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SUMMARY The goal of Chapter 13 has been to understand the static and dynamic properties of fluids. GENERAL PRINCIPLES Fluid Statics Gases Fluid Dynamics Ideal-fluid model Su mmary 431 Freely mov in g particles Compressible , " In co mpressible Smooth, lamin ar now No nviscous Density p Pressure mainly due to particle co llisions with walls liquids , , , Equation of continuity A) CiV Volume Ilow rate Q = - = vIA I = V2A 2 Ci , " , Loosely bou nd particles Ln compress ibl e Pressure due to the weight of the liquid Bernoulli's equation is a statement of energy co nservation: I ., I ., PI +"2PV I-+ pgYI = P 2+ "2PV2- +pgyZ p Hydrostatic press ure at depth d is p = Po + pgd Th e press ure is the same at a ll points on a horizontal line through a liquid (of one ki nd) in hydrostati c eq uilibrium Poiseuille's equation governs viscous flow through a tube: IMPORTANT CONCEPTS Density p = ",IV, where m is mass and V is volume. Pressure P = FIA, where F is force magnitude and A is the area on which the force acts. Press ure ex ists at a ll po ints in a fluid. Pressure pushes equaJ ly in a ll directions. Gauge pressure P g = P - I atm. Viscosity 'Y1 is th e property of a fluid thai makes it resist fl owin g. APPLICATIONS Buoyancy is th e upward force of a fluid on an obj ect immersed in the fluid. Archimedes ' principle: Th e magnitude of the buoyant force eq uals the weight of the fluid displaced by the object. Sink: Payg > Pr FB < we) Float: P ayg < Pr FB > we) P '''. Neutrally _ FB = Wo buoyant: P ayg - Pf Representing fluid flow Streamlines are the paths of individual fluid particles. The velociry of a fluid panicle t%:::: '-- where the are closer togclh cr. Barometers measure atmospheric pressure. Atmospheric press ure is related to the height of the li quid column by PalnJOS = pgll. , ..... . vg ! A Fluid elements co nta in a fi xed vol ume of flui d. Their shape may change as they move. Every flu id panicle thai up the element moves on i l.s Own strc:lmlinc. Manometers measure press ure. Th e pressure at the closed e nd o f the nlbe is p = I atm + pgll. p P' fIl l<» These two points are at the smne pressure p.

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SUMMARY The goal of Chapter 13 has been to understand the static
and dynamic properties of fluids.

GENERAL PRINCIPLES

Density p

Pressure mainl y due to particle colli s ions with walls

liquids

Ci ,

p

Hydrostati c pressure at depth d is p = Po + pgd

The pressure is the same at all points o n a hori zontal line through a liquid (of one kind) in hydrostatic equilibrium

Poiseuille's equation governs viscous flow through a tube:

IMPORTANT CONCEPTS

Density p = ",IV, where m is mass and V is vo lume.

Pressure P = FIA, where F is force magnitude and A is the area on wh ic h the force acts.

Pressure ex ists at all points in a fluid.

Pressure pushes equaJ ly in all d irect ions.

Gauge pressure Pg = P - I atm.

Viscosity 'Y1 is the property of a fluid thai makes it resist fl owing.

APPLICATIONS

Buoyancy is the upward force of a fluid on an object immersed in the flu id.

Archimedes' principle: The magni tude of the buoyant force equals the weigh t of the fluid displaced by the object.

Sink: Payg > P r FB < we)

L~j Float: P ayg < Pr FB > we)

P '''. Neutrally _ FB = Wo buoyant: P ayg - Pf

Representing fluid flow

The velociry of a fluid panicle i.~

,,,"g"t to,,,~/ "''''mlm'~

Barometers measure atmospheric pressure. Atmospheric pressure is related to the he ight of the liquid colum n by PalnJOS = pgll.

~! , ..... . vg

! A

Fluid elements contain a fi xed vol ume of flui d. Their shape may change as they move.

Every flu id panicle thai make~ up the element moves on il.s Own strc:lmlinc.

Manometers measure pressure . The pressure at the closed e nd of the nlbe is p = I atm + pgll.

p

432 CHAPTER 13 Fluids

!:!.!/ www.masteringphysics.com

Problem difficulty is labeled as I (straightforward) to 11111 (challenging).

QUESTIONS

Conceptual Questions

t. Which has the greater density, 1 g of mercury or 1000 g of water?

2. A 1 X 10- 3 111

3 chunk of material has a mass of 3 kg. a. What is the material's density? b. Would a 2 X 10- 3 m} chunk of the same material have the

same mass? Explain. c. Would a 2 X 10-] mJ chunk of the same material have the

same density? Explain. 3. You are give n an irregularly shaped chunk of material and

asked to find its density. List the specific steps that you would follow to do so.

4. Object 1 has an irregular shape . Its density is 4000 kg/mJ. a. Object 2 has the same shape and dimensions as object I ,

but it is twice as mass ive. What is the density of object 2? b. Object 3 has the same mass and the same shape as object I,

but its size in all three dimensions is twice that of object I. What is the density of object 3?

5. When yo u get a blood transfusion the bag of blood is held BIO above your body, but when you donate blood the collection bag

is held below. Why is thi s? 6. To explore the bottom of a 10-m-deep lake, your friend Tom

BIO proposes to get a long garden hose, put one end on land and the other in hi s mouth fo r breathing underwater, and descend into the depths. Susan, who overhears the conversation, reacts with horror and warns Tom that he will not be able to inhale when he is at the lake bottom. Why is Susan so worried?

7. Rank in order, from largest to smaUest, the pressures at A, B, and C in Figure QI3.7. Explain.

8. Refer to Figure Q13.7. Rank in order, from largest to smallest, the pressures at 0, E, and F. Explain.

9. Cylinders A and B contain liq- uids. The pressure PA at the bot tom of A is higher than the pressure PB althe bottom of B. Is

E o

~ II

c

the ratio PA Ips of the absolute pressures larger, smaller, or equallo the rati o of the gauge pressures? Explain.

10. In FigureQ I3. 10, Aand B arerec tangular tanks full of water. They have equal he ights and equal side lengths (the dimension into the page), but different widths. a. Compare the forces the water

DOl A Sides 8

FIGURE 013.10

exerts on the bottoms o f the tanks. Is FA larger. smaller, or equal to F B ? Explain.

b. Compare the forces the water exerts on the s ides of the tanks. Is FA larger, smaUer, or equal to FB? Explain.

Problems labeled INi integrate significant material from earlier

chapters; BlO are of biological or medical interest.

II. Heliu m-filled weather balloons are spherical whe n they reach very high altitudes. However, they are on ly partial.ly inflated with helium before they are released. Explain why thi s is done.

12. Water expands when heated. Suppose a beaker of water is heated from 10°C to 90°C. Does the pressure at the bottom of the beaker inc rease, decrease, or Slay the same? Explain.

13. In Figure QI3.13, is VA larger, ~

smaUer, or equal to PB? Explain. . . . .... 2m 14. A beaker o f water rests on a

scale. A metal ball is then low ered into the beaker using a string tied to the ball. The baJl doesn't touch the sides or bot- tom of the beaker, and no water sp iUs from the beaker. Does the scale reading decrease, increase, or stay the same? Ex plain.

o A

FIGURE 013.13

. o . B

15. Rank in order, from largest to smallest, the densities of objects A, B, and C in Figure QI3. IS. Explain.

8'--JB c

FIGURE 013 .15 FIGURE 013.16

16. Objects A, B, and C in Figure Q 13. 16 have the same volume. Rank in order, from largest to smallest, the s izes of the buoyant forces FA' FB, and Fe on A, B, and C. Explain.

