chapter 14 Fluid Mechanics
the weight of the boat (d) equal to the weight of the dis-
placed water (e) equal to the buoyant force on the boat
10. A small piece of steel is tied to a block of wood. When
the wood is placed in a tub of water with the steel on top,
half of the block is submerged. Now the block is inverted
so that the steel is under water. (i) Does the amount
of the block submerged (a) increase, (b) decrease, or
(c) remain the same? (ii) What happens to the water
level in the tub when the block is inverted? (a) It rises.
(b) It falls. (c) It remains the same.
11. A piece of unpainted porous wood barely floats in an
open container partly filled with water. The container
is then sealed and pressurized above atmospheric pres-
sure. What happens to the wood? (a) It rises in the
water. (b) It sinks lower in the water. (c) It remains at
the same level.
12. A person in a boat floating in a small pond throws an
anchor overboard. What happens to the level of the
pond? (a) It rises. (b) It falls. (c) It remains the same.
13. Rank the buoyant forces exerted on the following five
objects of equal volume from the largest to the smallest.
Assume the objects have been dropped into a swimming
pool and allowed to come to mechanical equilibrium.
If any buoyant forces are equal, state that in your rank-
ing. (a)a block of solid oak (b) an aluminum block (c) a
beach ball made of thin plastic and inflated with air
(d) an iron block (e) a thin-walled, sealed bottle of water
14. A water supply maintains a constant rate of flow for water
in a hose. You want to change the opening of the nozzle
so that water leaving the nozzle will reach a height that
is four times the current maximum height the water
reaches with the nozzle vertical. To do so, should you
(a) decrease the area of the opening by a factor of 16,
(b) decrease the area by a factor of 8, (c) decrease the
area by a factor of 4, (d)decrease the area by a factor of
2, or (e) give up because it cannot be done?
15. A glass of water contains floating ice cubes. When the ice
melts, does the water level in the glass (a) go up, (b) go
down, or (c) remain the same?
16. An ideal fluid flows through a horizontal pipe whose
diameter varies along its length. Measurements would
indicate that the sum of the kinetic energy per unit
volume and pressure at different sections of the pipe
would (a)decrease as the pipe diameter increases,
(b) increase as the pipe diameter increases, (c) increase
as the pipe diameter decreases, (d) decrease as the
pipe diameter decreases, or (e) remain the same as the
pipe diameter changes.
6. A solid iron sphere and a solid lead sphere of the
same size are each suspended by strings and are sub-
merged in a tank of water. (Note that the density of
lead is greater than that of iron.) Which of the fol-
lowing statements are valid? (Choose all correct state-
ments.) (a) The buoyant force on each is the same.
(b) The buoyant force on the lead sphere is greater
than the buoyant force on the iron sphere because lead
has the greater density. (c) The tension in the string
supporting the lead sphere is greater than the tension
in the string supporting the iron sphere. (d) The buoy-
ant force on the iron sphere is greater than the buoy-
ant force on the lead sphere because lead displaces
more water. (e) None of those statements is true.
7. Three vessels of different shapes are filled to the same
level with water as in Figure OQ14.7. The area of the
base is the same for all three vessels. Which of the fol-
lowing statements are valid? (Choose all correct state-
ments.) (a) The pressure at the top surface of vessel
A is greatest because it has the largest surface area.
(b) The pressure at the bottom of vessel A is greatest
because it contains the most water. (c)The pressure at
the bottom of each vessel is the same. (d) The force on
the bottom of each vessel is not the same. (e) At a given
depth below the surface of each vessel, the pressure on
the side of vessel A is greatest because of its slope.
8. One of the predicted problems due to global warm-
ing is that ice in the polar ice caps will melt and raise
sea levels everywhere in the world. Is that more of a
worry for ice (a)at the north pole, where most of the
ice floats on water; (b) at the south pole, where most
of the ice sits on land; (c)both at the north and south
pole equally; or (d) at neither pole?
9. A boat develops a leak and, after its passengers are res-
cued, eventually sinks to the bottom of a lake. When
the boat is at the bottom, what is the force of the lake
bottom on the boat? (a) greater than the weight of the
boat (b) equal to the weight of the boat (c) less than
denotes answer available in Student Solutions Manual/Study Guide
1. When an object is immersed in a liquid at rest, why is
the net force on the object in the horizontal direction
equal to zero?
