CHAPTER 13 Liquids

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Nov 29, 2013 (3 years and 19 days ago)

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CHAPTER

13



Liquids

Summary of Terms

Pressure



The ratio of force to the area over wh
ich that force is distributed:
Liquid
pressure = weight density × depth

Buoyant force



The net upward force that a fluid exerts on an immersed object.

Archimedes’
principle



An immersed body is buoyed up by a force equal to the weight of the fluid it
displaces.

Principle of
flotation



A floating object displaces a weight of fluid equal to its own weight.

Pascal’s principle



The pressure applied
to a motionless fluid confined in a container is
transmitted undiminished throughout the fluid.

Surface tension



The tendency of the surface of a liquid to contract in area and thus to behave
like a stretched elastic membrane.

Capillarity



The rise

of a liquid in a fine, hollow tube or in a narrow space.


Review Questions

1.


Give two examples of a fluid.

Pressure

2.


Distinguish between force and pressure.

Pressure in a Liquid

3.


What is the relationship between liquid pressure and the
depth of a
liquid? Between liquid pressure and density?

4.


If you swim beneath the surface in salt water, will the pressure be
greater than in freshwater at the same depth? Why or why not?

5.


How does water pressure one meter below the surface of a sma
ll pond
compare with water pressure one meter below the surface of a huge
lake?

6.


If you punch a hole in a container filled with water, in what direction
does the water initially flow outward from the container?

Buoyancy

7.


Why does buoyant force act

upward on an object submerged in
water?

8.


Why is there no horizontal buoyant force on a submerged object?

9.


How does the volume of a completely submerged object compare
with the volume of water displaced?

Archimedes’ Principle

10.


How does the
buoyant force on a submerged object compare with the
weight of water displaced?

11.


Distinguish between a submerged body and an immersed body.

12.


What is the mass of 1 L of water? What is its weight in newtons?

13.


If a 1
-
L container is immersed
halfway into water, what is the volume
of water displaced? What is the buoyant force on the container?

What Makes an Object Sink or Float?

14.


Is the buoyant force on a submerged object equal to the weight of the
object itself or equal to the weight of
the fluid displaced by the
object?

15.


There is a condition in which the buoyant force on an object does
equal the weight of the object. What is this condition?

16.


Does the buoyant force on a submerged object depend on the volume
of the object or the
weight of the object?

17.


Fill in the blanks: An object denser than water will ______ in water.
An object less dense than water will ______ in water. An object with
the same density of water will ____________ in water.

18.


How is the density of a fish
controlled? How is the density of a
submarine controlled?

Flotation

19.


It was emphasized earlier that buoyant force does not equal an
object’s weight but does equal the weight of displaced water. Now we
say buoyant force equals the object’s weight.
Isn’t this a grand
contradiction? Explain.

20.


What weight of water is displaced by a 100
-
ton ship? What is the
buoyant force that acts on a floating 100
-
ton ship?

Pascal’s Principle

21.


What happens to the pressure in all parts of a confined fluid if

the
pressure in one part is increased?

22.


If the pressure in a hydraulic press is increased by an additional 10
N/cm
2
, how much extra load will the output piston support if its
cross
-
sectional area is 50 cm
2
?

Surface Tension

23.


What geometrical
shape has the least surface area for a given
volume?

24.


What is the cause of surface tension?

Capillarity

25.


Distinguish between adhesive and cohesive forces.

26.


What determines how high water will climb in a capillary tube?


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Projects



1.

Place an egg in a pan of tap water. Then

dissolve salt in the water until the egg floats. How
does the density of an egg compare to that of tap water? To that of salt water?


2.

If you punch a couple of holes in the bottom of a water
-
filled container, water will spurt out
because of water
pressure. Now drop the container, and, as it freely falls, note that the water
no longer spurts out! If your friends don’t understand this, could you figure it out and then
explain it to them?



(Click image to enlarge)



3.

Float a water
-
soaked Ping
-
Pong ball in a can of water held more than a meter above a rigid
floor. Then drop the can. Careful inspection will show the ball pulled beneath the surface as
both the ball and the can drop.
(What does this say about surface tension?). More
dramatically, when the can makes impact with the floor, what happens to the ball, and why?
Try it and you’ll be astonished! (Caution: Unless you’re wearing safety goggles, keep your
head away from above the

can when it makes impact.)


4.

Soap greatly weakens the cohesive forces between water molecules. You can see this by
putting some oil in a bottle of water and shaking it so that the oil and water mix. Notice that
the oil and water quickly separate as so
on as you stop shaking the bottle. Now add some
soap to the mixture. Shake the bottle again and you will see that the soap makes a fine film
around each little oil bead and that a longer time is required for the oil to gather after you
stop shaking the bot
tle.