17. Refer to Figure QI3. 16. Now A, B, and C have the same den s ity, but st ill have the masses g iven in the figure. Rank in order, from largest to smallest, the sizes of the buoyant forces on A, B, and C. Ex plain.

18. When you stand on a bathroom scale, it reads 700 N. Suppose a g iant vac uum cleaner Slicks half the air out of the room, reduc ing the pressure to 0.5 atm. Would the scale read ing increase, decrease. or stay the same? Explain.

19. Suppose you stand on a bathroom scale that is on the botto m of a swimming pool. The water comes up to your waist. Does the scale read more, less, or the same as your true weight? Ex plain.

20. When you place an egg in water, it s inks. If you add sa lt to the water, after some time the egg noats. Ex plain.

21.

22.

BID

23.

24.

25.

26.

27.

28.

Submerged submarines contain tanks filled with water. To ri se to the surface, compressed air is used to force the water out of the tanks. Explain why thi s works. Fish can adjust their buoyancy with an organ cal led the swim biaddeJ: The swim bladder is a flexible gas- filled sac; the fish can increase or decrease the amount of gas in the swim bladder so that it stays neutrally buoyant-neither sinking nor floating. Suppose the fi sh is neutra lly buoyant at some depth and then goes deeper. What needs to happen to the volume of air in the swim bladder? Will the fish need to add or remove gas from the swim bladder to maintain its neu tral buoyancy? Figure Q13.23 shows two identi cal beakers filled to the same he ight with water. Bcaker B has a plastic sphere floatin g in it. Which beaker, with all its con-

tj- tents, weighs more? Or are they FIGURE 013 .23

equal? Explain.

8

A tub of water, filled to the brim, sits on a scale. Then a floating block of wood is placed in the tub, pushing some water over the rim. The water that overflows immediately runs off the scale. What happens to the read ing of the scale? Ships A and B have the same height and the same mass. Their cross-section profiles are shown in Figure QI3.25. Does one ship ride higher in the water (more height above the water line) than the other? IJ so, which one? Explain. Gas flows through a pipe, as shown in Figure Q 13.26. The pipe's constant outer diame ter is shown; you can't see into the pipe to know how the inner diameter changes.

uv A 8

FIGURE 013 .25

FIGURE 013.26

Rank in order, from largest to smallest, the gas speeds 1'1 to 1'3 at points 1, 2, and 3. Explain. Liquid flows through a pipe as shown in Figure QI3.27. The pipe's constant outer diameter is shown; you can' t see into the pipe to know how the inner diameter changes.

• _ Liquid inside _

FIGURE 013.27

Rank in order, from largest to smallest, the flow speeds VI to v 3 at poinlS 1, 2, and 3. Explain. A liquid with neg ligible viscosity flows through the pipe shown in Figure QI3.28. This is an overhead view. a. Rank in order, from largest to smal lest, the flow speeds V I to

1'4 at points I to 4. Explain. b. Rank in order, from largest to smallest, the pressures PI to v.

at points I to 4. Explain .

• · 2

Questions 433

29. Wind blows over the house shown in Figure QI3.29. A window on the ground floor is open. Is there an air flow through the house? If so, does the air flow in the window and out the chim ney, or in the chimney and out the window? Explain.

30. Two pipes have the same inner cross-sect ion area. One has a circular cross sect ion and the other has a rectangular cross sec tion with its height one-tenth its width. Through which pipe, if either, would it be easier to pump a viscous liquid? Explain.

Multiple-Choice Questions

31. I Figure Q13.31 shows a 100 g block of copper (p ~ 8900 kg/m3) and a 100 g block of aluminum (p = 2700 kg/m3) connected by a massless string that runs over two massless, frictionless pul- leys. The two blocks exactl y Copper

balance, since they have the FIGURE 013.31 same mass. Now suppose that

Aluminum

the whole system is submerged in water. What will happen? A. The copper block will fall , the aluminum block will rise. B. The aluminum block will fall, the copper block wiU rise.

32.

C. Nothing wiU change. D. Both blocks will rise. Mass A

~'=======dI~ I Ma~ses A and B rest on very light pistons that enclose a fluid , as shown in Figure QI3.32. There is no friction between the pistons and the

Area = I.Om2

cylinders they fit inside. FIGURE 013.32

Which of the following is true? A. Mass A is greater. B. Mass B is greater. C. Mass A and mass B are the same.

Area = 2.0 m~

33. I If yo u di ve underwater, you not ice an uncomfortable pres BID sure on your eardrums due to the increased pressure. The

human eardrum has an area of about 70 mm2 (7 X 10 5 m2) , and it can sustain a force of about 7 N without rupturing . .If your body had no means of balancing the extra pressure (which, in reality, it does), what would be the max imum depth you could dive without rupturing your eardrum? A. 0.3 m B. 1 111 C. 3 m D. 10m

34. II An 8.0 Ib bowling ball has a diameter of 8.5 inches. When lowered into waler, this ball will A. Float. B. Sink. C. Have neutral buoyancy.

35. I A basketball has a mass of 0.50 kg and a volume of 8.0 X 10- 3 111 3• What is the magnitude of the net force on a bas ketball when it is fully submerged in water? A. 4.9 N B. 74 N C. 78 N D. 83 N

36. I An object floats in water, with 75% of its volume submerged. What is it s approx imate density? A. 250 kg/m3 B. 750 kg/m3

C. 1000 kg/m' D. 1250 kg/m3 37. I A syringe is being used to

sq uirt water as shown in Figure QI3.37. The water is ejected from the nozzle at 10 mls. At what speed is the plunger of the sy ringe being depressed?

Radius = 1 em

.f FIGURE 013.37

\

A. 0.01 m/s B. 0.1 mls C. I mls D. 10 m/s

434 CHAPTER 13 Fluids

38. II Water n ows through a 4.0-cm-diamete r horizo ntal pipe at a speed of 1.3 m/s. The pi pe then narrows down to a diameter of 2.0 em. Ignori ng viscos ity, what is the pressure differe nce between the wide and narrow sections of the pipe? A. 850 Pa B. 3400 Pa C. 9300 Pa D. 12,700 Pa E. \3,500 Pa

PROBLEMS

Section 13.1 Fluids and Density

1. II A 100 mL beaker ho lds 120 g of liqu id . What is the liqu id's density in 5 1 uni ts?

2. 1 Conta iners A and B have equal vo lumes. Container A ho lds helium gas at 1.0 aIm pressure and 20°C. Container B is com pletely fill ed with a liquid whose mass is 7600 times the mass of he lium gas in container A. Identify the li quid in B.

3. II A ir e nclosed in a sphere has dens ity p = 1.4 kg/m3. What will the dens ity be if the rad ius o f the sphere is ha lved, com press ing the air within?

4. II Ai r e nclosed in a cyl inder has dens ity p = 1.4 kg/m3 .

a. What will be the de nsity of the air if the length of the cy li n der is doubled whi le the radius is unchanged?

b. What will be the dens ity of the air if the rad ius of the cy li n der is halved whi le the length is unchanged?

5. II a. 50 g of gasoli ne are mi xed with 50 g of water. W hat is the average de nsity of the mi xture?

b. 50 cm) of gasoli ne are mi xed with 50 cm) of water. What is the average dens ity o f the mi xture?