2. Two thin-walled drinking glasses having equal base
areas but different shapes, with very different cross-
sectional areas above the base, are filled to the same
level with water. According to the expression P 5 P
rgh, the pressure is the same at the bottom of both
glasses. In view of this equality, why does one weigh
more than the other?
3. Because atmospheric pressure is about 105 N/m2 and the
area of a person’s chest is about 0.13 m2, the force of the
14. Does a ship float higher in the water of an inland lake
or in the ocean? Why?
15. When ski jumpers are airborne (Fig. CQ14.15), they
bend their bodies forward and keep their hands at
their sides. Why?
16. Why do airplane pilots prefer to take off with the air-
plane facing into the wind?
17. Prairie dogs ventilate their burrows by building a mound
around one entrance, which is open to a stream of air
when wind blows from any direction. A second entrance
at ground level is open to almost stagnant air. How does
this construction create an airflow through the burrow?
18. In Figure CQ14.18, an airstream moves from right to
left through a tube that is constricted at the middle.
Three table-tennis balls are levitated in equilibrium
above the vertical columns through which the air
escapes. (a) Why is the ball at the right higher than the
one in the middle? (b)Why is the ball at the left lower
than the ball at the right even though the horizontal
tube has the same dimensions at these two points?
19. A typical silo on a farm has many metal bands wrapped
around its perimeter for support as shown in Figure
CQ14.19. Why is the spacing between successive bands
smaller for the lower portions of the silo on the left,
and why are double bands used at lower portions of the
silo on the right?
atmosphere on one’s chest is around 13 000 N. In view of
this enormous force, why don’t our bodies collapse?
4. A fish rests on the bottom of a bucket of water while
the bucket is being weighed on a scale. When the fish
begins to swim around, does the scale reading change?
Explain your answer.
5. You are a passenger on a spacecraft. For your survival
and comfort, the interior contains air just like that at
the surface of the Earth. The craft is coasting through
a very empty region of space. That is, a nearly perfect
vacuum exists just outside the wall. Suddenly, a mete-
oroid pokes a hole, about the size of a large coin, right
through the wall next to your seat. (a) What happens?
(b) Is there anything you can or should do about it?
6. If the airstream from a hair dryer is directed over a
table-tennis ball, the ball can be levitated. Explain.
7. A water tower is a common sight in many communities.
Figure CQ14.7 shows a collection of colorful water tow-
ers in Kuwait City, Kuwait. Notice that the large weight
of the water results in the center of mass of the system
being high above the ground. Why is it desirable for a
water tower to have this highly unstable shape rather
than being shaped as a tall cylinder?
8. If you release a ball while inside a freely falling eleva-
tor, the ball remains in front of you rather than falling
to the floor because the ball, the elevator, and you all
experience the same downward gravitational accelera-
tion. What happens if you repeat this experiment with
a helium-filled balloon?
9. (a) Is the buoyant force a conservative force? (b) Is a
potential energy associated with the buoyant force?
(c) Explain your answers to parts (a) and (b).
10. An empty metal soap dish barely floats in water. A bar
of Ivory soap floats in water. When the soap is stuck in
the soap dish, the combination sinks. Explain why.
11. How would you determine the density of an irregularly
12. Place two cans of soft drinks, one regular and one diet,
in a container of water. You will find that the diet drink
floats while the regular one sinks. Use Archimedes’s
principle to devise an explanation.
13. The water supply for a city is often provided from res-
ervoirs built on high ground. Water flows from the
reservoir, through pipes, and into your home when
you turn the tap on your faucet. Why does water flow
more rapidly out of a faucet on the first floor of a
building than in an apartment on a higher floor?
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chapter 14 Fluid Mechanics
Note: In all problems, assume the density of air is the
20°C value from Table 14.1, 1.20 kg/m3, unless noted
Section 14.1 Pressure
1. A large man sits on a four-legged chair with his feet off
the floor. The combined mass of the man and chair is
95.0 kg. If the chair legs are circular and have a radius
of 0.500 cm at the bottom, what pressure does each leg
exert on the floor?
2. The nucleus of an atom can be modeled as several pro-
tons and neutrons closely packed together. Each par-
ticle has a mass of 1.67 3 10227 kg and radius on the
order of 10215 m. (a) Use this model and the data pro-
vided to estimate the density of the nucleus of an atom.
(b) Compare your result with the density of a material
such as iron. What do your result and comparison sug-
gest concerning the structure of matter?
