Fundamentals of Physics

David Halliday, Robert Resnick , Jearl Walker

Chapter 14

Fluids - all with Video Answers

Educators

Chapter Questions

Problem 1

A fish maintains its depth in fresh water by adjusting the air content of porous bone or air sacs to make its average density the same as that of the water. Suppose that with its air sacs collapsed, a fish has a density of 1.08 $\mathrm{g} / \mathrm{cm}^{3} .$ To what fraction of its expanded body volume must the fish inflate the air sacs to reduce its density to that of water?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 2

A partially evacuated airtight container has a tight-fitting lid
of surface area 77 $\mathrm{m}^{2}$ and negligible mass. If the force required to
remove the lid is 480 $\mathrm{N}$ and the atmospheric pressure is $1.0 \times 10^{5}$
Pa, what is the internal air pressure?

Averell Hause

Carnegie Mellon University

Problem 3

Find the pressure increase in the fluid in a syringe when a
nurse applies a force of 42 $\mathrm{N}$ to the syringe's circular piston, which
has a radius of 1.1 $\mathrm{cm} .$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 4

Three liquids that will not mix are poured into a cylindrical container. The volumes and densities of the liquids are $0.50 \mathrm{L}, 2.6 \mathrm{g} / \mathrm{cm}^{3}$ ;
$0.25 \mathrm{L}, 1.0 \mathrm{g} / \mathrm{cm}^{3} ;$ and $0.40 \mathrm{L}, 0.80 \mathrm{g} / \mathrm{cm}^{3} .$ What is the force on the bottom of the container due to these liquids? One liter $=1 \mathrm{L}=$
1000 $\mathrm{cm}^{3} .$ (Ignore the contribution due to the atmosphere.)

Averell Hause

Carnegie Mellon University

Problem 5

An office window has dimensions 3.4 $\mathrm{m}$ by 2.1 $\mathrm{m.As}$ a
result of the passage of a storm, the outside air pressure drops to
0.96 atm, but inside the pressure is held at 1.0 atm. What net force
pushes out on the window?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 6

You inflate the front tires on your car to 28 psi. Later, you measure
your blood pressure. obtaining a reading of 120$/ 80$ . the readings being in mm Hg. In metric countries (which is to say, most of the world), these
pressures are customarily reported in kilopascals (kPa). In kilopascals,
what are (a) your tire pressure and (b) your blood pressure?

Averell Hause

Carnegie Mellon University

Problem 7

In 1654 Otto von Guericke, inventor of the air pump, gave a demonstration before the noblemen of the Holy Roman Empire in which two teams of eight horses could not pull apart two evacuated
brass hemispheres. (a) Assuming the hemispheres have (strong) thin walls, so that $R$ in Fig. $14-29$ may be considered both the inside
and outside radius, show that the force $\vec{F}$ required to pull apart
the hemispheres has magnitude $F=\pi R^{2} \Delta p,$ where $\Delta p$ is the difference between the pressures outside and inside the sphere.
(b) Taking $R$ as 30 $\mathrm{cm}$ , the inside pressure as 0.10 atm, and the outside pressure as 1.00 atm, find the force magnitude the teams of horses would have had to exert to pull apart the hemispheres.
(c) Explain why one team of horses could have proved the
point just as well if the hemispheres were attached to a sturdy wall.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 8

The bends during flight .Anyone who scuba dives is advised not to fly within the next 24 h because the air mixture for diving can introduce nitrogen to the bloodstream. Without allowing the nitrogen to come out of solution slowly, any sudden air-pressure reduction (such as during airplane ascent) can result
in the nitrogen forming bubbles in the blood, creating the bends,
which can be painful and even fatal. Military special operation
forces are especially at risk. What is the change in pressure on
such a special-op soldier who must scuba dive at a depth of 20 $\mathrm{m}$ in seawater one day and parachute at an altitude of 7.6 $\mathrm{km}$ the
next day? Assume that the average air density within the altitude
range is 0.87 $\mathrm{kg} / \mathrm{m}^{3}$ .

Averell Hause

Carnegie Mellon University

Problem 9

$Blood pressure in Argentinosaurus$ (a) If this long. necked, gigantic sauropod had a height of 21 $\mathrm{m}$ and a heart height of $9.0 \mathrm{m},$ what (hydrostatic) gauge pressure in its blood
was required at the heart such that the blood pressure at the brain was 80 torr (just enough to perfuse the brain with blood)? Assume the blood had a density of $1.06 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3} .$ (b) What was the blood pressure (in torr or mm Hg) at the feet?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 10

The plastic tube in Fig. $14-30$ has a cross-scctional area of 5.00 $\mathrm{cm}^{2} .$ The tube is
filled with water until the short arm (of length $d=0.800 \mathrm{m}$ ) is full. Then the short arm
is sealed and more water is gradually poured into the long arm. If the seal will pop off when
the force on it exceeds $9.80 \mathrm{N},$ what total
height of water in the long arm will put the seal on the verge of popping?

Supratim Pal

Numerade Educator

Problem 11

$Giraffe bending to drink$ . In a giraffe with its head 2.0 $\mathrm{m}$ above its heart, and its heart 2.0 $\mathrm{m}$ above its feet, the (hydrostatic)
gauge pressure in the blood at its heart is 250 torr. Assume that the gi-
raffe stands upright and the blood density is $1.06 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3} .$ In torr (or mm Hg), find the (gauge) blood pressure (a) at the brain (the
pressure is enough to perfuse the brain with blood, to keep the giraffe
from fainting) and (b) at the feet (the pressure must be countered by tight-fitting skin acting like a pressure stocking). (c) If the giraffe
were to lower its head to drink from a pond without splaying its legs
and moving slowly, what would be the increase in the blood pressure
in the brain? (Such action would probably be lethal.)

