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

Questions & Answers

what is temperature
Adeleye Reply
temperature is the measure of degree of hotness or coldness of a body. measured in kelvin
Ahmad
a characteristic which tells hotness or coldness of a body
babar
Average kinetic energy of an object
Kym
average kinetic energy of the particles in an object
Kym
A measure of the quantity of heat contained in an object which arises from the average kinetic energy of the constituent particles of that object. It can be measured using thermometers. It has a unit of kelvin in the thermodynamic scale and degree Celsius in the Celsius scale.
ibrahim
Mass of air bubble in material medium is negative. why?
Hrithik Reply
a car move 6m. what is the acceleration?
Umaru Reply
depends how long
Peter
What is the simplest explanation on the difference of principle, law and a theory
Kym Reply
how did the value of gravitational constant came give me the explanation
Varun Reply
how did the value of gravitational constant 6.67×10°-11Nm2kg-2
Varun
A steel ball is dropped onto a hard floor from a height of 1.50 m and rebounds to a height of 1.45 m. (a) Calculate its velocity just before it strikes the floor.
Kris Reply
9.8m/s?
Muhammad
Sqrt(2*1.5m*9.81m/s^2)
Richard
0.5m* mate.
Muhammad
0.05 I meant.
Muhammad
Guess your solution is correct considering the ball fall from 1.5m height initially.
Muhammad
Sqrt(2*1.5m*9.81m/s^2)
Deepak
How can we compare different combinations of capacitors?
Prakash Reply
find the dimension of acceleration if it's unit is ms-2
Happiness Reply
lt^-2
Ahmad
b=-2 ,a =1
Ahmad
M^0 L^1T^-2
Sneha
what is botany
Masha
it is a branch of science which deal with the study of plants animals and environment
Varun
what is work
Sunday Reply
a boy moving with an initial velocity of 2m\s and finally canes to rest with a velocity of 3m\s square at times 10se calculate it acceleration
Sunday
.
Abdul
6.6 lol 😁😁
Abdul
show ur work
Sunday
sorry..the answer is -10
Abdul
your question is wrong
Abdul
If the boy is coming to rest then how the hell will his final velocity be 3 it'll be zero
Abdul
re-write the question
Nicolas
men i -10 isn't correct.
Stephen
using v=u + at
Stephen
1/10
Happy
ya..1/10 is very correct..
Stephen
hnn
Happy
how did the value 6.67×10°-11Nm2kg2 came tell me please
Varun
Work is the product of force and distance
Kym
physicist
Michael
what is longitudinal wave
Badmus Reply
A longitudinal wave is wave which moves parallel or along the direction of propagation.
sahil
longitudinal wave in liquid is square root of bulk of modulus by density of liquid
harishree
Is British mathematical units the same as the United States units?(like inches, cm, ext.)
Nina Reply
We use SI units: kg, m etc but the US sometimes refer to inches etc as British units even though we no longer use them.
Richard
Thanks, just what I needed to know.
Nina
What is the advantage of a diffraction grating over a double slit in dispersing light into a spectrum?
Uditha Reply
can I ask questions?
Boniface Reply
yes.
Abdul
Yes
Albert
sure
Ajali
yeap
Sani
yesssss
bilal
hello guys
Ibitayo
when you will ask the question
Ana
anybody can ask here
bichu
is free energy possible with magnets?
joel
no
Mr.
you could construct an aparatus that might have a slightly higher 'energy profit' than energy used, but you would havw to maintain the machine, and most likely keep it in a vacuum, for no air resistance, and cool it, so chances are quite slim.
Mr.
calculate the force, p, required to just make a 6kg object move along the horizontal surface where the coefficient of friction is 0.25
Gbolahan
Yes ask
Albert
if a man travel 7km 30degree east of North then 10km east find the resultant displacement
Ajali Reply
11km
Dohn
disagree. Displacement is the hypotenuse length of the final position to the starting position. Find x,y components of each leg of journey to determine final position, then use final components to calculate the displacement.
Daniel
1.The giant star Betelgeuse emits radiant energy at a rate of 10exponent4 times greater than our sun, where as it surface temperature is only half (2900k) that of our sun. Estimate the radius of Betelgeuse assuming e=1, the sun's radius is s=7*10exponent8metres
James Reply
2. A ceramic teapot (e=0.20) and a shiny one (e=0.10), each hold 0.25 l of at 95degrees. A. Estimate the temperature rate of heat loss from each B. Estimate the temperature drop after 30mins for each. Consider only radiation and assume the surrounding at 20degrees
James

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