3. A 50.0-kg woman wearing high-heeled shoes is invited
into a home in which the kitchen has vinyl floor cover-
ing. The heel on each shoe is circular and has a radius
of 0.500cm. (a) If the woman balances on one heel,
what pressure does she exert on the floor? (b) Should
the home owner be concerned? Explain your answer.
4. Estimate the total mass of the Earth’s atmosphere.
(The radius of the Earth is 6.37 3 106 m, and atmo-
spheric pressure at the surface is 1.013 3 105 Pa.)
5. Calculate the mass of a solid gold rectangular bar that
has dimensions of 4.50 cm 3 11.0 cm 3 26.0 cm.
Section 14.2 Variation of Pressure with Depth
6. (a) A very powerful vacuum cleaner has a hose 2.86 cm
in diameter. With the end of the hose placed perpen-
dicularly on the flat face of a brick, what is the weight
of the heaviest brick that the cleaner can lift? (b) What
If? An octopus uses one sucker of diameter 2.86 cm on
each of the two shells of a clam in an attempt to pull
the shells apart. Find the greatest force the octopus
can exert on a clamshell in salt water 32.3 m deep.
7. The spring of the pressure gauge shown in Figure
P14.7 has a force constant of 1 250 N/m, and the piston
has a diameter of 1.20 cm. As the gauge is lowered into
water in a lake, what change in depth causes the piston
to move in by 0.750 cm?
8. The small piston of a hydraulic lift (Fig. P14.8) has a
cross-sectional area of 3.00 cm2, and its large piston
has a cross-sectional area of 200 cm2. What downward
force of magnitude F
must be applied to the small
piston for the lift to raise a load whose weight is F
= 15.0 kN
9. What must be the contact area between a suction cup
(completely evacuated) and a ceiling if the cup is to
support the weight of an 80.0-kg student?
10. A swimming pool has dimensions 30.0 m 3 10.0 m and a
flat bottom. When the pool is filled to a depth of 2.00 m
with fresh water, what is the force exerted by the water
on (a) the bottom? (b) On each end? (c) On each side?
11. (a) Calculate the absolute pressure at the bottom of
a freshwater lake at a point whose depth is 27.5 m.
Assume the density of the water is 1.00 3 103 kg/m3
and that the air above is at a pressure of 101.3 kPa.
(b) What force is exerted by the water on the window
of an underwater vehicle at this depth if the window is
circular and has a diameter of 35.0cm?
12. Why is the following situation impossible? Figure P14.12
shows Superman attempting to drink cold water
The problems found in this
chapter may be assigned
online in Enhanced WebAssign
full solution available in the Student
Solutions Manual/Study Guide
Analysis Model tutorial available in
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17. Review. Piston in Figure P14.17 has a diameter of
0.250 in. Piston has a diameter of 1.50 in. Determine
the magnitude F of the force necessary to support the
500-lb load in the absence of friction.
18. Review. A solid sphere of brass (bulk modulus of
14.0 3 1010 N/m2) with a diameter of 3.00 m is thrown
into the ocean. By how much does the diameter of the
sphere decrease as it sinks to a depth of 1.00 km?
Section 14.3 Pressure Measurements
19. Normal atmospheric pressure is 1.013 3 105 Pa. The
approach of a storm causes the height of a mercury
barometer to drop by 20.0 mm from the normal height.
What is the atmospheric pressure?
20. The human brain and spinal cord are immersed in the
cerebrospinal fluid. The fluid is normally continuous
between the cranial and spinal cavities and exerts a
pressure of 100 to 200 mm of H
O above the prevail-
ing atmospheric pressure. In medical work, pressures
are often measured in units of millimeters of H
because body fluids, including the cerebrospinal fluid,
typically have the same density as water. The pressure
of the cerebrospinal fluid can be measured by means
of a spinal tap as illustrated in Figure P14.20. A hollow
tube is inserted into the spinal column, and the height
to which the fluid rises is observed. If the fluid rises
to a height of 160 mm, we write its gauge pressure as
160 mm H
O. (a) Express this pressure in pascals, in
atmospheres, and in millimeters of mercury. (b)Some
conditions that block or inhibit the flow of cerebrospi-
nal fluid can be investigated by means of Queckenstedt’s
test. In this procedure, the veins in the patient’s neck
are compressed to make the blood pressure rise in the
brain, which in turn should be transmitted to the cere-
brospinal fluid. Explain how the level of fluid in the
spinal tap can be used as a diagnostic tool for the con-
dition of the patient’s spine.
through a straw of length , 5 12.0 m. The walls of the
tubular straw are very strong and do not collapse. With
his great strength, he achieves maximum possible suc-
tion and enjoys drinking the cold water.