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 12

The maximum depth $d_{\text { max }}$ that a diver can snorkel is set by the density of the water and the fact that human lungs can function against a maximum pressure difference (between inside and
outside the chest cavity) of 0.0550 atm. What is the difference in $d_{\max }$ for fresh water and the water of the Dead Sea (the saltiest natural
water in the world, with a density of $1.5 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3} ) ?$

Averell Hause

Carnegie Mellon University

Problem 13

At a depth of $10.9 \mathrm{km},$ the Challenger Deep in the
Marianas Trench of the Pacific Ocean is the deepest site in any
ocean. Yet, in $1960,$ Donald Walsh and Jacques Piccard reached the Challenger Deep in the bathyscaph Trieste. Assuming that
seawater has a uniform density of 1024 $\mathrm{kg} / \mathrm{m}^{3}$ , approximate the
hydrostatic pressure (in atmospheres) that the Trieste had to withstand. (Even a slight defect in the Trieste structure would
have been disastrous.)

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 14

Calculate the hydrostatic difference in blood pressure between the brain and the foot in a person of height 1.83 m. The density of blood is $1.06 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3} .$

Averell Hause

Carnegie Mellon University

Problem 15

What gauge pressure must a machine produce in order to suck
mud of density 1800 $\mathrm{kg} / \mathrm{m}^{3}$ up a tube by a height of 1.5 $\mathrm{m} ?$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 16

Snorkeling by humans and elephants. When a person snorkels, the lungs are connected
directly to the atmosphere through the snorkel tube and thus are at atmospheric pressure. In atmospheres, what is the difference $\Delta p$ between this internal air pressure and the water pressure against the body if the length of the snorkel tube is (a) 20 $\mathrm{cm}$ (standard situation) and (b) 4.0 $\mathrm{m}$ (probably lethal situation)? In the latter, the pressure difference causes
blood vessels on the walls of the lungs to rupture, releasing blood into the lungs. As depicted in Fig. $14-31,$ an elephant can safely snorkel through its trunk while swimming with its lungs 4.0 $\mathrm{m}$ below the water surface because the membrane around its lungs
contains connective tissue that holds and protects the blood vessels, preventing rupturing.

Averell Hause

Carnegie Mellon University

Problem 17

Crew members attempt to escape from a damaged submarine 100 $\mathrm{m}$ below the surface. What force must be applied to a pop-out hatch, which is 1.2 $\mathrm{m}$ by 0.60 $\mathrm{m},$ to push it out at
that depth? Assume that the density of the ocean water is 1024
$\mathrm{kg} / \mathrm{m}^{3}$ and the internal air pressure is at 1.00 $\mathrm{atm.}$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 18

In Fig. $14-32,$ an open tube of length
$L=1.8 \mathrm{m}$ and cross-sectional area $A=$
4.6 $\mathrm{cm}^{2}$ is fixed to the top of a cylindrical barrel of diameter $D=1.2 \mathrm{m}$ and height $H=$ 1.8 m. The barrel and tube are filled with water (to the top of the tube). Calculate
the ratio of the hydrostatic force on the bottom of the barrel to the gravitational force on the water contained in the barrel. Why is that ratio not equal to 1.0$?$ (You need
not consider the atmospheric pressure.)

Averell Hause

Carnegie Mellon University

Problem 19

A large aquarium of height 5.00 $\mathrm{m}$ is filled with fresh water to a depth of
2.00 $\mathrm{m} .$ One wall of the aquarium consists of thick plastic 8.00 $\mathrm{m}$ wide. By how much does the total force on that wall increase if the aquarium is next filled to a depth of
4.00 $\mathrm{m} ?$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 20

The L-shaped fish tank shown in
Fig. $14-33$ is filled with water and is open at
the top. If $d=5.0 \mathrm{m},$ what is the (total)
force exerted by the water (a) on face $A$
and (b) on face $B$ ?

Averell Hause

Carnegie Mellon University

Problem 21

Two identical cylindrical vessels with their bases at the same level each contain a liquid of density $1.30 \times 10^{3}$ $\mathrm{kg} / \mathrm{m}^{3} .$ The area of each base is 4.00 $\mathrm{cm}^{2}$
but in one vessel the liquid height is 0.854
$\mathrm{m}$ and in the other it is 1.560 $\mathrm{m} .$ Find the
work done by the gravitational force in cqualizing the levels when the two vessels are connected.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 22

$g-L O C$ in dogfights. When a pilot takes a tight tum at high speed in a modem fighter airplane, the blood pressure at the brain
level decreases, blood no longer perfuses the brain, and the blood in
the brain drains If the heart maintains the (hydrostatic) gauge pressure in the aorta at 120 torr (or mm Hg) when the pilot undergoes a horizontal centripetal acceleration of $4 g,$ what is the blood pressure (in torr) at the brain, 30 cm radially inward from the heart? The perfusion torr) at the brain, 30 $\mathrm{cm}$ radially inward from the heart? The perfusion
in the brain is small enough that the vision switches to black and white and narrows to "tunnel vision" and the pilot can undergo $g-\operatorname{LOC}(" g-$
induced loss of consciousness"). Blood density is 1.06 $\times 10^{3} \mathrm{kg} / \mathrm{m}^{3} .$

Averell Hause

Carnegie Mellon University

Problem 23

In analyzing certain geological features, it is often appropriate to assume that the pressure
at some horizontal level of compensation, deep inside Earth, is the same over a large region and is
equal to the pressure due to the gravitational force on the overlying material. Thus, the pressure on
the level of compensation is given by the fluid pressure formula. This model requires, for one thing, that mountains have roots of continental rock extending into the denser mantle (Fig. $14-34$ ). Consider a mountain of height $H=6.0 \mathrm{km}$
on a continent of thickness $T=32 \mathrm{km}$ . The continental rock has
a density of $2.9 \mathrm{g} / \mathrm{cm}^{3},$ and beneath this rock the mantle has a density of 3.3 $\mathrm{g} / \mathrm{cm}^{3}$ . Calculate the depth $D$ of the root. (Hint: Set
the pressure at points $a$ and $b$ equal; the depth $y$ of the level of
compensation will cancel out.)