6. III Eth y l alco hol has been added to 200 mL of water in a con taine r that has a mass o f 150 g whe n e mpty. The resulti ng con tainer and liquid mi xtu re has a mass o f 5 12 g. What vo lume of alcohol was added to the water?

7. II The average density of the body of a fi sh is 1080 kg/m3. To keep BID from s inking, the fi sh increases its volume by in flati ng an internal

air bladder, known as a swim bladder, with air. By what percen t must the fi sh increase its volume 10 be neutrall y buoyant in fresh water? Use the Table 13. 1 value fo r the density of air at 20°C.

Section 13.2 Pressure

8. II The deepest po int in the ocean is II km be low sea level, deeper than Mt. Everest is tall. What is the pressure in atmos pheres at thi s depth?

9. II a. What vol ume o f water has the same mass as 8.0 m3 o f eth yl alcohol?

b. If thi s volume of water completely fi lls a cubic tank. what is the pressure at the bottom?

10. III A 1.0-m-di ameter vat of liquid is 2.0 m deep. The pressure at the bOllom of the vat is 1.3 atm. What is the mass o f the liquid in the vat?

II . III A 35-cm-ta ll , 5.0-cm-di ameter cy lindrical beaker is fill ed to its brim with water. What is the downward force of the water on the bottom o f the beaker?

l 2. II The gauge pressure at the bottom of a cyl inder of liquid is PI> = OAO atm. The liq uid is poured into ano ther cyl inde r with twice the radius of the fi rst cy li nder. What is the gauge pressure at the botto m of the second cyli nde r?

13. 11111 A researc h submari ne has a 20-cm-diameter window 8.0 cm thick. The manufac turer says the window can withstand forces up

39. II A IS-m-long garde n hose has an inner d iameter of 2.5 em. One end is connec ted to a spigot; 200C waler fl ows from the other end at a rate of 1.2 Us. What is the gauge press ure at the spigot end of the hose? A. 1900 Pa B. 2700 Pa C. 4200 Pa D. 5800 Pa E. 7300 Pa

to 1.0 X 106 N. What is the submarine's max imum safe depth in seawate r? The pressure ins ide the submari ne is ma intained at I .Oatm.

14. 111 11 The highest that George can Sli ck water up a very long straw BID is 2.0 m. (This is a typical value.) What is the lowest pressu re

that he can maintai n in hi s mo uth? IS. III The two 60-cm-di ame te r cy li nders in

Fig ure P I3 .1 5, closed at one end, open at the other, are joined to form a s ingle cy linder, then the air ins ide is re moved. a. How much force does the atmosphere exert

on the fl at end of each cyl inder? b. S uppose the upper cyl inder is bo lted to a

sturdy ceiling. How many 100 kg footba ll players wo uld need to hang from the lower cy linder to pull the two cyli nders apart?

FIGURE P13.15

Section 13.3 Measuring and Using Pressure

16. 1111 What is the gas press ure ins ide the box shown in Fi gure PI 3.16'

P~ .. y(cm)

FIGURE P13.16 FIGURE P13.17

17. II The conta ine r shown in Figure P1 3 .1 7 is fi lled with o il. It is opcn 10 the atmosphere on the left. a. What is the pressure at po int A? b. What is the pressure di fference between po ints A and B?

Between po ints A and C? 18. III Glyceri n is po ured into an ope n U-s haped tube until the

height in both sides is 20 cm. Ethyl alcoho l is then poured into one ann until the he ight o f the alcoho l co lumn is 20 Clll. The two liquids do not mix. What is the d iffe rence in hcight between the top surface of the glyce ri n and the to p surface o f the alco ho l?

19 . II I A V-shaped tube. open to the air o n both ends. contains mer cury. Water is poured in to the le ft arm until the water co lumn is 10.0 cm deep. How far upward from its initi al positio n does the mercury in the right arm ri se?

20. I What is the height of a water barometer at atmospheric pressure?

21. III Postural hypotension is the occurrence of low (systolic) BID blood pressure when standing up too quickly from a reclined

position, causing faintin g or lightheadedness. For most people, a systolic pressure less than 90 mm Hg is cons idered low. If the blood pressure in your brain is 120 mm when you are lying down, what would it be when you stand up? Assume that your brain is 40 cm from your heart and that P = 1060 kg/m3 for your blood. No te: NonnaJly, your blood vessels constrict and expand to keep your brain blood pressure stable when you change your posture.

Section 13.4 Buoyancy

22. II A 6.00-cm-diameter sphere with a mass of 89.3 g is neu trally buoyant in a liquid. Identify the liquid.

23. III A cargo barge is loaded in a saltwater harbor for a trip up a freshwater river. Tfthe rectangular barge is 3.0 m by'20.0 m and sits 0.80 m deep in the harbor, how deep wiJi it sit in the river?

24. II A 10 em X 10 cm X 10 em wood block with a density of 700 kg/mJ floats in water. a. What is the distance from the top of the block to the water

if the water is fresh? b. If it 's seawater?

25. 11 What is the tension in the string in Figure PI 3.25?

FIGURE P13 .25

l00cm3 of I~~ --1.-1V -11 aluminum, density ~ P Al = 2700 kg/m3

Ethyl alcohol

26. 1 A 10 em X 10 em X 10 em block of steel (Pslcd=

7900 kg/m3) is suspended from a spring scale. The scale is in newtons. a. What is the scale reading if the block is in air? b. What is the scale reading after the block has been lowered

into a beaker of oil and is completely submerged? 27. 111 Styrofoam has a density of 300 kg/m3. What is the maximum

mass that can hang without sinking from a 50-cm-diameter Sty rofoam sphere in water? Assume the volume of the mass is neg ligible compared to that of the sphere.

28. 1111 Calculate the buoyant force due to the surrounding air on a man weighing 800 N. Assume hi s average density is the same as that of water.

Section 13.5 Fluids in Motion

29. III Ri ver Pascal with a volume flow rate of 5.0 X 105 Us joins with Ri ve r Archimedes, which carries 10.0 X 105 Us. to form the Bernoulli River. The Bernoulli Ri ver is ISO m wide and 10 m deep. What is the speed of the water in the Bernoulli River?

30. JI Water flowing through a 2.0-cm-diameter pipe can fill a 300 L bathtub in 5.0 min. What is the speed of the water in the pipe?

3 1. 1111 A pump is used to empty a 6000 L wading pool. The water exits the 2.5-em-diameter hose at a speed of2.l m/s. How long will it take to empty the pool ?

Problems 435

32. II A I.O-em-diameter pipe widens to 2.0 cm. then narrows to 0.50 cm. Liquid flows through the first segment at a speed of 4.0 m/s. a. What are the speeds in the second and third segments? h. What is the volume fl ow rate through the pipe?

Section 13.6 Fluid Dynamics

33. II What does the top pressure gauge in Figure P1 3.33 read?

1.0em View from above

20m~ A O ~ ~~ 3.0em

FIGURE P13.33 FIGURE P13.34

34. 11111 The 3.0-cm-diameter water line in Figure P13 .34 splits into two I.O-cm-diameter pipes. All pipes me circular and at the same e levation . At point A, the water speed is 2.0 m/s and the gauge pressure is 50 kPa. What is the gauge pressure at point B?

35. III A rectangular trough, 2.0 m long, 0.60 m wide, and 0.45 m deep, is completely full of water. One end of the trough has a small drain plug right at the bottom edge. When you pull the plug, at what speed does water emerge from the hole?