This is how soap works in cleaning. It breaks the surface tension around each particle of
dirt so that the water can reach the particles and surround them. The dirt is carried away in
rinsing. Soap is a good cleaner only in the presence of water.



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One
-
Step Calculations

Pressure = weight
density × depth

(Neglect the pressure due to the atmosphere in the calculations below.)

1.


Calculate the water pressure at the bottom of the 100
-
m
-
high water
tower shown in Figure 13.2.


2.


Calculate the water pressure at the base of a dam when the dept
h of
water behind the dam is 100 m.


3.


The top floor of a building is 50 m above the basement. Calculate how
much greater the water pressure is in the basement compared with the
pressure at the top floor.


4.


Water pressure at the bottom of a 1
-
m
-
tall closed barrel is 98 kPa.
What is the pressure at the bottom of the barrel when a 5
-
m pipe filled
with water is inserted into the top of the barrel?


Exercises

1.


What common liquid covers more than two
-
thirds
of our planet,
makes up 60% of our bodies, and sustains our lives and lifestyles in
countless ways?

2.


Which is more likely to hurt

being stepped on by a 200
-
lb man
wearing loafers or being stepped on by a 100
-
lb woman wearing high
heels?

3.


Which do

you suppose exerts more pressure on the ground

an
elephant or a lady standing on spike heels? (Which will be more likely
to make dents in a linoleum floor?) Approximate a rough calculation
for each.

4.


Stand on a bathroom scale and read your weight. Wh
en you lift one
foot up so that you’re standing on one foot, does the reading change?
Does a scale read force or pressure?

5.


Why are persons who are confined to bed less likely to develop
bedsores on their bodies if they use a waterbed rather than an
ordinary
mattress?

6.


You know that a sharp knife cuts better than a dull knife. Do you
know why this is so? Defend your answer.

7.


If water faucets upstairs and downstairs are turned fully on, will more
water per second flow out of the upstairs
faucets or the downstairs
faucets?

8.


The photo shows physics instructor Marshall Ellenstein walking
barefoot on broken glass bottles in his class. What physics concept is
Marshall demonstrating, and why is he careful that the broken pieces
are small an
d numerous? (The Band
-
Aids on his feet are for humor!)



(Click image to enlarge)


9.


Why does your body get more rest when you’re lying down than it
does when you’re sitting? And why is blood pressure measured in the
upper arm, at the elevation of your heart? Is blood pressure in your
legs greater?

10.


When standing, blood pressure in y
our legs is greater than in your
upper body. Would this be true for an astronaut in orbit? Defend your
answer.

11.


How does water pressure 1 meter beneath the surface of a lake
compare with water pressure 1 meter beneath the surface of a
swimming pool?

12.


Which teapot holds more liquid?




(Click image to enlarge)


13.


The sketch shows a reservoir that supplies water to a farm. It is made
of wood and is reinforced with metal hoops. (a) Why is it elevated?
(b) Why are the hoops closer together near the

bottom part of the
tank?



(Click image to enlarge)


14.


A block of aluminum with a volume of 10 cm
3

is placed in a beaker
of water filled to the brim. Water overflows. The same is done in
another beaker with a 10
-
cm
3

block of lead. Does the lead displace
more, less, or the same amount of water?

15.


A block of aluminum with a mass of 1 kg is placed in

a beaker of
water filled to the brim. Water overflows. The same is done in
another beaker with a 1
-
kg block of lead. Does the lead displace
more, less, or the same amount of water?

16.


A block of aluminum with a weight of 10 N is placed in a beaker of
water filled to the brim. Water overflows. The same is done in
another beaker with a 10
-
N block of lead. Does the lead displace
more, less, or the same amount of water? (Why are your answers to
this exercise and to Exercise 15 different from your answer to

Exercise 14?)

17.


In 1960, the U.S. Navy’s bathyscaphe Trieste (a submersible)
descended to a depth of nearly 11 kilometers in the Marianas Trench
near the Philippines in the Pacific Ocean. Instead of a large viewing
window, it was a small circular win
dow 15 centimeters in diameter.
What is your explanation for so small a window?

18.


There is a story about Pascal in which it is said that he climbed a
ladder and poured a small container of water into a tall, thin, vertical
pipe inserted into a wooden
barrel full of water below. The barrel
burst when the water in the pipe reached about 12 m. This was all the
more intriguing because the weight of added water in the tube was
very small. What two physical principles was Pascal demonstrating?

19.


There i
s a legend of a Dutch boy who bravely held back the whole
North Sea by plugging a hole in a dike with his finger. Is this possible
and reasonable? (See also Problem 4.)

20.


If you’ve wondered about the flushing of toilets on the upper floors of
city sky
scrapers, how do you suppose the plumbing is designed so
that there is not an enormous impact of sewage arriving at the
basement level? (Check your speculations with someone who is
knowledgeable about architecture.)

21.


Why does water “seek its own
level”?