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 24

In Fig. $14-35,$ water stands at depth $D=35.0 \mathrm{m}$ behind the vertical upstream face of a dam of width $W=314 \mathrm{m} .$ Find $(\mathrm{a})$ the net horizontal force on the dam from
the gauge pressure of the water and (b) the net torque due to that force about a horizontal line through $O$ parallel to the (long) width of the dam. This torque tends to rotate the dam around that line, which would cause the dam to fail. (c) Find the moment arm of the torque.

Averell Hause

Carnegie Mellon University

Problem 25

In one observation, the column in a mercury barometer (as is
shown in Fig. $14-5 a$ ) has a measured height $h$ of 740.35 $\mathrm{mm}$ . The temperature is $-5.0^{\circ} \mathrm{C}$ at which temperature the density of mercury $\rho$ is $1.3608 \times 10^{4} \mathrm{kg} / \mathrm{m}^{3} .$ The free-fall acceleration $g$ at the site of the barometer is 9.7835 $\mathrm{m} / \mathrm{s}^{2} .$ What is the atmospheric pressure at that site in pascals and in torr (which is the common unit for barometer readings)?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 26

$\cdot 26$ To suck lemonade of density 1000 $\mathrm{kg} / \mathrm{m}^{3}$ up a straw to a maxi-
mum height of $4.0 \mathrm{cm},$ what minimum gauge pressure (in atmo-
spheres) must you produce in your lungs?

Averell Hause

Carnegie Mellon University

Problem 27

What would be the height of the atmosphere if the
air density (a) were uniform and (b) decreased linearly to zero with height? Assume that at sea level the air pressure is 1.0 atm
and the air density is 1.3 $\mathrm{kg} / \mathrm{m}^{3} .$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 28

A piston of cross-sectional
area $a$ is used in a hydraulic press to
exert a small force of magnitude $f$ on
the enclosed liquid. A connecting
pipe leads to a larger piston of crosssectional area $A($ Fig. $14-36)$ . (a) What
force magnitude $F$ will the larger piston sustain without moving? (b) If
the piston diameters are 3.80 $\mathrm{cm}$ and $53.0 \mathrm{cm},$ what force magnitude on the small piston will balance a 20.0
kN force on the large piston?

Donald Albin

Numerade Educator

Problem 29

In Fig. $14-37,$ a spring of spring
constant $3.00 \times 10^{4} \mathrm{N} / \mathrm{m}$ is between a
rigid beam and the output piston of a
hydraulic lever. An empty container
with negligible mass sits on the input piston. The input piston has area $A_{b}$ , and
the output piston has area 18.0$A_{i}$
Initially the spring is at its rest length.
How many kilograms of sand must be (slowly) poured into the container to compress the spring by 5.00 $\mathrm{cm} ?$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 30

A 5.00 $\mathrm{kg}$ object is released from rest while fully submerged
in a liquid. The liquid displaced by the submerged object has a
mass of 3.00 $\mathrm{kg} .$ How far and in what direction does the object move in 0.200 s, assuming that it moves freely and that the drag
force on it from the liquid is negligible?

Averell Hause

Carnegie Mellon University

Problem 31

A block of wood floats in fresh water with two-thirds of its
volume $V$ submerged and in oil with 0.90$V$ submerged. Find the density of (a) the wood and (b) the oil.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 32

In Fig. $14-38,$ a cube of edge
length $L=0.600 \mathrm{m}$ and mass 450 $\mathrm{kg}$
is suspended by a rope in an open
tank of liquid of density 1030 $\mathrm{kg} / \mathrm{m}^{3}$ . Find (a) the magnitude of the total
downward force on the top of the cube from the liquid and the atmosphere, assuming atmospheric pressure is 1.00 atm, ( b ) the magnitude of the total upward force on the bot-
tom of the cube, and (c) the tension in the rope. (d) Calculate the magnitude of the buoyant force on
the cube using Archimedes' principle. What relation exists among
all these quantities?

Averell Hause

Carnegie Mellon University

Problem 33

An iron anchor of density 7870 $\mathrm{kg} / \mathrm{m}^{3}$ appears 200 $\mathrm{N}$
lighter in water than in air. (a) What is the volume of the anchor?
(b) How much does it weigh in air?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 34

A boat floating in fresh water displaces water weighing 35.6 $\mathrm{kN}$ . (a) What is the weight of the water this boat displaces
when floating in salt water of density $1.10 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3} ?$ (b) What is
the difference between the volume of fresh water displaced and
the volume of salt water displaced?

Averell Hause

Carnegie Mellon University

Problem 35

Three children, each of weight 356 $\mathrm{N}$ ,
make a log raft by lashing together logs of
diameter 0.30 $\mathrm{m}$ and length 1.80 $\mathrm{m} .$ How
many logs will be needed to keep them afloat in fresh water? Take the density of the logs to be 800 $\mathrm{kg} / \mathrm{m}^{3}$ .