Seclion 13.7 Viscosity and Poiseuille's Equation

36. 11111 What pressure difference is required between the ends of a 2.0-m-long, I.O-mm-diameter hori zontal tube for 40°C water to flow through it at an average speed of 4.0 m/s?

37. 11111 Water fl ows at 0.25 Us through a 10-m-Iong garden hose 2.5 em in diameter that is lying flat on the ground. The tempera ture of the water is 20°e. What is the gauge pressure of the water where it enters the hose?

38. II Figure P1 3.38 shows a water-filled syr inge with a 4.0-em-long needle. What is the gauge pressure of the water at the point P, where the needle meets the wider chamber of the syringe?

General Problems

FIGURE P13.38

39. III The density of go ld is 19,300 kg/m3. 197 g of go ld is shaped into a cube. What is the length of each edge?

40. II The density of copper is 8920 kg/mJ . How many moles are in

INT a 2.0 em X 2.0 em X 2.0 cm cube of copper? 4 1. III The density of aluminum is 2700 kg/m3. How many atoms INT are in a 2.0 em X 2.0 em X 2.0 em cube of al uminum? 42. II A 50-cm-thick layer of oil floats on a 120-cm-thick layer of

waler. What is the pressure at the bottom of the water layer? 43. 11111 An oil layer floats on 85 cm of water in a tank. The absolute

pressure at the bottom of the tank is 1 J 2.0 kPa. How thick is the oil?

44. II The little Dutch boy saved Holland by sticking hi s fin ger in the leaking dike. If the water level was 2.5 m above hi s fin ger, estimate the force of the water on hi s finger.

436 CHAPTER 13 Fluids

45. II a. In Figure PI3.45, how much force does the fluid exert on the end of the cyl inder at A?

b. How much force does the fluid exert on the end of the cyl inder at B?

10 kg tloating piston 4.0cm

IOOcm

60 em

FIGURE P13.45

46. III A friend asks you how much pressure is in your car tires. You know that the tire man ufacturer recommends 30 psi, but it 's been a while since you've checked. YOLI can't find a tire gauge in the car, but you do find the owner's manual and a ruler. For tunatel y, yo u've just finished taking physics, so YOLI tell yo ur friend, " \ don't know, but I can figure it out." From the owner's manual you find that the car's mass is 1500 kg. It seems reason able to assume that each lire supports one-fourth o f the weight. With the ruler you find that the tires are 15 em wide and the flat tened segment of the tire in contact with the road is 13 cm long. What answer will you give your friend ?

47. III A diver 50 m deep in 10°C fresh water ex hales a 1.0-cm INT di ameter bubble. What is the bubble's diameter just as it reaches

the surface of the lake. where the water temperature is 20°C? Hint: Assume that the air bubble is always in thermal equilib rium with the surrounding water.

48. II A 6.0-em-tall cy linder floats in water with its ax is perpendic ular to the surface. The length of the cylinde r above water is 2.0 cm. What is the cy linder 's mass density?

49. II A sphere completely submerged in water is tethered to the bottom with a string. The tension in the string is one-third the weight of the sphere. What is the density of the sphere?

50. II You need to determine the density of a ceramic statue. If you suspend it from a spring scale, the scale reads 28.4 N. If you then lower the statue into a tub of water so that it is completely submerged, the scale reads 17.0 N. What is the dens ity?

51. II A 5.0 kg rock whose density is 4800 kg/m3 is suspended by a string suc h that half of the rock' s volume is under water. What is the tension in the string?

52. II A flat slab of styrofoam, with a density of 32 kglm\ float s on a lake. What minimum vol ume must the s lab ha ve so thai a 40 kg boy can sil on the slab without it s inking?

53. 1111 A 2.0 mL syringe has an inner diameter of 6.0 mm, a needle BIO inner diameter of 0.25 mm, and a plunger pad diameter (where

you place your finger) of 1.2 em. A nurse uses the syringe to inject medicine into a patien t whose blood pressure is 140/ 100.

Assume the liquid is an ideal fluid. a. What is the minimum force the nurse needs to app ly to the

sy ringe? b. The nurse empties the syri nge in 2.0 s. What is the flow

speed of the medicine through the needle? 54. III A child 's water pisto l shoots water through a I.O-mm-diame INT ter hole . If the pistol is fired hori zontally 70 e m above the

f,'l'ound, a squ irt hits the ground 1.2 m away. What is the vo lume flow rate during the squirt? Ignore air res istance.

55. II The leaves of a tree lose wate r to the atmosphere v ia the BIO process of transpiration. A particular tree loses water at the rate

of 3 X 10 8 m3/s; thi s water is replenished by the upward flow of sap through vesse ls in the trunk. Thi s tree 's trunk conta ins about 2000 vessels, each 100,um in diameter. What is the speed of the sap flowing in the vessels?

56. II A hurri cane wind blows across a 6.00 m X 15.0 m flat roof at a speed of 130 km/h. a. Is the air pressure above the roof higher or lower than the

pressure ins ide the hOLl se? Explain. b. What is the pressure difference? c . How much force is exerted on the roof? If the roof cannot

withstand thi s much force, will it "blow in" or "blow out"? 57. 11111 Water flows from the pipe shown in Figure P13.57 with a

speed of 4.0 m/s. a. What is the water pressure as it ex its into the air? b. What is the height II of the standing co lumn of water?

FIGURE P13 .57

ID em-

58. 1111 Air flows through the tube shown in Figure PI 3.58. Assume that air is an ideal fluid. a. What are the air speeds VI and V2 at points I and 2? b. What is the volume fl ow rate?

4.0mm

FtGURE P13.58 FIGURE P13.59

59. III Air flows through the tube shown in Figure P13.59 at a rate INT of 1200 cm 3/s. Assume that air is an ideal fluid . What is the

height" of mercury in the ri ght s ide of the U-tube? 60. III Water flows at 5 .0 U s through a hori zontal pipe that narrows

smoothly from 10 cm diameter to 5.0 cm diameter. A pressure gauge in the narrow section reads 50 kPa. What is the reading of

61. INT

a pressure gauge in the wide section? III The mercury manometer shown in Figure P1 3.6 1 is att- ached to a gas cel l. The mer- cury height II is 120 mm when the ce ll is placed in an ice- water mi xture. The mercury height drops to 30 mm when

Gas cell

the device is carried into an industrial freezer. What is the FIGURE P13 .61 freezer temperature? Hint: The right tube of the manometer is much narrower than the le ft tube. What reasonable assumption can YOll make about the gas volu me?

62. 11111 Figure P1 3.62 shows a sect ion of a long tube that narrows near its open end to a diameter of 1.0 mm. Water at 200 e flows out of the open end at 0.020 Us. What is the

FIGURE P13.62

gauge pressure at point P, where the diameter is 4.0 mm? 63. II Smoking tobacco is bad for your c irculatory health. In an BIO attempt to maintain the blood's capacity to deliver oxygen, the

body increases its red blood cell production, and thi s increases the viscosity of the blood. In addition, nicotine from tobacco causes arteries to constrict.

For a nonsmoker, with blood viscos ity of 2.S X 10-3 Pa· s, normal blood flow requires a pressure difference of 8.0 mm Hg between the two ends of an artery. If thi s person were to smoke regularly, hi s blood viscos ity would increase to 2.7 X 10- 3 Pa· s, and the arterial diameter would constrict to 90% of it s normal value. What pressure difference would be needed to maintain the same blood flow?