13. For the cellar of a new house, a hole is dug in the
ground, with vertical sides going down 2.40 m. A con-
crete foundation wall is built all the way across the
9.60-m width of the excavation. This foundation wall
is 0.183 m away from the front of the cellar hole. Dur-
ing a rainstorm, drainage from the street fills up the
space in front of the concrete wall, but not the cellar
behind the wall. The water does not soak into the clay
soil. Find the force the water causes on the founda-
tion wall. For comparison, the weight of the water is
given by 2.40 m 3 9.60 m 3 0.183 m 3 1 000 kg/m3 3
9.80 m/s2 5 41.3 kN.
14. A container is filled to a depth of 20.0 cm with water.
On top of the water floats a 30.0-cm-thick layer of oil
with specific gravity 0.700. What is the absolute pres-
sure at the bottom of the container?
15. Review. The tank in Figure P14.15 is filled with water
of depth d 5 2.00 m. At the bottom of one sidewall is a
rectangular hatch of height h 5 1.00 m and width w 5
2.00m that is hinged at the top of the hatch. (a) Deter-
mine the magnitude of the force the water exerts
on the hatch. (b)Find the magnitude of the torque
exerted by the water about the hinges.
Problems 15 and 16.
16. Review. The tank in Figure P14.15 is filled with water of
depth d. At the bottom of one sidewall is a rectangular
hatch of height h and width w that is hinged at the top
of the hatch. (a) Determine the magnitude of the force
the water exerts on the hatch. (b) Find the magnitude
of the torque exerted by the water about the hinges.
chapter 14 Fluid Mechanics
scale and submerged in water, the scale reads 3.50 N
(Fig. P14.26). Find the density of the object.
Problems 26 and 27.
27. A 10.0-kg block of metal measuring 12.0 cm by 10.0 cm
by 10.0 cm is suspended from a scale and immersed in
water as shown in Figure P14.26b. The 12.0-cm dimen-
sion is vertical, and the top of the block is 5.00 cm below
the surface of the water. (a) What are the magnitudes of
the forces acting on the top and on the bottom of the
block due to the surrounding water? (b) What is the
reading of the spring scale? (c)Show that the buoyant
force equals the difference between the forces at the top
and bottom of the block.
28. A light balloon is filled with 400 m3 of helium at atmo-
spheric pressure. (a) At 0°C, the balloon can lift a pay-
load of what mass? (b) What If? In Table 14.1, observe
that the density of hydrogen is nearly half the density
of helium. What load can the balloon lift if filled with
29. A cube of wood having an edge dimension of 20.0 cm
and a density of 650 kg/m3 floats on water. (a) What
is the distance from the horizontal top surface of the
cube to the water level? (b) What mass of lead should
be placed on the cube so that the top of the cube will
be just level with the water surface?
30. The United States possesses the ten largest warships
in the world, aircraft carriers of the Nimitz class. Sup-
pose one of the ships bobs up to float 11.0 cm higher
in the ocean water when 50 fighters take off from it in
a time interval of 25 min, at a location where the free-
fall acceleration is 9.78m/s2. The planes have an aver-
age laden mass of 29 000kg. Find the horizontal area
enclosed by the waterline of the ship.
31. A plastic sphere floats in water with 50.0% of its vol-
ume submerged. This same sphere floats in glycerin
with 40.0% of its volume submerged. Determine the
densities of (a)the glycerin and (b) the sphere.
32. A spherical vessel used for deep-sea exploration has a
radius of 1.50 m and a mass of 1.20 3 104 kg. To dive,
the vessel takes on mass in the form of seawater. Deter-
mine the mass the vessel must take on if it is to descend
at a constant speed of 1.20 m/s, when the resistive force
on it is 1 100 N in the upward direction. The density of
seawater is equal to 1.03 3 103 kg/m3.
33. A wooden block of volume 5.24 3 1024 m3 floats in
water, and a small steel object of mass m is placed on
top of the block. When m 5 0.310 kg, the system is in
21. Blaise Pascal duplicated Torricelli’s barometer using a
red Bordeaux wine, of density 984 kg/m3, as the work-
ing liquid (Fig. P14.21). (a) What was the height h of
the wine column for normal atmospheric pressure?