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 36

In Fig. $14-39 a,$ a rectangular block is gradually pushed
face-down into a liquid. The block
has height $d ;$ on the bottom and
top the face area is $A=5.67 \mathrm{cm}^{2}$ Figure $14-39 b$ gives the apparent
weight $W_{\text { app of the block as a func- }}$
tion of the depth $h$ of its lower face. The scale on the vertical axis
is set by $W_{s}=0.20 \mathrm{N}$ . What is the
density of the liquid?

Averell Hause

Carnegie Mellon University

Problem 37

A hollow spherical iron shell floats almost completely submerged in water. The outer diameter is $60.0 \mathrm{cm},$ and the density of iron is 7.87 $\mathrm{g} / \mathrm{cm}^{3} .$ Find the inner diameter.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 38

A small solid ball is released from rest while fully submerged in a liquid and then its kinetic energy is measured when it has moved 4.0 $\mathrm{cm}$ in the liquid. Figure $14-40$ gives
the results after many liquids are used: The kinetic energy $K$ is plotted versus the liquid density $\rho_{\text { iq }},$ and $K_{s}=1.60 \mathrm{J}$
sets the scale on the vertical axis.What are (a) the density and (b) the volume of the ball?

Averell Hause

Carnegie Mellon University

Problem 39

A hollow sphere of inner radius 8.0 $\mathrm{cm}$ and outer
radius 9.0 $\mathrm{cm}$ floats half-submerged in a liquid of density 80 $\mathrm{kg} / \mathrm{m}^{3}$ .
(a) What is the mass of the sphere? (b) Calculate the density of the
material of which the sphere is made.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 40

Lurking alligators. An alligator waits for prey by floating with
only the top of its head exposed, so that the prey cannot easily see it. One way it can adjust the extent of sinking is by controlling the size of its lungs Another way may be by swallowing stones (gastrolithes) that then reside in the stomach. Figure $14-41$ shows a highly simplified
model (a "rhombohedron gater") of mass 130 kg that roams with its head partially exposed. The top head surface has area 0.20 $\mathrm{m}^{2} .$ If the
alligator were to swallow stones with a total mass of 1.0$\%$ of its body
mass (a typical amount), how far would it sink?

Averell Hause

Carnegie Mellon University

Problem 41

What fraction of the volume of an iceberg (density 917 $\mathrm{kg} / \mathrm{m}^{3} )$ would be visible if the iceberg floats (a) in the ocean (salt water, density 1024 $\mathrm{kg} / \mathrm{m}^{3}$ and $(\mathrm{b})$ in a river (fresh water, density 1000 $\mathrm{kg} / \mathrm{m}^{3} )$ ? (When salt water freezes to form ice, the salt is excluded. So, an iceberg could provide fresh water to a community.)

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 42

A flotation device is in the shape of a right cylinder, with a
height of 0.500 $\mathrm{m}$ and a face area of 4.00 $\mathrm{m}^{2}$ on top and bottom, and
its density is 0.400 times that of fresh water. It is initially held fully
submerged in fresh water, with its top face at the water surface. Then it is allowed to ascend gradually until it begins to float. How much
work does the buoyant force do on the device during the ascent?

Averell Hause

Carnegie Mellon University

Problem 43

$\cdot 43$ When researchers find a reasonably complete fossil of a dinosaur, they can determine the mass and weight of the living dinosaur with a scale model sculpted from
plastic and based on the dimensions of the fossil bones. The scale of the model is $1 / 20 ;$ that is, lengths are 1$/ 20$ actual length, areas are $(1 / 20)^{2}$ actual
areas, and volumes are $(1 / 20)^{3}$ actual volumes. First, the model is suspended from one arm of a balance
and weights are added to the other arm until equilibrium is
reached. Then the model is fully submerged in water and enough
weights are removed from the second arm to reestablish equilibrium (Fig. $14-42 )$ . For a model of a particular $T$ rex fossil, 637.76 $\mathrm{g}$
had to be removed to reestablish equilibrium. What was the volume of (a) the model and (b) the actual T.rex? (c) If the density of T. rex was approximately the density of water, what was its mass?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 44

A wood block (mass $3.67 \mathrm{kg},$ density 600 $\mathrm{kg} / \mathrm{m}^{3}$ ) is fitted
with lead (density 1.14 $\times 10^{4} \mathrm{kg} / \mathrm{m}^{3} )$ so that it floats in water with 0.900 of its volume submerged. Find the lead mass if the lead is fit-
ted to the block's (a) top and (b) bottom.

Averell Hause

Carnegie Mellon University

Problem 45

$\cdot 445 \bigoplus$ An iron casting containing a number of cavities weighs
6000 $\mathrm{N}$ in air and 4000 $\mathrm{N}$ in water. What is the total cavity volume
in the casting? The density of solid iron is 7.87 $\mathrm{g} / \mathrm{cm}^{3} .$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 46

Suppose that you release a small ball from rest at a depth
of 0.600 m below the surface in a pool of water. If the density of the
ball is 0.300 that of water and if the drag force on the ball from the
water is negligible, how high above the water surface will the ball shoot as it emerges from the water? (Neglect any transfer of energy to the splashing and waves produced by the emerging ball.)

Averell Hause

Carnegie Mellon University

Problem 47

The volume of air space in the passenger compartment of an
1800 $\mathrm{kg}$ car is 5.00 $\mathrm{m}^{3} .$ The volume of the motor and front wheels is
$0.750 \mathrm{m}^{3},$ and the volume of the rear wheels, gas tank, and trunk is 0.800 $\mathrm{m}^{3} ;$ water cannot enter these two regions. The car rolls into a lake, (a)
At first, no water enters the passenger compartment. How much of the
car, in cubic meters, is below the water surface with the car floating (Fig. $14-43 )$ (b) As water slowly cnters, the car sinks. How many cubic
meters of water are in the car as it disappears below the water surface?
(The car, with a heavy load in the trunk, remains horizontal.)