64. III A stitf, IO-cm-Iong tube with an inner diameter of 3.0 mm is auached to a small hole in the side of a tall beaker. The tube sticks out horizon tally. The beaker is fill ed with 200e water to a level 45 cm above the hole, and it is continually topped off to maintain thai level. What is the vo lume flow rate through the tube?

Passage Problems

Blood Pressure and Blood Flow BID

The blood pressure at your heart is approximately 100 mm Hg. As blood is pumped from the left ventri cle of your heart, it flows through the aorta, a single large blood vessel with a diameter of about 2.5 cm. The speed of blood flow in the aorta is about 60 cm/s. Any change in pressure as blood flows in the aorta is due to the change in height: the

Stop to Think 13.1: PI = pz = P3. Density depends only on what the object is made of, not how big the pieces are.

Stop to Think 13.2: C. These are all open tubes, so the liquid ri ses to the same height in all three despite their different shapes.

Stop to Think 13.3: B. The weight of the displaced water equals the weight of the ice cube. When the ice cube melts and turns into water, that amount of water will exactl y filJ the volume that the ice cube is now displacing.

Stop to Think 13.4: 1 cmJjs out. The fluid is incompressible, so the sum of what flow s in must match the sum of what flow s out. 13 cm3/s is known to be flowing in while 12 cm3/s flows out. An additional I c1113/s must now out to achieve balance.

Problems 437

vesse l is large enough that viscoLi s drag is not a major factor. As the blood moves through the circulatory system, it flows into succes sive ly smaller and smaller blood vessels until it reaches the capillaries. Blood flows in the capillari es at the much lower speed of approxi mately 0.7 mm/s. The diameter of capillaries and other small blood vessels is so small that viscous drag is a major factor.

65. I There is a limit to how long yo ur neck can be. If your neck were too long, no blood would reach your brain! What is the maximum height a person ' s brain could be above his heart , given the noted pressure and assuming that there are no valves or supplementary pumping mechanisms in the neck? The den sity of blood is 1060 kg/m). A. 0.97 m B. 1.3 m C. 9.7 m D. 13 III

66. I Because the flow speed in your capillaries is much less than in the aorta, the total cross-section area of the capillaries con sidered together must be much larger than that of the aorta. Given the flow speeds noted , the total area of the capillaries considered together is equi valent to the cross-sec tion area of a si ngle vessel of approximately what diameter? A. 2Scm B. SOcm C. 7Scm D. 100 cm

67. I Suppose that in response to some stimulus a small blood ves sel narrows to 90% its original diameter. [f there is no change in the pressure ac ross the vessel, what is the ratio ofthe new vol ume flow rate to the original now rate? A. 0.66 B. 0.73 C. 0.81 D. 0.90

68. I Sustained exerc ise can increase the blood flow rate of the heart by a factor of 5 with only a modest increase in blood pres sure. This is a large change in fl ow. Although several factors come into play, which of the follo wing physiological changes would most plausibly account for such a large increase in fl ow with a small change in pressure? A. A decrease in the viscos ity of the blood B. Dilation of the smaller blood vessels to larger diameters C. Dilation of the aorta to larger diameter D. An increase in the oxygen carried by the blood

Stop to Think 13.5: 112 > h4 > IlJ > "l' The liquid level is higher where the pressure is lower. The pressure is lower where the flow speed is higher. The flow speed is highest in the narrowest tube, zero in the open air.

GENERAL PRINCIPLES

Density p

Pressure mainl y due to particle colli s ions with walls

liquids

Ci ,

p

Hydrostati c pressure at depth d is p = Po + pgd

The pressure is the same at all points o n a hori zontal line through a liquid (of one kind) in hydrostatic equilibrium

Poiseuille's equation governs viscous flow through a tube:

IMPORTANT CONCEPTS

Density p = ",IV, where m is mass and V is vo lume.

Pressure P = FIA, where F is force magnitude and A is the area on wh ic h the force acts.

Pressure ex ists at all points in a fluid.

Pressure pushes equaJ ly in all d irect ions.

Gauge pressure Pg = P - I atm.

Viscosity 'Y1 is the property of a fluid thai makes it resist fl owing.

APPLICATIONS

Buoyancy is the upward force of a fluid on an object immersed in the flu id.

Archimedes' principle: The magni tude of the buoyant force equals the weigh t of the fluid displaced by the object.

Sink: Payg > P r FB < we)

L~j Float: P ayg < Pr FB > we)

P '''. Neutrally _ FB = Wo buoyant: P ayg - Pf

Representing fluid flow

The velociry of a fluid panicle i.~

,,,"g"t to,,,~/ "''''mlm'~

Barometers measure atmospheric pressure. Atmospheric pressure is related to the he ight of the liquid colum n by PalnJOS = pgll.

~! , ..... . vg

! A

Fluid elements contain a fi xed vol ume of flui d. Their shape may change as they move.

Every flu id panicle thai make~ up the element moves on il.s Own strc:lmlinc.

Manometers measure pressure . The pressure at the closed e nd of the nlbe is p = I atm + pgll.

p

432 CHAPTER 13 Fluids

!:!.!/ www.masteringphysics.com

Problem difficulty is labeled as I (straightforward) to 11111 (challenging).

QUESTIONS

Conceptual Questions

t. Which has the greater density, 1 g of mercury or 1000 g of water?

2. A 1 X 10- 3 111

3 chunk of material has a mass of 3 kg. a. What is the material's density? b. Would a 2 X 10- 3 m} chunk of the same material have the

same mass? Explain. c. Would a 2 X 10-] mJ chunk of the same material have the

same density? Explain. 3. You are give n an irregularly shaped chunk of material and

asked to find its density. List the specific steps that you would follow to do so.

4. Object 1 has an irregular shape . Its density is 4000 kg/mJ. a. Object 2 has the same shape and dimensions as object I ,

but it is twice as mass ive. What is the density of object 2? b. Object 3 has the same mass and the same shape as object I,

but its size in all three dimensions is twice that of object I. What is the density of object 3?

5. When yo u get a blood transfusion the bag of blood is held BIO above your body, but when you donate blood the collection bag

is held below. Why is thi s? 6. To explore the bottom of a 10-m-deep lake, your friend Tom

BIO proposes to get a long garden hose, put one end on land and the other in hi s mouth fo r breathing underwater, and descend into the depths. Susan, who overhears the conversation, reacts with horror and warns Tom that he will not be able to inhale when he is at the lake bottom. Why is Susan so worried?

7. Rank in order, from largest to smaUest, the pressures at A, B, and C in Figure QI3.7. Explain.

8. Refer to Figure Q13.7. Rank in order, from largest to smallest, the pressures at 0, E, and F. Explain.

9. Cylinders A and B contain liq- uids. The pressure PA at the bot tom of A is higher than the pressure PB althe bottom of B. Is

E o

~ II

c

the ratio PA Ips of the absolute pressures larger, smaller, or equallo the rati o of the gauge pressures? Explain.

10. In FigureQ I3. 10, Aand B arerec tangular tanks full of water. They have equal he ights and equal side lengths (the dimension into the page), but different widths. a. Compare the forces the water

DOl A Sides 8

FIGURE 013.10

exerts on the bottoms o f the tanks. Is FA larger. smaller, or equal to F B ? Explain.

b. Compare the forces the water exerts on the s ides of the tanks. Is FA larger, smaUer, or equal to FB? Explain.

Problems labeled INi integrate significant material from earlier

chapters; BlO are of biological or medical interest.