(b) Would you expect the vacuum above the column to
be as good as for mercury?
22. Mercury is poured into a U-tube as shown in Figure
P14.22a. The left arm of the tube has cross-sectional
of 10.0cm2, and the right arm has a cross-
sectional area A
of 5.00 cm2. One hundred grams of
water are then poured into the right arm as shown in
Figure P14.22b. (a)Determine the length of the water
column in the right arm of the U-tube. (b) Given that
the density of mercury is 13.6 g/cm3, what distance h
does the mercury rise in the left arm?
23. A backyard swimming pool with a circular base of
diameter 6.00 m is filled to depth 1.50 m. (a) Find the
absolute pressure at the bottom of the pool. (b) Two
persons with combined mass 150 kg enter the pool and
float quietly there. No water overflows. Find the pres-
sure increase at the bottom of the pool after they enter
the pool and float.
24. A tank with a flat bottom of area A and vertical sides is
filled to a depth h with water. The pressure is P
top surface. (a) What is the absolute pressure at the bot-
tom of the tank? (b) Suppose an object of mass M and
density less than the density of water is placed into the
tank and floats. No water overflows. What is the result-
ing increase in pressure at the bottom of the tank?
Section 14.4 Buoyant Forces and archimedes’s Principle
25. A table-tennis ball has a diameter of 3.80 cm and aver-
age density of 0.084 0 g/cm3. What force is required to
hold it completely submerged under water?
26. The gravitational force exerted on a solid object is
5.00 N. When the object is suspended from a spring
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fiduciary marks are to be placed along the rod to indi-
cate densities of 0.98 g/cm3, 1.00 g/cm3, 1.02 g/cm3,
1.04 g/cm3, . . . , 1.14 g/cm3. The row of marks is to start
0.200 cm from the top end of the rod and end 1.80 cm
from the top end. (a) What is the required length of the
rod? (b) What must be its average density? (c) Should
the marks be equally spaced? Explain your answer.
38. On October 21, 2001, Ian Ashpole of the United King-
dom achieved a record altitude of 3.35 km (11 000ft)
powered by 600 toy balloons filled with helium. Each
filled balloon had a radius of about 0.50 m and an esti-
mated mass of 0.30 kg. (a) Estimate the total buoyant
force on the 600balloons. (b) Estimate the net upward
force on all 600balloons. (c) Ashpole parachuted to
the Earth after the balloons began to burst at the high
altitude and the buoyant force decreased. Why did the
39. How many cubic meters of helium are required to lift
a light balloon with a 400-kg payload to a height of
8 000 m? Take r
5 0.179 kg/m3. Assume the balloon
maintains a constant volume and the density of air
decreases with the altitude z according to the expres-
e2z/8 000, where z is in meters and r
1.20 kg/m3 is the density of air at sea level.
Section 14.5 Fluid Dynamics
Section 14.6 Bernoulli’s Equation
40. Water flowing through a garden hose of diameter
2.74 cm fills a 25-L bucket in 1.50 min. (a) What is the
speed of the water leaving the end of the hose? (b) A
nozzle is now attached to the end of the hose. If the
nozzle diameter is one-third the diameter of the hose,
what is the speed of the water leaving the nozzle?
41. A large storage tank, open at the top and filled with
water, develops a small hole in its side at a point 16.0 m
below the water level. The rate of flow from the leak
is found to be 2.50 3 1023m3/min. Determine (a) the
speed at which the water leaves the hole and (b) the
diameter of the hole.
42. Water moves through a constricted pipe in steady, ideal
flow. At the lower point shown in Figure P14.42, the
pressure is P
5 1.75 3 104 Pa and the pipe diameter
is 6.00cm. At another point y 5 0.250 m higher, the
pressure is P
5 1.203 104 Pa and the pipe diameter is
3.00 cm. Find the speed of flow (a) in the lower section
and (b) in the upper section. (c) Find the volume flow
rate through the pipe.
43. Figure P14.43 on page 442 shows a stream of water in
steady flow from a kitchen faucet. At the faucet, the
equilibrium and the top of the wooden block is at the
level of the water. (a) What is the density of the wood?
(b) What happens to the block when the steel object is
replaced by an object whose mass is less than 0.310 kg?