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 48

Figure $14-44$ shows an iron ball suspended by thread of negligible mass from an upright cylinder that floats partially submerged in water. The cylinder has a height of $6.00 \mathrm{cm},$ a face area of $12.0 .$ $\mathrm{cm}^{2}$ on the top and bottom, and a density of
$0.30 \mathrm{g} / \mathrm{cm}^{3},$ and 2.00 $\mathrm{cm}$ of its height is above
the water surface. What is the radius of the iron
ball?

Supratim Pal

Numerade Educator

Problem 49

Canal effect. Figure $14-45$ shows an anchored barge that extends across a canal by distance
$d=30 \mathrm{m}$ and into the water by distance $b=12 \mathrm{m} .$ The canal has a
width $D=55 \mathrm{m},$ a water depth $H=14 \mathrm{m},$ and a uniform water-flow
speed $v_{i}=1.5 \mathrm{m} / \mathrm{s} .$ Assume that the
flow around the barge is uniform. As the water passes the bow, the water
level undergoes a dramatic dip known as the canal effect. If the dip has depth $h=0.80 \mathrm{m},$ what is the water speed alongside the boat
through the vertical cross sections at (a) point $a$ and (b) point $b$ ?
The erosion due to the speed increase is a common concern to hydraulic engineers.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 50

Figure $14-46$ shows two sections of an old pipe system that runs through a hill, with
distances $d_{A}=d_{B}=30 \mathrm{m}$ and $D=110 \mathrm{m} .$ On each side of
the hill, the pipe radius is 2.00 $\mathrm{cm} .$ However, the radius of the pipe inside the hill is no longer known. To determine it, hydraulic engineers first establish that water
flows through the left and right sections at 2.50 $\mathrm{m} / \mathrm{s}$ . Then they release a dye in the water at point $A$ and find that it takes 88.8 s to
reach point $B$ . What is the average radius of the pipe within the hill?

Averell Hause

Carnegie Mellon University

Problem 51

A garden hose with an internal diameter of 1.9 $\mathrm{cm}$ is
connected to a (stationary) lawn sprinkler that consists merely of
a container with 24 holes, each 0.13 $\mathrm{cm}$ in diameter. If the water
in the hose has a speed of 0.91 $\mathrm{m} / \mathrm{s}$ , at what speed does it leave the
sprinkler holes?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 52

Two streams merge to form a river. One stream has a width
of $8.2 \mathrm{m},$ depth of $3.4 \mathrm{m},$ and current speed of 2.3 $\mathrm{m} / \mathrm{s}$ . The other
stream is 6.8 $\mathrm{m}$ wide and 3.2 $\mathrm{m}$ deep, and flows at 2.6 $\mathrm{m} / \mathrm{s}$ . If the
river has width 10.5 $\mathrm{m}$ and speed $2.9 \mathrm{m} / \mathrm{s},$ what is its depth?

Averell Hause

Carnegie Mellon University

Problem 53

Water is pumped steadily out of a flooded basement at
5.0 $\mathrm{m} / \mathrm{s}$ through a hose of radius $1.0 \mathrm{cm},$ passing through a window
3.0 $\mathrm{m}$ above the waterline. What is the pump's power?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 54

The water flowing through a 1.9 $\mathrm{cm}$ (inside diameter) pipe
flows out through three 1.3 $\mathrm{cm}$ pipes. (a) If the flow rates in the three smaller pipes are $26,19,$ and 11 $\mathrm{L} / \mathrm{min}$ , what is the flow rate in
the 1.9 $\mathrm{cm}$ pipe? (b) What is the ratio of the spced in the 1.9 $\mathrm{cm}$ pipe
to that in the pipe carrying 26 $\mathrm{L} / \mathrm{min}$ ?

Averell Hause

Carnegie Mellon University

Problem 55

How much work is done by pressure in forcing 1.4 $\mathrm{m}^{3}$ of
water through a pipe having an internal diameter of 13 $\mathrm{mm}$ if the
difference in pressure at the two ends of the pipe is 1.0 atm?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 56

Suppose that two tanks, 1 and $2,$ each with a large opening at
the top, contain different liquids. A small hole is made in the side of
each tank at the same depth $h$ below the liquid surface, but the
hole in tank 1 has half the cross-sectional area of the hole in $\operatorname{tank} 2$ .
(a) What is the ratio $\rho_{1} / \rho_{2}$ of the densities of the liquids if the mass
flow rate is the same for the two holes? (b) What is the ratio
$R_{V 1} / R_{V 2}$ of the volume flow rates from the two tanks? (c) At one instant, the liquid in tank 1 is 12.0 $\mathrm{cm}$ above the hole. If the tanks are
to have equal volume flow rates, what height above the hole must
the liquid in tank 2 be just then?

Averell Hause

Carnegie Mellon University

Problem 57

A cylindrical tank with a large diameter is filled
with water to a depth $D=0.30 \mathrm{m} .$ A hole of cross-sectional area
$A=6.5 \mathrm{cm}^{2}$ in the bottom of the tank allows water to drain out. (a) What is the drainage rate in cubic meters per second? (b) At what
distance below the bottom of the tank is the cross-sectional area of
the stream equal to one-half the area of the hole?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 58

The intake in Fig. $14-47$ has
cross-sectional area of 0.74 $\mathrm{m}^{2}$ and
water flow at 0.40 $\mathrm{m} / \mathrm{s} .$ At the outlet,
distance $D=180 \mathrm{m}$ below the intake, the cross-sectional area is
smaller than at the intake and the
water flows out at 9.5 $\mathrm{m} / \mathrm{s}$ into
equipment. What is the pressure difference between inlet and outlet?