II. Heliu m-filled weather balloons are spherical whe n they reach very high altitudes. However, they are on ly partial.ly inflated with helium before they are released. Explain why thi s is done.

12. Water expands when heated. Suppose a beaker of water is heated from 10°C to 90°C. Does the pressure at the bottom of the beaker inc rease, decrease, or Slay the same? Explain.

13. In Figure QI3.13, is VA larger, ~

smaUer, or equal to PB? Explain. . . . .... 2m 14. A beaker o f water rests on a

scale. A metal ball is then low ered into the beaker using a string tied to the ball. The baJl doesn't touch the sides or bot- tom of the beaker, and no water sp iUs from the beaker. Does the scale reading decrease, increase, or stay the same? Ex plain.

o A

FIGURE 013.13

. o . B

15. Rank in order, from largest to smallest, the densities of objects A, B, and C in Figure QI3. IS. Explain.

8'--JB c

FIGURE 013 .15 FIGURE 013.16

16. Objects A, B, and C in Figure Q 13. 16 have the same volume. Rank in order, from largest to smallest, the s izes of the buoyant forces FA' FB, and Fe on A, B, and C. Explain.

17. Refer to Figure QI3. 16. Now A, B, and C have the same den s ity, but st ill have the masses g iven in the figure. Rank in order, from largest to smallest, the sizes of the buoyant forces on A, B, and C. Ex plain.

18. When you stand on a bathroom scale, it reads 700 N. Suppose a g iant vac uum cleaner Slicks half the air out of the room, reduc ing the pressure to 0.5 atm. Would the scale read ing increase, decrease. or stay the same? Explain.

19. Suppose you stand on a bathroom scale that is on the botto m of a swimming pool. The water comes up to your waist. Does the scale read more, less, or the same as your true weight? Ex plain.

20. When you place an egg in water, it s inks. If you add sa lt to the water, after some time the egg noats. Ex plain.

21.

22.

BID

23.

24.

25.

26.

27.

28.

Submerged submarines contain tanks filled with water. To ri se to the surface, compressed air is used to force the water out of the tanks. Explain why thi s works. Fish can adjust their buoyancy with an organ cal led the swim biaddeJ: The swim bladder is a flexible gas- filled sac; the fish can increase or decrease the amount of gas in the swim bladder so that it stays neutrally buoyant-neither sinking nor floating. Suppose the fi sh is neutra lly buoyant at some depth and then goes deeper. What needs to happen to the volume of air in the swim bladder? Will the fish need to add or remove gas from the swim bladder to maintain its neu tral buoyancy? Figure Q13.23 shows two identi cal beakers filled to the same he ight with water. Bcaker B has a plastic sphere floatin g in it. Which beaker, with all its con-

tj- tents, weighs more? Or are they FIGURE 013 .23

equal? Explain.

8

A tub of water, filled to the brim, sits on a scale. Then a floating block of wood is placed in the tub, pushing some water over the rim. The water that overflows immediately runs off the scale. What happens to the read ing of the scale? Ships A and B have the same height and the same mass. Their cross-section profiles are shown in Figure QI3.25. Does one ship ride higher in the water (more height above the water line) than the other? IJ so, which one? Explain. Gas flows through a pipe, as shown in Figure Q 13.26. The pipe's constant outer diame ter is shown; you can't see into the pipe to know how the inner diameter changes.

uv A 8

FIGURE 013 .25

FIGURE 013.26

Rank in order, from largest to smallest, the gas speeds 1'1 to 1'3 at points 1, 2, and 3. Explain. Liquid flows through a pipe as shown in Figure QI3.27. The pipe's constant outer diameter is shown; you can' t see into the pipe to know how the inner diameter changes.

• _ Liquid inside _

FIGURE 013.27

Rank in order, from largest to smallest, the flow speeds VI to v 3 at poinlS 1, 2, and 3. Explain. A liquid with neg ligible viscosity flows through the pipe shown in Figure QI3.28. This is an overhead view. a. Rank in order, from largest to smal lest, the flow speeds V I to

1'4 at points I to 4. Explain. b. Rank in order, from largest to smallest, the pressures PI to v.

at points I to 4. Explain .

• · 2

Questions 433

29. Wind blows over the house shown in Figure QI3.29. A window on the ground floor is open. Is there an air flow through the house? If so, does the air flow in the window and out the chim ney, or in the chimney and out the window? Explain.

30. Two pipes have the same inner cross-sect ion area. One has a circular cross sect ion and the other has a rectangular cross sec tion with its height one-tenth its width. Through which pipe, if either, would it be easier to pump a viscous liquid? Explain.

Multiple-Choice Questions

31. I Figure Q13.31 shows a 100 g block of copper (p ~ 8900 kg/m3) and a 100 g block of aluminum (p = 2700 kg/m3) connected by a massless string that runs over two massless, frictionless pul- leys. The two blocks exactl y Copper

balance, since they have the FIGURE 013.31 same mass. Now suppose that

Aluminum

the whole system is submerged in water. What will happen? A. The copper block will fall , the aluminum block will rise. B. The aluminum block will fall, the copper block wiU rise.

32.

C. Nothing wiU change. D. Both blocks will rise. Mass A

~'=======dI~ I Ma~ses A and B rest on very light pistons that enclose a fluid , as shown in Figure QI3.32. There is no friction between the pistons and the

Area = I.Om2

cylinders they fit inside. FIGURE 013.32

Which of the following is true? A. Mass A is greater. B. Mass B is greater. C. Mass A and mass B are the same.

Area = 2.0 m~

33. I If yo u di ve underwater, you not ice an uncomfortable pres BID sure on your eardrums due to the increased pressure. The

human eardrum has an area of about 70 mm2 (7 X 10 5 m2) , and it can sustain a force of about 7 N without rupturing . .If your body had no means of balancing the extra pressure (which, in reality, it does), what would be the max imum depth you could dive without rupturing your eardrum? A. 0.3 m B. 1 111 C. 3 m D. 10m

34. II An 8.0 Ib bowling ball has a diameter of 8.5 inches. When lowered into waler, this ball will A. Float. B. Sink. C. Have neutral buoyancy.

35. I A basketball has a mass of 0.50 kg and a volume of 8.0 X 10- 3 111 3• What is the magnitude of the net force on a bas ketball when it is fully submerged in water? A. 4.9 N B. 74 N C. 78 N D. 83 N

36. I An object floats in water, with 75% of its volume submerged. What is it s approx imate density? A. 250 kg/m3 B. 750 kg/m3

C. 1000 kg/m' D. 1250 kg/m3 37. I A syringe is being used to

sq uirt water as shown in Figure QI3.37. The water is ejected from the nozzle at 10 mls. At what speed is the plunger of the sy ringe being depressed?

Radius = 1 em

.f FIGURE 013.37

\

A. 0.01 m/s B. 0.1 mls C. I mls D. 10 m/s

434 CHAPTER 13 Fluids

38. II Water n ows through a 4.0-cm-diamete r horizo ntal pipe at a speed of 1.3 m/s. The pi pe then narrows down to a diameter of 2.0 em. Ignori ng viscos ity, what is the pressure differe nce between the wide and narrow sections of the pipe? A. 850 Pa B. 3400 Pa C. 9300 Pa D. 12,700 Pa E. \3,500 Pa

PROBLEMS

Section 13.1 Fluids and Density

1. II A 100 mL beaker ho lds 120 g of liqu id . What is the liqu id's density in 5 1 uni ts?

2. 1 Conta iners A and B have equal vo lumes. Container A ho lds helium gas at 1.0 aIm pressure and 20°C. Container B is com pletely fill ed with a liquid whose mass is 7600 times the mass of he lium gas in container A. Identify the li quid in B.