22.


Suppose that you wish to lay a level foundation for a home on hilly
and bushy terrain. How can you use a garden hose filled with water to
determine equal elevations for distant points?

23.


When you are bathing on a stony beach, why do the

stones hurt your
feet less when you’re standing in deep water?

24.


If liquid pressure were the same at all depths, would there be a
buoyant force on an object submerged in the liquid? Explain.

25.


A can of diet soda floats in water, whereas a can of

regular soda
sinks. Explain this phenomenon first in terms of density, then in terms
of weight versus buoyant force.

26.


Why will a block of iron float in mercury but sink in water?

27.


The mountains of the Himalayas are slightly less dense than the

mantle material upon which they “float.” Do you suppose that, like
floating icebergs, they are deeper than they are high?

28.


Why is a high mountain composed mostly of lead an impossibility on
the planet Earth?

29.


How much force is needed to push a

nearly weightless but rigid 1
-
L
carton beneath a surface of water?

30.


Why will a volleyball held beneath the surface of water have more
buoyant force than if it is floating?

31.


Why does an inflated beach ball pushed beneath the surface of water
sw
iftly shoot above the water surface when released?

32.


Why is it inaccurate to say that heavy objects sink and that light
objects float? Give exaggerated examples to support your answer.

33.


Why is the buoyant force on a submerged submarine
appreciably
greater than the buoyant force on it while it is floating?

34.


A piece of iron placed on a block of wood makes it float lower in the
water. If the iron were instead suspended beneath the wood, would it
float as low, lower, or higher? Defend
your answer.




(Click image to enlarge)


35.


Compared with an empty ship, would a ship loaded with a cargo of
Styrofoam sink deeper into the water or rise in the water? Defend
your answer.

36.


If a submarine starts to sink,
will it continue to sink to the bottom if
no changes are made? Explain.

37.


A barge filled with scrap iron is in a canal lock. If the iron is thrown
overboard, does the water level at the side of the lock rise, fall, or
remain unchanged? Explain.

38.


Would the water level in a canal lock go up or down if a battleship in
the lock sank?

39.


Will a rock gain or lose buoyant force as it sinks deeper in water? Or
will the buoyant force remain the same at greater depths? Defend
your answer.

40.


Will a

swimmer gain or lose buoyant force as she swims deeper in the
water? Or will her buoyant force remain the same at greater depths?
Defend your answer, and contrast it with your answer to Exercise 39.

41.


A balloon is weighted so that it is barely able
to float in water. If it is
pushed beneath the surface, will it return to the surface, stay at the
depth to which it is pushed, or sink? Explain. (Hint: Does the
balloon’s density change?)




(Click image to enlarge)


42.


The density of a rock doesn’t change when it is submerged in water,
but your density changes when you are submerged. Explain.

43.


In answering the question of why bodies float higher

in salt water
than in freshwater, your friend replies that the reason is that salt water
is denser than freshwater. (Does your friend often answer questions
by reciting only factual statements that relate to the answers but don’t
provide any concrete reas
ons?) How would you answer the same
question?

44.


A ship sailing from the ocean into a freshwater harbor sinks slightly
deeper into the water. Does the buoyant force on the ship change? If
so, does it increase or decrease?

45.


Suppose that you are gi
ven the choice between two life preservers
that are identical in size, the first a light one filled with Styrofoam and
the second a very heavy one filled with gravel. If you submerge these
life preservers in the water, upon which will the buoyant force be
greater? Upon which will the buoyant force be ineffective? Why are
your answers different?

46.


The weight of the human brain is about 15 N. The buoyant force
supplied by fluid around the brain is about 14.5 N. Does this mean
that the weight of fluid surrounding the brain is at least 14.5 N?
Defend your answer.

47.


The relative densities of water,

ice, and alcohol are 1.0, 0.9, and 0.8,
respectively. Do ice cubes float higher or lower in a mixed alcoholic
drink? What comment can you make about a cocktail in which the ice
cubes lie submerged at the bottom of the glass?

48.


When an ice cube in a
glass of water melts, does the water level in the
glass rise, fall, or remain unchanged? Does your answer change if the
ice cube has many air bubbles? How about if the ice cube contains
many grains of heavy sand?

49.


When the wooden block is placed in
the beaker, what happens to the
scale reading? Answer the same question for an iron block.




(Click image to enlarge)


50.


A half
-
filled bucket of water is on a spring scale. Will the reading of
the scale increase or remain the same if a fish is placed in the bucket?
(Will your answer be different if the bucket is initially filled to the
brim?)

51.


The weight of the contain
er of water, as shown in a, is equal to the
weight of the stand and the suspended solid iron ball. When the
suspended ball is lowered into the water, as shown in b, the balance is
upset. Will the additional weight needed on the right side to restore
balanc
e be greater than, equal to, or less than the weight of the solid
iron ball?