Averell Hause

Carnegie Mellon University

Problem 59

Water is moving with a speed of 5.0 $\mathrm{m} / \mathrm{s}$ through a pipe
with a cross-sectional area of 4.0 $\mathrm{cm}^{2} .$ The water gradually descends
10 $\mathrm{m}$ as the pipe cross-sectional area increases to 8.0 $\mathrm{cm}^{2} .$ (a) What
is the speed at the lower level? (b) If the pressure at the upper level
is $1.5 \times 10^{5} \mathrm{Pa}$ , what is the pressure at the lower level?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 60

Models of torpedoes are sometimes tested in a horizontal pipe of
flowing water, much as a wind tunnel is used to test model airplanes.
Consider a circular pipe of internal diameter 25.0 $\mathrm{cm}$ and a torpedo
model aligned along the long axis of the pipe. The model has a 5.00 $\mathrm{cm}$ diameter and is to be tested with water flowing past it at 2.50 $\mathrm{m} / \mathrm{s}$ (a)
With what speed must the water flow in the part of the pipe that is
unconstricted by the model? (b) What will the pressure difference be
between the constricted and unconstricted parts of the pipe?

Averell Hause

Carnegie Mellon University

Problem 61

A water pipe having a 2.5 $\mathrm{cm}$ inside diameter carries water into the basement of a house at a speed of 0.90 $\mathrm{m} / \mathrm{s}$ and a pressure of 170 $\mathrm{kPa}$ . If the pipe tapers to 1.2 $\mathrm{cm}$ and rises to the second
floor 7.6 $\mathrm{m}$ above the input point, what are the (a) speed and
(b) water pressure at the second floor?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 62

A pitot tube (Fig. 14.48 ) is used to determine the airspeed of an airplane. It consists of an outer tube with a number ofsmall holes $B$ (four are shown) that allow air into the tube; that
tube is connected to one arm of a U. -tube. The other arm of the U-tube is connected to hole $A$ at the front end of the device, which
points in the direction the plane is headed. At $A$ the air becomes
stagnant so that $v_{A}=0 .$ At $B$ , however, the speed of the air presumably equals the airspeed $v$ of the plane. (a) Use Bernoulli's equation
to show that
$$v=\sqrt{\frac{2 \rho g h}{\rho_{\text { air }}}}$$
where $\rho$ is the density of the liquid in the U.tube and $h$ is the differ-
ence in the liquid levels in that tube. (b) Suppose that the tube con-
tains alcohol and the level difference $h$ is 26.0 $\mathrm{cm} .$ What is the
plane's speed relative to the air? The density of the air is 1.03 $\mathrm{kg} / \mathrm{m}^{3}$
and that of alcohol is 810 $\mathrm{kg} / \mathrm{m}^{3} .$

Averell Hause

Carnegie Mellon University

Problem 63

A pitot tube (see Problem 62$)$ on a high-altitude aircraft measures a differential pressure of 180 $\mathrm{Pa}$ . What is the aircraft's
airspeed if the density of the air is 0.031 $\mathrm{kg} / \mathrm{m}^{3} ?$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 64

In Fig. $14-49,$ water flows through a horizontal pipe and then out
into the atmosphere at a speed $v_{1}=15$
$\mathrm{m} / \mathrm{s} .$ The diameters of the left and right sections of the pipe are 5.0 $\mathrm{cm}$ and 3.0
$\mathrm{cm} .$ (a) What volume of water flows into the atmosphere during a 10 $\mathrm{min}$ period? In the left section of the
pipe, what are $(\mathrm{b})$ the speed $v_{2}$ and (c) the gauge pressure?

Averell Hause

Carnegie Mellon University

Problem 65

A venturi meter is used to measure the flow speed of a fluid in a pipe. The meter is connected between two
sections of the pipe (Fig. $14-50 ) ;$ the cross-sectional area $A$ of the
entrance and exit of the meter matches the pipe's cross-sectional
area. Between the entrance and exit, the fluid flows from the pipe with speed $V$ and then through a narrow "throat" of crosssectional area $a$ with speed $v$ . A manometer connects the wider
portion of the meter to the narrower portion. The change in the
fluid's speed is accompanied by a change $\Delta p$ in the fluid's pressure, which causes a height difference $h$ of the liquid in the two arms of
the manometer. (Here $\Delta p$ means pressure in the throat minus pressure in the pipe.) (a) By applying Bernoulli's equation and the
equation of continuity to points 1 and 2 in Fig. $14-50$ , show that
$$V=\sqrt{\frac{2 a^{2} \Delta p}{\rho\left(a^{2}-A^{2}\right)}}$$
where $\rho$ is the density of the fluid. (b) Suppose that the fluid is
fresh water, that the cross-sectional areas are 64 $\mathrm{cm}^{2}$ in the pipe
and 32 $\mathrm{cm}^{2}$ in the throat, and that the pressure is 55 $\mathrm{kPa}$ in the pipe
and 41 $\mathrm{kPa}$ in the throat. What is the rate of water flow in cubic
meters per second?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 66

Consider the venturi tube of Problem 65 and Fig. $14-50$ without the manometer. Let $A$ equal 5$a$ . Suppose the pressure $p_{1}$ at $A$
is 2.0 atm. Compute the values of (a) the speed $V$ at $A$ and (b) the
speed $v$ at $a$ that make the pressure $p_{2}$ at $a$ equal to zero. (c) Compute the corresponding volume flow rate if the diameter at $A$ is 5.0 $\mathrm{cm}$ .
The phenomenon that occurs at $a$ when $p_{2}$ falls to nearly zero is
known as cavitation. The water vaporizes into small bubbles.