(c) What happens to the block when the steel object
is replaced by an object whose mass is greater than
34. The weight of a rectangular block of low-density mate-
rial is 15.0 N. With a thin string, the center of the hori-
zontal bottom face of the block is tied to the bottom of
a beaker partly filled with water. When 25.0% of the
block’s volume is submerged, the tension in the string is
10.0 N. (a) Find the buoyant force on the block. (b) Oil
of density 800 kg/m3 is now steadily added to the bea-
ker, forming a layer above the water and surround-
ing the block. The oil exerts forces on each of the
four sidewalls of the block that the oil touches. What
are the directions of these forces? (c) What happens
to the string tension as the oil is added? Explain how
the oil has this effect on the string tension. (d) The
string breaks when its tension reaches 60.0 N. At this
moment, 25.0% of the block’s volume is still below the
water line. What additional fraction of the block’s vol-
ume is below the top surface of the oil?
35. A large weather balloon whose mass is 226 kg is filled
with helium gas until its volume is 325 m3. Assume the
density of air is 1.20 kg/m3 and the density of helium is
0.179 kg/m3. (a) Calculate the buoyant force acting on
the balloon. (b) Find the net force on the balloon and
determine whether the balloon will rise or fall after it
is released. (c) What additional mass can the balloon
support in equilibrium?
36. A hydrometer is an instrument used to determine liquid
density. A simple one is sketched in Figure P14.36. The
bulb of a syringe is squeezed and released to let the
atmosphere lift a sample of the liquid of interest into a
tube containing a calibrated rod of known density. The
rod, of length L and average density r
, floats partially
immersed in the liquid of density r. A length h of the
rod protrudes above the surface of the liquid. Show
that the density of the liquid is given by
Problems 36 and 37.
37. Refer to Problem 36 and Figure P14.36. A hydrometer is
to be constructed with a cylindrical floating rod. Nine
chapter 14 Fluid Mechanics
water must be pumped if it is to arrive at the village?
(b) If 4 500 m3 of water is pumped per day, what is
the speed of the water in the pipe? Note: Assume the
free-fall acceleration and the density of air are con-
stant over this range of elevations. The pressures you
calculate are too high for an ordinary pipe. The water
is actually lifted in stages by several pumps through
48. In ideal flow, a liquid of density 850 kg/m3 moves from
a horizontal tube of radius 1.00 cm into a second hori-
zontal tube of radius 0.500 cm at the same elevation as
the first tube. The pressure differs by DP between the
liquid in one tube and the liquid in the second tube.
(a) Find the volume flow rate as a function of DP. Eval-
uate the volume flow rate for (b) DP 5 6.00 kPa and
(c) DP 5 12.0 kPa.
49. The Venturi tube discussed in Example 14.8 and shown
in Figure P14.49 may be used as a fluid flowmeter.
Suppose the device is used at a service station to mea-
sure the flow rate of gasoline (r 5 7.00 3 102 kg/m3)
through a hose having an outlet radius of 1.20 cm. If
the difference in pressure is measured to be P
1.20 kPa and the radius of the inlet tube to the meter
is 2.40 cm, find (a) the speed of the gasoline as it leaves
the hose and (b) the fluid flow rate in cubic meters per
50. Review. Old Faithful Geyser in Yellowstone National
Park erupts at approximately one-hour intervals,
and the height of the water column reaches 40.0 m
(Fig. P14.50). (a)Model the rising stream as a series
of separate droplets. Analyze the free-fall motion of
diameter of the stream is 0.960 cm. The stream fills a
125-cm3 container in 16.3 s. Find the diameter of the
stream 13.0 cm below the opening of the faucet.
44. A village maintains a large tank with an open top, con-
taining water for emergencies. The water can drain
from the tank through a hose of diameter 6.60 cm. The
hose ends with a nozzle of diameter 2.20 cm. A rubber
stopper is inserted into the nozzle. The water level in
the tank is kept 7.50 m above the nozzle. (a) Calculate
the friction force exerted on the stopper by the nozzle.
(b) The stopper is removed. What mass of water flows
from the nozzle in 2.00h? (c) Calculate the gauge pres-
sure of the flowing water in the hose just behind the
45. A legendary Dutch boy saved Holland by plugging a
hole of diameter 1.20 cm in a dike with his finger. If
the hole was 2.00 m below the surface of the North Sea
(density 1 030 kg/m3), (a) what was the force on his fin-
ger? (b) If he pulled his finger out of the hole, during
what time interval would the released water fill 1 acre
of land to a depth of 1ft? Assume the hole remained
constant in size.