(Click image to enlarge)


52.


If the gravitational field of the Earth were to increase, would a fish
float to the surface, sink, or stay at the same depth?

53.


What would you experience when swimming in water in an orbiting
space habitat where simulated gravity is
g? Would you float in
the water as you do on Earth?

54.


We say that the shape of a l
iquid is that of its container. But, with no
container and no gravity, what is the natural shape of a blob of water?
Why?

55.


If you release a Ping
-
Pong ball beneath the surface of water, it will
rise to the surface. Would it do the same if it were insi
de a big blob of
water floating weightless in an orbiting spacecraft?

56.


So you’re on a run of bad luck, and you slip quietly into a small, quiet
pool as hungry crocodiles lurking at the bottom are relying on
Pascal’s principle to help them to detect a

tender morsel. What does
Pascal’s principle have to do with their delight at your arrival?

57.


In the hydraulic arrangement shown, the larger piston has an area that
is fifty times that of the smaller piston. The strong man hopes to exert
enough force
on the large piston to raise the 10 kg that rest on the
small piston. Do you think he will be successful? Defend your
answer.




(Click image to enlarge)


58.


In the hydraulic arrangement shown in Figure 13.22, the
multiplication of force is equal to the ratio of the areas of the large
and small pistons. Some people are surprised to learn
that the area of
the liquid surface in the reservoir of the arrangement shown in Figure
13.23 is immaterial. What is your explanation to clear up this
confusion?

59.


Why will hot water leak more readily than cold water through small
leaks in a car radia
tor?

60.


On the surface of a pond, it is common to see water striders, insects
that can “walk” on the surface of water without sinking. What physics
concept explains their ability?



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©


2010 Pearson
Education, Inc.

Problems

1.


The depth of water behind the Hoover Dam in Nevada is 220 m. What
is the water pressure at the base of this dam? (Neglect the pressure
due to the atmosphere.)

2.


A 6
-
kg piece of metal displaces 1 liter of water when
submerged.
What is its density?

3.


A rectangular barge, 5 m long and 2 m wide, floats in freshwater. (a)
Find how much deeper it floats when its load is a 400
-
kg horse. (b) If
the barge can only be pushed 15 cm deeper into the water before
water overflo
ws to sink it, how many 400
-
kg horses can it carry?

4.


A dike in Holland springs a leak through a hole of area 1 cm
2

at a
depth of 2 m below the water surface. How much force must a boy
apply to the hole with his thumb to stop the leak? Could he do it?

5.


A merchant in Kathmandu sells you a solid
-
gold 1
-
kg statue for a very
reasonable price. When you return home, you wonder whether or not
you got a bargain, so you lower the statue into a container of water
and measure the volume of displaced water.
What volume will verify
that it’s pure gold?

6.


When a 2.0
-
kg object is suspended in water, it “masses” 1.5 kg. What
is the density of the object?

7.


An ice cube measures 10 cm on a side and floats in water. One cm
extends above water level. If you s
haved off the 1
-
cm part, how many
cm of the remaining ice would extend above water level?

8.


A swimmer wears a heavy belt to make her average density exactly
equal to the density of water. Her mass, including the belt, is 60 kg.
(a) What is the swimmer’
s weight in newtons? (b) What is the
swimmer’s volume in m
3
? (c) At a depth of 2 m below the surface of a
pond, what buoyant force acts on the swimmer? What net force acts
on her?

9.


A vacationer floats lazily in the ocean with 90% of his body below the

surface. The density of the ocean water is 1,025 kg/m
3
. What is the
vacationer’s average density?

10.


In the hydraulic pistons shown in the sketch, the small piston has a
diameter of 2 cm. The larger piston has a diameter of 6 cm. How
much more force can the larger piston exert compared with the force
applied to the smaller piston?



Solutions to Chapter

13 Exercises



1.

Water.



2.

Pressure would be appreciably greater by the woman, which would hurt you more.



3.

A woman with spike heels exerts considerably more pressure on the ground than an elephant! Example: A 500
-
N woman with 1
-
cm
2
spike heels put
s half her weight on each foot, distributed (let’s say) half on her heel and
half on her sole. So the pressure exerted by each heel will be (125 N/1 cm
2
) = 125 N/cm
2
. A 20,000
-
N elephant
with 1000 cm
2

feet exerting
1
/
4

its weight on each foot produces (500
0N/1000 cm
2
) = 5N/cm
2
; about 25 times
less pressure. (So a woman with spike heels will make greater dents in a new linoleum floor than an elephant
will.)



4.

The

scale measures force, not pressure, and is calibrated to read your weight. That’s why your weight on the
scale is the same whether you stand on one foot or both.



5.

There is less pressure with a waterbed due to the greater contact area.



6.

A sharp kni
fe cuts better than a dull knife because it has a thinner cutting area which results in more cutting
pressure for a given force.