Averell Hause

Carnegie Mellon University

Problem 67

In Fig. $14-51,$ the fresh water behind a reservoir dam has depth $D=15 \mathrm{m} .$ A horizontal pipe 4.0 $\mathrm{cm}$ in diameter passes
through the dam at depth $d=6.0 \mathrm{m} .$ A plug secures the pipe opening. (a) Find the magnitude of the frictional force between plug and pipe wall. (b) The plug is removed. What
water volume exits the pipe in 3.0 $\mathrm{h} ?$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 68

Fresh water flows horizontally from pipe section 1 of cross-sectional
area $A_{1}$ into pipe section 2 of cross-sectional area $A_{2 .}$ Figure $14-52$ gives a plot
of the pressure difference $p_{2}-p_{1}$ versus the inverse area squared $A_{1}^{-2}$ that
would be expected for a volume flow rate of a certain value if the water flow were laminar under all circumstances. The scale on the vertical axis is set by $\Delta p_{s}=300 \mathrm{kN} / \mathrm{m}^{2}$ . For the conditions
of the figure, what are the values of
(a) $A_{2}$ and $(\mathrm{b})$ the volume flow rate?

Averell Hause

Carnegie Mellon University

Problem 69

A liquid of density 900 $\mathrm{kg} / \mathrm{m}^{3}$ flows through a horizontal pipe that has a cross-sectional area of $1.90 \times 10^{-2} \mathrm{m}^{2}$ in region $A$ and a
cross-sectional area of $9.50 \times 10^{-2} \mathrm{m}^{2}$ in region $B .$ The pressure
difference between the two regions is $7.20 \times 10^{3} \mathrm{Pa}$ . What are (a)
the volume flow rate and $(\mathrm{b})$ the mass flow rate?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 70

In Fig. $14-53,$ water flows steadily from the left pipe section
(radius $r_{1}=2.00 R$ ), through the middle section (radius $R ),$ and into the right section (radius $r_{3}=3.00 R )$ . The speed of the water in the middle section is 0.500 $\mathrm{m} / \mathrm{s} .$ What is the net work done on 0.400 $\mathrm{m}^{3}$ of the water as it moves from the left section to the right section?

Averell Hause

Carnegie Mellon University

Problem 71

Figure $14-54$ shows a stream of water flowing through a hole at depth $h=10 \mathrm{cm}$ in a tank holding water to height $H=40 \mathrm{cm} .$ (a) At what distance $x$ does the stream strike the
floor? (b) At what depth should a second hole be made to give the same value of $x ?($ c) At what depth should a hole be made to maximize $x ?$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 72

A very simplified schematic of the rain drainage system for a home is shown in Fig. $14-55$ .
Rain falling on the slanted roof runs off into gutters around the
roof edge. it then drans through downspouts (only one is
shown) into a main drainage pipe $M$ below the basement, which carries the water to an
even larger pipe below the street. In Fig. $14-55,$ a floor drain in the basement is also connected to drainage pipe $M .$ Suppose the following apply:
(1) the downspouts have height $h_{1}=11 \mathrm{m},$ (2) the floor drain has height $h_{2}=$
$1.2 \mathrm{m},(3)$ pipe $M$ has radius 3.0 $\mathrm{cm},(4)$ the house has side width $w=30 \mathrm{m}$ and front length $L=60 \mathrm{m},(5)$ all the water striking the roof goes through pipe $M,(6)$ the initial speed
of the water in a downspout is negligible, and $(7)$ the wind speed is
negligible (the rain falls vertically). At what rainfall rate, in centimeters per hour, will water from
pipe $M$ reach the height of the floor drain and threaten to flood
the basement?

Averell Hause

Carnegie Mellon University

Problem 73

About one-third of the body of a person floating in the
Dead Sea will be above the waterline. Assuming that the human
body density is 0.98 $\mathrm{g} / \mathrm{cm}^{3}$ , find the density of the water in the
Dead Sea. (Why is it so much greater than 1.0 $\mathrm{g} / \mathrm{cm}^{3} ? )$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 74

A simple open $U$ -tube contains mercury. When 11.2 $\mathrm{cm}$ of
water is poured into the right arm of the tube, how high above its
initial level does the mercury rise in the left arm?

Averell Hause

Carnegie Mellon University

Problem 75

If a bubble in sparkling water accelerates upward at the
rate of 0.225 $\mathrm{m} / \mathrm{s}^{2}$ and has a radius of $0.500 \mathrm{mm},$ what is its mass? Assume that the drag force on the bubble is negligible.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 76

Suppose that your body has a uniform density of 0.95
times that of water. (a) If you float in a swimming pool, what fraction of your body's volume is above the water surface?
Quicksand is a fluid produced when water is forced up into sand, moving the sand grains away from one another so they are no longer locked together by friction. Pools of quicksand can form when
water drains underground from hills into valleys where there are longer locked together by friction. Pools of quicksand can form when
water drains underground from hills into valleys where there
sand pockets. (b) If you float in a deep pool of quicksand that has a
density 1.6 times that of water, what fraction of your body's volume is
above the quicksand surface? (c) Are you unable to breathe?

Averell Hause

Carnegie Mellon University

Problem 77

A glass ball of radius 2.00 $\mathrm{cm}$ sits at the bottom of a container
of milk that hat has a density of 1.03 $\mathrm{g} / \mathrm{cm}^{3} .$ The normal force on the
ball from the container's lower surface has magnitude $9.48 \times 10^{-2} \mathrm{N}$ .
What is the mass of the ball?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 78

Caught in an avalanche, a skier is fully submerged in flowing snow of density 96 $\mathrm{kg} / \mathrm{m}^{3} .$ Assume that the average density
of the skier, clothing, and sking equipment is 1020 $\mathrm{kgm}^{3} .$ What
percentage of the gravitational force on the skier is offset by the
buoyant force from the snow?