46. Water falls over a dam of height h with a mass flow rate
of R, in units of kilograms per second. (a) Show that
the power available from the water is
P 5 Rgh
where g is the free-fall acceleration. (b) Each hydro-
electric unit at the Grand Coulee Dam takes in water at
a rate of 8.50 3 105 kg/s from a height of 87.0 m. The
power developed by the falling water is converted to
electric power with an efficiency of 85.0%. How much
electric power does each hydroelectric unit produce?
47. Water is pumped up from the Colorado River to sup-
ply Grand Canyon Village, located on the rim of the
canyon. The river is at an elevation of 564 m, and the
village is at an elevation of 2 096 m. Imagine that
the water is pumped through a single long pipe 15.0 cm
in diameter, driven by a single pump at the bottom
end. (a) What is the minimum pressure at which the
4.00 m 3 1.50 m. Assume the density of the air to be
constant at 1.20 kg/m3. The air inside the building is at
atmospheric pressure. What is the total force exerted
by air on the windowpane? (b) What If? If a second
skyscraper is built nearby, the airspeed can be espe-
cially high where wind passes through the narrow sepa-
ration between the buildings. Solve part (a) again with
a wind speed of 22.4 m/s, twice as high.
55. A hypodermic syringe contains a medicine with the
density of water (Fig. P14.55). The barrel of the syringe
has a cross-sectional area A 5 2.50 3 1025 m2, and the
needle has a cross-sectional area a 5 1.00 3 1028 m2.
In the absence of a force on the plunger, the pressure
everywhere is 1.00 atm. A force F
of magnitude 2.00 N
acts on the plunger, making medicine squirt hori-
zontally from the needle. Determine the speed of the
medicine as it leaves the needle’s tip.
56. Decades ago, it was thought that huge herbivorous
dinosaurs such as Apatosaurus and Brachiosaurus habit-
ually walked on the bottom of lakes, extending their
long necks up to the surface to breathe. Brachiosaurus
had its nostrils on the top of its head. In 1977, Knut
Schmidt-Nielsen pointed out that breathing would be
too much work for such a creature. For a simple model,
consider a sample consisting of 10.0 L of air at absolute
pressure 2.00 atm, with density 2.40 kg/m3, located at
the surface of a freshwater lake. Find the work required
to transport it to a depth of 10.3 m, with its tempera-
ture, volume, and pressure remaining constant. This
energy investment is greater than the energy that can
be obtained by metabolism of food with the oxygen in
that quantity of air.
57. (a) Calculate the absolute pressure at an ocean depth of
1 000 m. Assume the density of seawater is 1 030 kg/m3
and the air above exerts a pressure of 101.3 kPa. (b) At
this depth, what is the buoyant force on a spherical
submarine having a diameter of 5.00 m?
58. In about 1657, Otto von Guericke, inventor of the air
pump, evacuated a sphere made of two brass hemi-
spheres (Fig. P14.58). Two teams of eight horses each
could pull the hemispheres apart only on some trials
and then “with greatest difficulty,” with the resulting
one of the droplets to determine the speed at which
the water leaves the ground. (b) What If? Model the
rising stream as an ideal fluid in streamline flow.
Use Bernoulli’s equation to determine the speed of
the water as it leaves ground level. (c) How does the
answer from part (a) compare with the answer from
part (b)? (d) What is the pressure (above atmospheric)
in the heated underground chamber if its depth is
175 m? Assume the chamber is large compared with
the geyser’s vent.
Section 14.7 other applications of Fluid Dynamics
51. An airplane is cruising at altitude 10 km. The pressure
outside the craft is 0.287 atm; within the passenger
compartment, the pressure is 1.00 atm and the temper-
ature is 208C. A small leak occurs in one of the window
seals in the passenger compartment. Model the air as
an ideal fluid to estimate the speed of the airstream
flowing through the leak.
52. An airplane has a mass of 1.60 3 104 kg, and each wing
has an area of 40.0 m2. During level flight, the pressure
on the lower wing surface is 7.00 3 104 Pa. (a) Suppose
the lift on the airplane were due to a pressure differ-
ence alone. Determine the pressure on the upper wing
surface. (b) More realistically, a significant part of the
lift is due to deflection of air downward by the wing.