7.

More water will flow from a downstairs open faucet because of the greater pressure. Since pressure depends
on depth, the downstairs faucet is effectively “deeper” than the upstairs faucet. The pressure downstairs is
greater by an amount = weight density


depth, where the depth is the vertical distance between faucets.



8.

The concept of pressure is being demonstrated. He is careful that the pieces are small and numerous so that
his weight is applied over a large area of contact. Then the sharp glass provi
des insufficient pressure to cut the
feet.



9.

Your body gets more rest when lying than when sitting or standing because when lying, the heart does not have
to pump blood to the heights that correspond to standing or sitting. Blood pressure is normally gr
eater in the
lower parts of your body simply because the blood is “deeper” there. Since your upper arms are at the same
level as your heart, the blood pressure in your upper arms will be the same as the blood pressure in your heart.



10.

No, in orbit ther
e are no pressure differences due to gravity.



11.

Both are the same, for pressure depends on depth.



12.

The water can be no deeper than the spouts, which are at the same height, so both teapots hold the same
amount of liquid.



13.

(a) The reservoir is

elevated so as to produce suitable water pressure in the faucets that it serves. (b) The
hoops are closer together at the bottom because the water pressure is greater at the bottom. Closer to the top,
the water pressure is not as great, so less reinforcem
ent is needed there.



14.

Both blocks have the same volume and therefore displace the same amount of water.



15.

A one
-
kilogram block of aluminum is larger than a one
-
kilogram block of lead. The aluminum therefore
displaces more water.


16.

A 10
-
N block
of aluminum is larger than a 10
-
N block of lead. The aluminum therefore displaces more water.
Only in Exercise 10 were the volumes of the block equal. In this and the preceding exercise, the aluminum
block is larger. (These exercises serve only to emphasiz
e the distinctions between volume, mass, and weight.)



17.

The smaller the window area, the smaller the crushing force of water on it.



18.

One, that water pressure depends on depth. The other, that the pressure due to the column of water is
transmitted

to all parts of the barrel.



19.

From a physics point of view, the event was quite reasonable, for the force of the ocean on his finger would
have been quite small. This is because the pressure on his finger has only to do with the depth of the water,
specifically the distance of the leak below the sea level

not the weight of the ocean. For a numerical example,
see Problem 4.



20.

A typical plumbing design involves short sections of pipe bent at 45
-
degree angles between vertical sections
two
-
stories lo
ng. The sewage therefore undergoes a succession of two
-
story falls which results in a moderate
momentum upon reaching the basement level.



21.

Water seeking its own level is a consequence of pressure depending on depth. In a bent U
-
tube full of water, for

example, the water in one side of the tube tends to push water up the other side until the pressures at the same
depth in each tube are equal. If the water levels were not the same, there would be more pressure at a given
level in the fuller tube, which w
ould move the water until the levels were equal.



22.

The use of a water
-
filled garden hose as an elevation indicator is a practical example of water seeking its own
level. The water surface at one end of the hose will be at the same elevation above sea l
evel as the water
surface at the other end of the hose.



23.

In deep water, you are buoyed up by the water displaced and as a result, you don’t exert as much pressure
against the stones on the bottom. When you are up to your neck in water, you hardly feel

the bottom at all.



24.

Buoyant force is the result of differences in pressure; if there are no pressure differences, there is no buoyant
force. This can be illustrated by the following example: A Ping
-
Pong ball pushed beneath the surface of water
will n
ormally float back to the surface when released. If the container of water is in free fall, however, a
submerged Ping
-
Pong ball will fall with the container and make no attempt to reach the surface. In this case
there is no buoyant force acting on the ball

because there are no pressure differences

the local effects of
gravity are absent.



25.

The diet drink is less dense than water, whereas the regular drink is denser than water. (Water with dissolved
sugar is denser than pure water.) Also, the weight of t
he can is less than the buoyant force that would act on it if
totally submerged. So it floats, where buoyant force equals the weight of the can.



26.

Mercury is more dense (13.6 g/cm
3
) than iron. A block of iron will displace its weight and still be parti
ally
above the mercury surface. Hence it floats in mercury. In water it sinks because it cannot displace its weight.




27.

Mountain ranges are very similar to icebergs: Both float in a denser medium, and extend farther down into that
medium than they extend above it. Mountains, like icebergs, are bigger than they appear to be. The concept of
floating mountains is
isostacy

Arch
imedes’ principle for rocks.


28.

A mostly
-
lead mountain would be more dense than the mantle and would sink in it. Guess where most of the
iron in the world is. In the Earth’s center!



29.

The force needed will be the weight of 1 L of water, which is 9.8
N. If the weight of the carton is not negligible,
then the force needed would be 9.8 N minus the carton’s weight, for then the carton would be “helping” to push
itself down.