Averell Hause

Carnegie Mellon University

Problem 79

An object hangs from a spring balance. The balance registers
30 $\mathrm{N}$ in air, 20 $\mathrm{N}$ when this object is immersed in water, and 24 $\mathrm{N}$
when the object is immersed in another liquid of unknown density. What is the density of that other liquid?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 80

In an experiment, a rectangular block with height $h$ is allowed
to float in four separate liquids. In the first liquid, which is water, it
floats fully submerged. In liquids $A, B,$ and $C,$ it floats with heights $h / 2,2 h / 3,$ and $h / 4$ above the liquid surface, respectively. What are
the relative densities (the densities relative to that of water) of
(a) $A,($ b) $B,$ and $(c) C ?$

Averell Hause

Carnegie Mellon University

Problem 81

Figure $14-30$ shows a modified U-tube: the right arm is
shorter than the left arm. The open end of the right arm is height
$d=10.0 \mathrm{cm}$ above the laboratory bench. The radius throughout
the tube is 1.50 $\mathrm{cm} .$ Water is gradually poured into the open end of the left arm until the water begins to flow out the open end of the
right arm. Then a liquid of density 0.80 $\mathrm{gm}^{3}$ is gradually added to
the left arm until its height in that arm is 8.0 $\mathrm{cm}$ (it does not mix
with the water). How much water flows out of the right arm?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 82

What is the acceleration of a rising hot-air balloon if the ratio
of the air density outside the balloon to that inside is 1.39$?$ Neglect
the mass of the balloon fabric and the basket.

Averell Hause

Carnegie Mellon University

Problem 83

Figure 14.56 shows a siphon, which is a device for removing liquid from a container.
Tube $A B C$ must initially be filled, but once this has been done, liquid will flow through the tube until the liquid surface in the container is level with the tube opening at $A .$ The liquid has density 1000 $\mathrm{kg} / \mathrm{m}^{3}$ and negligible viscosity. The distances shown are $h_{1}=25 \mathrm{cm}, d=$ $12 \mathrm{cm},$ and $h_{2}=40 \mathrm{cm} .$ (a) With what speed does the liquid emerge from the tube at $C ?$ (b) If the atmospheric pressure is 1.0 $\times 10^{5} \mathrm{Pa}$ what is the pressure in the liquid at the topmost point $B ?$ (c) Theoretically, what is the greatest possible height $h_{1}$ that a siphon can lift water?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 84

When you cough, you expel air at high speed through the trachea and upper bronchi so that the air will remove excess mucus
lining the pathway. You produce the high speed by this procedure: You
breathe in a large amount of air, trap it by closing the glottis (the narrow opening in the larynx), increase the air pressure by contracting
the lungs, partially collapse the trachea and upper bronchi to narrow the pathway, and then expel the air through the pathway by suddenly
reopening the glottis. Assume that during the expulsion the volume flow rate is $7.0 \times 10^{-3} \mathrm{m}^{3} / \mathrm{s}$ . What multiple of 343 $\mathrm{m} / \mathrm{s}$ (the speed of
sound $v_{\mathrm{s}}$ is the airspeed through the trachea if the trachea diameter
(a) remains its normal value of 14 $\mathrm{mm}$ and (b) contracts to 5.2 $\mathrm{mm}$ ?

Averell Hause

Carnegie Mellon University

Problem 85

A tin can has a total volume of 1200 $\mathrm{cm}^{3}$
and a mass of 130 $\mathrm{g}$ . How many grams of lead
shot of density 11.4 $\mathrm{g} / \mathrm{cm}^{3}$ could it carry without sinking in water?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 86

The tension in a string holding a solid
block below the surface of a liquid (of density
greater than the block) is $T_{0}$ when the container
(Fig. $14-57$ ) is at rest. When the container is given an upward acceleration of 0.250$g$ , what
multiple of $T_{0}$ gives the tension in the string?

Averell Hause

Carnegie Mellon University

Problem 87

What is the minimum area (in square meters) of the top surface of an ice slab 0.441 m thick floating on fresh water that will
hold up a 938 kg automobile? Take the densities of ice and fresh
water to be 917 $\mathrm{kg} / \mathrm{m}^{3}$ and $998 \mathrm{kg} / \mathrm{m}^{3},$ respectively.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 88

A 8.60 kg sphere of radius 6.22 $\mathrm{cm}$ is at a depth of 2.22 $\mathrm{km}$ in
seawater that has an average density of 1025 $\mathrm{kg} / \mathrm{m}^{3} .$ What are the
(a) gauge pressure, (b) total pressure, and (c) corresponding total force compressing the sphere's surface? What are (d) the magnitude of the buoyant force on the sphere and (c) the magnitude of
the sphere's acceleration if is free to move? Take atmospheric
pressure to be $1.01 \times 10^{5}$ Pa.

Averell Hause

Carnegie Mellon University

Problem 89

(a) For seawater of density $1.03 \mathrm{g} / \mathrm{cm}^{3},$ find the weight of water on top of a submarine at a depth of 255 $\mathrm{m}$ if the horizontal
cross-sectional hull area is 2200.0 $\mathrm{m}^{2} .$ (b) In atmospheres, what water pressure would a diver experience at this depth?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 90

The sewage outlet of a house constructed on a slope is 6.59 $\mathrm{mbe}$ -
low street level. If the sewer is 2.16 $\mathrm{m}$ below street level, find the minimum pressure difference that must be created by the sewage pump to
transfer waste of average density 1000.00 $\mathrm{kg} / \mathrm{m}^{3}$ from outlet to sewer.

Averell Hause

Carnegie Mellon University