if this is the equation or formula your looking for; (x-h)²=4c(y-k) (x-h)²=-4c(y-k) (y-k)²=-4c(x-h) (y-k)²=4c(x-h)
Kristine
If the parabola opens downward, the equation will be (x-h)²=-4c(y-k). If the parabola opens upward, (x-h)²=4c(y-k). If the parabola opens to the left it will be -4c and if the parabola opens to the right it will be 4c
Kristine