30. When the ball is held beneath the surface, it displaces a greater weight of
water.



31.

The buoyant force on the ball beneath the surface is much greater than the force of gravity on the ball,
producing a large net force and large acceleration.



32.

Heavy objects may or may not sink, depending on their densities (a heavy log flo
ats while a small rock sinks, or
a boat floats while a paper clip sinks, for example). People who say that heavy objects sink really mean that
dense objects sink. Be careful to distinguish between how heavy an object is and how dense it is.



33.

While flo
ating, BF equals the weight of the submarine. When submerged, BF equals the submarine’s weight
plus

the weight of water taken into its ballast tanks. Looked at another way, the submerged submarine
displaces a greater weight of water than the same submarine

floating.



34.

The block of wood would float higher if the piece of iron is suspended below it rather than on top of it. By the law
of flotation: The iron
-
and
-
wood unit displaces its combined weight and the same volume of water whether the
iron is on top

or the bottom. When the iron is on the top, more wood is in the water; when the iron is on the
bottom, less wood is in the water. Or another explanation is that when the iron is below

submerged

buoyancy on it reduces its weight and less of the wood is pul
led beneath the water line.



35.

When a ship is empty its weight is least and it displaces the least water and floats highest. Carrying a load of
anything increases its weight and makes it float lower. It will float as low carrying a few tons of Styrofoam as it
will carrying the same numb
er of tons of iron ore. So the ship floats lower in the water when loaded with
Styrofoam than when empty. If the Styrofoam were outside the ship, below water line, then the ship would float
higher as a person would with a life preserver.



36.

A sinking su
bmarine will continue to sink to the bottom so long as the density of the submarine is greater than
the density of the surrounding water. If nothing is done to change the density of the submarine, it will continue to
sink because the density of water is pr
actically constant. In practice, water is sucked into or blown out of a
submarine’s tanks to adjust its density to match the density of the surrounding water.


37.

The water level will fall. This is because the iron will displace a greater amount of water
while being supported
than when submerged. A floating object displaces its weight of water, which is more than its own volume, while
a submerged object displaces only its volume. (This may be illustrated in the kitchen sink with a dish floating in
a dishpa
n full of water. Silverware in the dish takes the place of the scrap iron. Note the level of water at the
side of the dishpan, and then throw the silverware overboard. The floating dish will float higher and the water
level at the side of the dishpan will
fall. Will the volume of the silverware displace enough water to bring the
level to its starting point? No, not as long as it is denser than water.)



38.

For the same reason as in the previous exercise, the water level will fall. (Try this one in your kit
chen sink also.
Note the water level at the side of the dishpan when a bowl floats in it. Tip the bowl so it fills and submerges,
and you’ll see the water level at the side of the dishpan fall.)


39.

Bouyant force will remain unchanged on the sinking rock
because it displaces the same weight of water at any
depth.



40.

Bouyant

force on a sinking swimmer will decrease as she sinks. This is because her body, unlike the rock in the
previous exercise, will be compressed by the greater pressure of greater depths.



41.

The balloon will sink to the bottom because its density increas
es with depth. The balloon is compressible, so the
increase in water pressure beneath the surface compresses it and reduces its volume, thereby increasing its
density. Density is further increased as it sinks to regions of greater pressure and compression.

This sinking is
understood also from a buoyant force point of view. As its volume is reduced by increasing pressure as it
descends, the amount of water it displaces becomes less. The result is a decrease in the buoyant force that
initially was sufficient
to barely keep it afloat.



42.

You are compressible, whereas a rock is not, so when you are submerged, the water pressure tends to
squeeze in on you and reduce your volume. This increases your density. (Be careful when swimming

at
shallow depths you may s
till be less dense than water and be buoyed to the surface without effort, but at
greater depths you may be pressed to a density greater than water and you’ll have to swim to the surface.)



43.

A body floats higher in denser fluid because it does not have

to sink as far to displace a weight of fluid equal to
its own weight. A smaller volume of the displaced denser fluid is able to match the weight of the floating body.



44.

The buoyant force does not change. The buoyant force on a floating object is alway
s equal to that object’s
weight, no matter what the fluid.



45.

Since both preservers are the same size, they will displace the same amount of water when submerged and be
buoyed up with equal forces. Effectiveness is another story. The amount of buoyant f
orce exerted on the heavy
gravel
-
filled preserver is much less than its weight. If you wear it, you’ll sink. The same amount of buoyant force
exerted on the lighter Styrofoam preserver is greater than its weight and it will keep you afloat. The
amount

of
t
he force and the
effectiveness

of the force are two different things.



46.