3. II A ir e nclosed in a sphere has dens ity p = 1.4 kg/m3. What will the dens ity be if the rad ius o f the sphere is ha lved, com press ing the air within?

4. II Ai r e nclosed in a cyl inder has dens ity p = 1.4 kg/m3 .

a. What will be the de nsity of the air if the length of the cy li n der is doubled whi le the radius is unchanged?

b. What will be the dens ity of the air if the rad ius of the cy li n der is halved whi le the length is unchanged?

5. II a. 50 g of gasoli ne are mi xed with 50 g of water. W hat is the average de nsity of the mi xture?

b. 50 cm) of gasoli ne are mi xed with 50 cm) of water. What is the average dens ity o f the mi xture?

6. III Eth y l alco hol has been added to 200 mL of water in a con taine r that has a mass o f 150 g whe n e mpty. The resulti ng con tainer and liquid mi xtu re has a mass o f 5 12 g. What vo lume of alcohol was added to the water?

7. II The average density of the body of a fi sh is 1080 kg/m3. To keep BID from s inking, the fi sh increases its volume by in flati ng an internal

air bladder, known as a swim bladder, with air. By what percen t must the fi sh increase its volume 10 be neutrall y buoyant in fresh water? Use the Table 13. 1 value fo r the density of air at 20°C.

Section 13.2 Pressure

8. II The deepest po int in the ocean is II km be low sea level, deeper than Mt. Everest is tall. What is the pressure in atmos pheres at thi s depth?

9. II a. What vol ume o f water has the same mass as 8.0 m3 o f eth yl alcohol?

b. If thi s volume of water completely fi lls a cubic tank. what is the pressure at the bottom?

10. III A 1.0-m-di ameter vat of liquid is 2.0 m deep. The pressure at the bOllom of the vat is 1.3 atm. What is the mass o f the liquid in the vat?

II . III A 35-cm-ta ll , 5.0-cm-di ameter cy lindrical beaker is fill ed to its brim with water. What is the downward force of the water on the bottom o f the beaker?

l 2. II The gauge pressure at the bottom of a cyl inder of liquid is PI> = OAO atm. The liq uid is poured into ano ther cyl inde r with twice the radius of the fi rst cy li nder. What is the gauge pressure at the botto m of the second cyli nde r?

13. 11111 A researc h submari ne has a 20-cm-diameter window 8.0 cm thick. The manufac turer says the window can withstand forces up

39. II A IS-m-long garde n hose has an inner d iameter of 2.5 em. One end is connec ted to a spigot; 200C waler fl ows from the other end at a rate of 1.2 Us. What is the gauge press ure at the spigot end of the hose? A. 1900 Pa B. 2700 Pa C. 4200 Pa D. 5800 Pa E. 7300 Pa

to 1.0 X 106 N. What is the submarine's max imum safe depth in seawate r? The pressure ins ide the submari ne is ma intained at I .Oatm.

14. 111 11 The highest that George can Sli ck water up a very long straw BID is 2.0 m. (This is a typical value.) What is the lowest pressu re

that he can maintai n in hi s mo uth? IS. III The two 60-cm-di ame te r cy li nders in

Fig ure P I3 .1 5, closed at one end, open at the other, are joined to form a s ingle cy linder, then the air ins ide is re moved. a. How much force does the atmosphere exert

on the fl at end of each cyl inder? b. S uppose the upper cyl inder is bo lted to a

sturdy ceiling. How many 100 kg footba ll players wo uld need to hang from the lower cy linder to pull the two cyli nders apart?

FIGURE P13.15

Section 13.3 Measuring and Using Pressure

16. 1111 What is the gas press ure ins ide the box shown in Fi gure PI 3.16'

P~ .. y(cm)

FIGURE P13.16 FIGURE P13.17

17. II The conta ine r shown in Figure P1 3 .1 7 is fi lled with o il. It is opcn 10 the atmosphere on the left. a. What is the pressure at po int A? b. What is the pressure di fference between po ints A and B?

Between po ints A and C? 18. III Glyceri n is po ured into an ope n U-s haped tube until the

height in both sides is 20 cm. Ethyl alcoho l is then poured into one ann until the he ight o f the alcoho l co lumn is 20 Clll. The two liquids do not mix. What is the d iffe rence in hcight between the top surface of the glyce ri n and the to p surface o f the alco ho l?

19 . II I A V-shaped tube. open to the air o n both ends. contains mer cury. Water is poured in to the le ft arm until the water co lumn is 10.0 cm deep. How far upward from its initi al positio n does the mercury in the right arm ri se?

20. I What is the height of a water barometer at atmospheric pressure?

21. III Postural hypotension is the occurrence of low (systolic) BID blood pressure when standing up too quickly from a reclined

position, causing faintin g or lightheadedness. For most people, a systolic pressure less than 90 mm Hg is cons idered low. If the blood pressure in your brain is 120 mm when you are lying down, what would it be when you stand up? Assume that your brain is 40 cm from your heart and that P = 1060 kg/m3 for your blood. No te: NonnaJly, your blood vessels constrict and expand to keep your brain blood pressure stable when you change your posture.

Section 13.4 Buoyancy

22. II A 6.00-cm-diameter sphere with a mass of 89.3 g is neu trally buoyant in a liquid. Identify the liquid.

23. III A cargo barge is loaded in a saltwater harbor for a trip up a freshwater river. Tfthe rectangular barge is 3.0 m by'20.0 m and sits 0.80 m deep in the harbor, how deep wiJi it sit in the river?

24. II A 10 em X 10 cm X 10 em wood block with a density of 700 kg/mJ floats in water. a. What is the distance from the top of the block to the water

if the water is fresh? b. If it 's seawater?

25. 11 What is the tension in the string in Figure PI 3.25?

FIGURE P13 .25

l00cm3 of I~~ --1.-1V -11 aluminum, density ~ P Al = 2700 kg/m3

Ethyl alcohol

26. 1 A 10 em X 10 em X 10 em block of steel (Pslcd=

7900 kg/m3) is suspended from a spring scale. The scale is in newtons. a. What is the scale reading if the block is in air? b. What is the scale reading after the block has been lowered

into a beaker of oil and is completely submerged? 27. 111 Styrofoam has a density of 300 kg/m3. What is the maximum

mass that can hang without sinking from a 50-cm-diameter Sty rofoam sphere in water? Assume the volume of the mass is neg ligible compared to that of the sphere.

28. 1111 Calculate the buoyant force due to the surrounding air on a man weighing 800 N. Assume hi s average density is the same as that of water.

Section 13.5 Fluids in Motion

29. III Ri ver Pascal with a volume flow rate of 5.0 X 105 Us joins with Ri ve r Archimedes, which carries 10.0 X 105 Us. to form the Bernoulli River. The Bernoulli Ri ver is ISO m wide and 10 m deep. What is the speed of the water in the Bernoulli River?

30. JI Water flowing through a 2.0-cm-diameter pipe can fill a 300 L bathtub in 5.0 min. What is the speed of the water in the pipe?

3 1. 1111 A pump is used to empty a 6000 L wading pool. The water exits the 2.5-em-diameter hose at a speed of2.l m/s. How long will it take to empty the pool ?

Problems 435

32. II A I.O-em-diameter pipe widens to 2.0 cm. then narrows to 0.50 cm. Liquid flows through the first segment at a speed of 4.0 m/s. a. What are the speeds in the second and third segments? h. What is the volume fl ow rate through the pipe?