Does the inclusion of this force mean that the pressure
in part (a) is higher or lower? Explain.
53. A siphon is used to drain water from a tank as illus-
trated in Figure P14.53. Assume steady flow without
friction. (a)If h 5 1.00 m, find the speed of outflow at
the end of the siphon. (b) What If? What is the limita-
tion on the height of the top of the siphon above the
end of the siphon? Note: For the flow of the liquid to be
continuous, its pressure must not drop below its vapor
pressure. Assume the water is at 20.08C, at which the
vapor pressure is 2.3kPa.
54. The Bernoulli effect can have important consequences
for the design of buildings. For example, wind can
blow around a skyscraper at remarkably high speed,
creating low pressure. The higher atmospheric pres-
sure in the still air inside the buildings can cause win-
dows to pop out. As originally constructed, the John
Hancock Building in Boston popped windowpanes
that fell many stories to the sidewalk below. (a) Sup-
pose a horizontal wind blows with a speed of 11.2 m/s
outside a large pane of plate glass with dimensions
chapter 14 Fluid Mechanics
balance with the use of counterweights of density r.
Representing the density of air as r
and the balance
reading as F9
, show that the true weight F
63. Water is forced out of a fire extinguisher by air pres-
sure as shown in Figure P14.63. How much gauge air
pressure in the tank is required for the water jet to have
a speed of 30.0m/s when the water level is 0.500 m
below the nozzle?
64. Review. Assume a certain liquid, with density
1 230kg/m3, exerts no friction force on spherical
objects. A ball of mass 2.10 kg and radius 9.00 cm is
dropped from rest into a deep tank of this liquid from a
height of 3.30m above the surface. (a) Find the speed at
which the ball enters the liquid. (b) Evaluate the magni-
tudes of the two forces that are exerted on the ball as it
moves through the liquid. (c) Explain why the ball
moves down only a limited distance into the liquid and
calculate this distance. (d)With what speed will the ball
pop up out of the liquid? (e) How does the time interval
, during which the ball moves from the surface
down to its lowest point, compare with the time interval
for the return trip between the same two points?
(f) What If? Now modify the model to suppose the liq-
uid exerts a small friction force on the ball, opposite in
direction to its motion. In this case, how do the time
compare? Explain your answer
with a conceptual argument rather than a numerical
65. Review. A light spring of constant k 5 90.0 N/m is
attached vertically to a table (Fig. P14.65a). A 2.00-g
balloon is filled with helium (density 5 0.179 kg/m3)
sound likened to a cannon firing. Find the force F
required to pull the thin-walled evacuated hemispheres
apart in terms of R, the radius of the hemispheres; P,
the pressure inside the hemispheres; and atmospheric
59. A spherical aluminum ball of mass 1.26 kg contains an
empty spherical cavity that is concentric with the ball.
The ball barely floats in water. Calculate (a) the outer
radius of the ball and (b) the radius of the cavity.
60. A helium-filled balloon (whose envelope has a mass of
5 0.250 kg) is tied to a uniform string of length , 5
2.00m and mass m 5 0.050 0 kg. The balloon is spheri-
cal with a radius of r 5 0.400m. When released in air
of temperature 208C and density r
5 1.20 kg/m3, it
lifts a length h of string and then remains stationary as
shown in Figure P14.60. We wish to find the length of
string lifted by the balloon. (a)When the balloon
remains stationary, what is the appropriate analysis
model to describe it? (b) Write a force equation for
the balloon from this model in terms of the buoyant
force B, the weight F
of the balloon, the weight F
the helium, and the weight F
of the segment of string
of length h. (c) Make an appropriate substitution for
each of these forces and solve symbolically for the
of the segment of string of length h in terms
, r, r
, and the density of helium r
. (d) Find
the numerical value of the mass m
. (e) Find the length
61. Review. Figure P14.61 shows a valve separating a res-
ervoir from a water tank. If this valve is opened, what
is the maximum height above point B attained by the
water stream coming out of the right side of the tank?
Assume h5 10.0m, L 5 2.00 m, and u 5 30.0°, and
assume the cross-sectional area at A is very large com-
pared with that at B.
62. The true weight of an object can be measured in a
vacuum, where buoyant forces are absent. A measure-
ment in air, however, is disturbed by buoyant forces. An
object of volume V is weighed in air on an equal-arm
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