No, there does not have to actually be 14.5 N of fluid in the skull to supply a buoyant force of 14.5 N on the
brain. To say that the buoyant force is 14.5 N is to say that the
brain is taking up the space that 14.5 N of fluid
would occupy if fluid instead of the brain were there. The amount of fluid in excess of the fluid that immediately
surrounds the brain does not contribute to the buoyancy on the brain. (A ship floats the sa
me in the middle of
the ocean as it would if it were floating in a small lock just barely larger than the ship itself. As long as there is
enough water to press against the hull of the ship, it will float. It is not important that the amount of water in th
is
tight
-
fitting lock weigh as much as the ship

think about that, and don’t let a literal word explanation “a floating
object displaces a weight of fluid equal to its own weight” and the idea it represents confuse you.)



47.

Ice cubes will float lower in
a mixed drink because the mixture of alcohol and water is less dense than water. In
a less dense liquid a greater volume of liquid must be displaced to equal the weight of the floating ice. In pure
alcohol, the volume of alcohol equal to that of the ice cu
bes weighs less than the ice cubes, and buoyancy is
less than weight and ice cubes will sink. Submerged ice cubes in a cocktail indicate that it is predominantly
alcohol.



48.

When the ice cube melts the water level at the side of the glass is unchanged (
neglecting temperature effects).
To see this, suppose the ice cube to be a 5 gram cube; then while floating it will displace 5 grams of water. But
when melted it becomes the same 5 grams of water. Hence the water level is unchanged. The same occurs
when th
e ice cube with the air bubbles melts. Whether the ice cube is hollow or solid, it will displace as much
water floating as it will melted. If the ice cube contains grains of heavy sand, however, upon melting, the water
level at the edge of the glass will d
rop. This is similar to the case of the scrap iron of Exercise 38.



49.

The total weight on the scale is the same either way, so the scale reading will be the same whether or not the
wooden block is outside or floating in the beaker. Likewise for an iron
block, where the scale reading shows the
total weight of the system.



50.

If water doesn’t overflow, the reading on the scale will increase by the ordinary weight of the fish. However, if
the bucket is brim filled so a volume of water equal to the volume

of the fish overflows, then the reading will not
change. We assume here that the fish and water have the same density.



51.

When the ball is submerged (but not touching the bottom of the container), it is supported partly by the buoyant
force on the left

and partly by the string connected to the right side. So the left pan must increase its upward
force to provide the buoyant force in addition to whatever force it provided before, and the right pan’s upward
force decreases by the same amount, since it now

supports a ball lighter by the amount of the buoyant force. To
bring the scale back to balance, the additional weight that must be put on the right side will equal twice the
weight of

water displaced by the submerged ball. Why twice? Half of the added wei
ght makes up for the loss of
upward force on the right, and the other half for the equal gain in upward force on the left. (If each side initially
weighs 10 N and the left side gains 2 N to become 12 N, the right side loses 2 N to become 8 N. So an
additio
nal weight of 4 N, not 2 N, is required on the right side to restore balance.) Because the density of water
is less than half the density of the iron ball, the restoring weight, equal to twice the buoyant force, will still be
less than the weight of the ba
ll.



52.

If the gravitational field of the Earth increased, both water and fish would increase in weight and weight density
by the same factor, so the fish would stay at its prior level in water.



53.

Both you and the water would have half the weight de
nsity as on Earth, and you would float with the same
proportion of your body above the water as on Earth. Water splashed upward with a certain initial speed would
rise twice as high, since it would be experiencing only half the “gravity force.” Waves on th
e water surface would
move more slowly than on Earth (at about 70% as fast since v
wave

~


g
).



54.

Because of surface tension, which tends to minimize the surface of a blob of water, its shape without gravity
and other distorting forces will be a
sphere

the shape with the least surface area for a given volume.



55.

A Ping
-
Pong ball in water in a zero
-
g

environment would experience no buoy
ant force. This is because
buoyancy depends on a pressure difference on different sides of a sub
merged body. In thi
s weightless state, no
pressure difference would exist because no water pressure exists. (See the answer to Exercise 20, and Home
Project 2.)



56.

Part of whatever pressure you add to the water is transmitted to the hungry crocodiles, via Pascal’s princip
le. If
the water were confined, that is, not open to the atmosphere, the crocs would receive every bit of pressure you
exert. But even if you were able to slip into the pool to quietly float without exerting pressure via swimming
strokes, your displacement

of water raises the water level in the pool. This ever
-
so
-
slight rise, and
accompanying ever
-
so
-
slight increase in pressure at the bottom of the pool, is an ever
-
so
-
welcome signal to the
hungry crocodiles.


57.

The strong man will be unsuccessful. He will

have to push with 50 times the weight of the 10

kilograms. The
hydraulic arrangement is arranged to his disadvantage. Ordinarily, the input force is applied against the smaller
piston and the output force is exerted by the large piston

this arrangement is

just the opposite.



58.

In Figure 13.21, the increased pressure in the reservoir is a result of the applied force distributed over the input
piston area. This increase in pressure is transmitted to the output piston. In Figure 13.23, however, the pressur
e
increase is supplied by the mechanical pump, which has nothing to do with the area of fluid interface between
the compressed air and the liquid.



59.

When water is hot, the molecules are moving more rapidly and do not cling to one another as well as whe
n they
are slower moving, so the surface tension is less. The lesser surface tension of hot water allows it to pass more
readily through small openings.



60.

Surface tension accounts for the walking of water striders, needles that appear to float, and eve
n razor blades
that also appear to float. In these cases the weights of the objects are less than the restoring forces in the water
surface that tends to resist stretching.



Chapter 13 Problem Solutions



1.

Pressure = weight density


depth = 10,000 N/m
3


220 m = 2,200,000 N/m
2

=

2200

kPa
(or for density = 9800
N/m
3
, pressure = 2160 kPa.




2.

Density =
m/V

= 6 kg/1 liter =
6 kg/liter
. (Since there are 1000 liters in 1 cubic meter,
density may be
expressed in units kg/m
3
. Density = 6 kg/1 liter


1000 liter/m
3

= 6000

kg/m
3
, six times the density of
water.)



3.

(a) The volume of the extra water displaced will weigh as much as the 400
-
kg horse. And the volume of extra
water displaced will also equal the area of the barge t
imes the extra depth. That is,



V = Ah
, where
A

is the horizontal area of the barge; Then
h =
V
A

.




Now A = 5m


2m = 10 m
2
; to find the volume V of barge pushed into the water by the horse’s weight, which
equals the volume of water
displaced, we know that



density =
m
V


. Or from this,
V

=
m
density

=
400kg
1000kg/m
3

= 0.4 m
3
.



So
h =
V
A

=
0.4 m
3
10 m
2

=
0.04 m
, which is 4 cm deeper.



(b) If eac
h horse will push the barge 4 cm deeper, the question becomes: How many 4
-
cm
increments will
make 15 cm? 15/4 = 3.75, so 3 horses can be carried without sinking.
4

horses

will sink the barge.



4.

First you must find the pressure. It is weight density


de
pth = (10,000 N/m
3
)(2 m) =
20,000 N/m
2
, or 20,000
Pa. Force is pressure


area, and 1 cm
2

= 10
-
4

m
2
, so F = (20,000

N/m
2
)(10
-
4

m
2
) =
2 N
. It would be
easy for the boy to exert this force. It is about the weight of a notebook or a small box of cereal.
(Note: Air
pressure is not figured into this calculation because its effect in pushing down on the water from above is
canceled by its effect in pushing from outside the hole against the leaking water.)



5.

From Table 12.1 the density of gold is 19.3 g/cm
3
. Your gold has a mass of 1000 grams, so
1000 g
V

= 19.3
g/cm
3
. Solving for V,



V =
1000 g
19.3 g/cm
3

=
51.8 cm
3
.



6.

Density =
mass
volume

=
2.0 kg
volume of
(
2.0
-

1.5
)

kg of water

=
2.0 kg
0.5 l





= 4 kg/liter. And since 1 liter = 10
3

cm
3

= 10
-
3

m
3
, density =
4,000 kg/m
3
.




(Or this can be reasoned as follows: The buoyant force on the object is the force needed to support 0.5 kg, so
0.5 kg of water is displaced. Since
density is mass/volume, volume is mass/density, and displaced volume =
(0.5 kg)/(1000 kg/m
3
) = 5


10
-
4

m
3
. The object’s volume is the same as the volume it displaces, so the object’s
density is mass/volume =



(2 kg)/(5


10
-
4

m
3
) = 4000 kg/m
3
, four
times the density of water.)



7.

10% of ice extends above water. So 10% of the 9
-
cm thick ice would float above the water line;
0.9 cm
. So the
ice pops up. Interestingly, when mountains erode they become lighter and similarly pop up! Hence it takes a
long

time for mountains to wear away.


8.

(a) Weight = m
g

= (60 kg)(10 m/s
2
) =
600 N

(or 588 N if 9.8 m/s
2

is used).



(b) She has the same density as water, 1000 kg/m
3
. Since density = mass/volume, volume = mass/density, so
volume = (60 kg)/(1000 kg/m
3
) =
0
.06 m
3
.



(c) Buoyant force = weight of water displaced = 600 N (or 588 N). Her weight balances the buoyant force, so net
force =
0
.



9.

The displaced water, with a volume 90 percent of the vacationer’s volume, weighs the same as the vacationer
(to provi
de a buoyant force equal to his weight). Therefore his density is
90 percent of the water’s density.
Vacationer’s density = (0.90)(1,025 kg/m
3
) =
923

kg/m
3
.



10.

The relative areas are as the squares of the diameters; 6
2
/2
2

= 36/4 = 9. The larger piston can lift 9 times the
input force to the smaller piston.