Section 13.6 Fluid Dynamics

33. II What does the top pressure gauge in Figure P1 3.33 read?

1.0em View from above

20m~ A O ~ ~~ 3.0em

FIGURE P13.33 FIGURE P13.34

34. 11111 The 3.0-cm-diameter water line in Figure P13 .34 splits into two I.O-cm-diameter pipes. All pipes me circular and at the same e levation . At point A, the water speed is 2.0 m/s and the gauge pressure is 50 kPa. What is the gauge pressure at point B?

35. III A rectangular trough, 2.0 m long, 0.60 m wide, and 0.45 m deep, is completely full of water. One end of the trough has a small drain plug right at the bottom edge. When you pull the plug, at what speed does water emerge from the hole?

Seclion 13.7 Viscosity and Poiseuille's Equation

36. 11111 What pressure difference is required between the ends of a 2.0-m-long, I.O-mm-diameter hori zontal tube for 40°C water to flow through it at an average speed of 4.0 m/s?

37. 11111 Water fl ows at 0.25 Us through a 10-m-Iong garden hose 2.5 em in diameter that is lying flat on the ground. The tempera ture of the water is 20°e. What is the gauge pressure of the water where it enters the hose?

38. II Figure P1 3.38 shows a water-filled syr inge with a 4.0-em-long needle. What is the gauge pressure of the water at the point P, where the needle meets the wider chamber of the syringe?

General Problems

FIGURE P13.38

39. III The density of go ld is 19,300 kg/m3. 197 g of go ld is shaped into a cube. What is the length of each edge?

40. II The density of copper is 8920 kg/mJ . How many moles are in

INT a 2.0 em X 2.0 em X 2.0 cm cube of copper? 4 1. III The density of aluminum is 2700 kg/m3. How many atoms INT are in a 2.0 em X 2.0 em X 2.0 em cube of al uminum? 42. II A 50-cm-thick layer of oil floats on a 120-cm-thick layer of

waler. What is the pressure at the bottom of the water layer? 43. 11111 An oil layer floats on 85 cm of water in a tank. The absolute

pressure at the bottom of the tank is 1 J 2.0 kPa. How thick is the oil?

44. II The little Dutch boy saved Holland by sticking hi s fin ger in the leaking dike. If the water level was 2.5 m above hi s fin ger, estimate the force of the water on hi s finger.

436 CHAPTER 13 Fluids

45. II a. In Figure PI3.45, how much force does the fluid exert on the end of the cyl inder at A?

b. How much force does the fluid exert on the end of the cyl inder at B?

10 kg tloating piston 4.0cm

IOOcm

60 em

FIGURE P13.45

46. III A friend asks you how much pressure is in your car tires. You know that the tire man ufacturer recommends 30 psi, but it 's been a while since you've checked. YOLI can't find a tire gauge in the car, but you do find the owner's manual and a ruler. For tunatel y, yo u've just finished taking physics, so YOLI tell yo ur friend, " \ don't know, but I can figure it out." From the owner's manual you find that the car's mass is 1500 kg. It seems reason able to assume that each lire supports one-fourth o f the weight. With the ruler you find that the tires are 15 em wide and the flat tened segment of the tire in contact with the road is 13 cm long. What answer will you give your friend ?

47. III A diver 50 m deep in 10°C fresh water ex hales a 1.0-cm INT di ameter bubble. What is the bubble's diameter just as it reaches

the surface of the lake. where the water temperature is 20°C? Hint: Assume that the air bubble is always in thermal equilib rium with the surrounding water.

48. II A 6.0-em-tall cy linder floats in water with its ax is perpendic ular to the surface. The length of the cylinde r above water is 2.0 cm. What is the cy linder 's mass density?

49. II A sphere completely submerged in water is tethered to the bottom with a string. The tension in the string is one-third the weight of the sphere. What is the density of the sphere?

50. II You need to determine the density of a ceramic statue. If you suspend it from a spring scale, the scale reads 28.4 N. If you then lower the statue into a tub of water so that it is completely submerged, the scale reads 17.0 N. What is the dens ity?

51. II A 5.0 kg rock whose density is 4800 kg/m3 is suspended by a string suc h that half of the rock' s volume is under water. What is the tension in the string?

52. II A flat slab of styrofoam, with a density of 32 kglm\ float s on a lake. What minimum vol ume must the s lab ha ve so thai a 40 kg boy can sil on the slab without it s inking?

53. 1111 A 2.0 mL syringe has an inner diameter of 6.0 mm, a needle BIO inner diameter of 0.25 mm, and a plunger pad diameter (where

you place your finger) of 1.2 em. A nurse uses the syringe to inject medicine into a patien t whose blood pressure is 140/ 100.

Assume the liquid is an ideal fluid. a. What is the minimum force the nurse needs to app ly to the

sy ringe? b. The nurse empties the syri nge in 2.0 s. What is the flow

speed of the medicine through the needle? 54. III A child 's water pisto l shoots water through a I.O-mm-diame INT ter hole . If the pistol is fired hori zontally 70 e m above the

f,'l'ound, a squ irt hits the ground 1.2 m away. What is the vo lume flow rate during the squirt? Ignore air res istance.

55. II The leaves of a tree lose wate r to the atmosphere v ia the BIO process of transpiration. A particular tree loses water at the rate

of 3 X 10 8 m3/s; thi s water is replenished by the upward flow of sap through vesse ls in the trunk. Thi s tree 's trunk conta ins about 2000 vessels, each 100,um in diameter. What is the speed of the sap flowing in the vessels?

56. II A hurri cane wind blows across a 6.00 m X 15.0 m flat roof at a speed of 130 km/h. a. Is the air pressure above the roof higher or lower than the

pressure ins ide the hOLl se? Explain. b. What is the pressure difference? c . How much force is exerted on the roof? If the roof cannot

withstand thi s much force, will it "blow in" or "blow out"? 57. 11111 Water flows from the pipe shown in Figure P13.57 with a

speed of 4.0 m/s. a. What is the water pressure as it ex its into the air? b. What is the height II of the standing co lumn of water?

FIGURE P13 .57

ID em-

58. 1111 Air flows through the tube shown in Figure PI 3.58. Assume that air is an ideal fluid. a. What are the air speeds VI and V2 at points I and 2? b. What is the volume fl ow rate?

4.0mm

FtGURE P13.58 FIGURE P13.59

59. III Air flows through the tube shown in Figure P13.59 at a rate INT of 1200 cm 3/s. Assume that air is an ideal fluid . What is the

height" of mercury in the ri ght s ide of the U-tube? 60. III Water flows at 5 .0 U s through a hori zontal pipe that narrows

smoothly from 10 cm diameter to 5.0 cm diameter. A pressure gauge in the narrow section reads 50 kPa. What is the reading of

61. INT

a pressure gauge in the wide section? III The mercury manometer shown in Figure P1 3.6 1 is att- ached to a gas cel l. The mer- cury height II is 120 mm when the ce ll is placed in an ice- water mi xture. The mercury height drops to 30 mm when

Gas cell

the device is carried into an industrial freezer. What is the FIGURE P13 .61 freezer temperature? Hint: The right tube of the manometer is much narrower than the le ft tube. What reasonable assumption can YOll make about the gas volu me?

62. 11111 Figure P1 3.62 shows a sect ion of a long tube that narrows near its open end to a diameter of 1.0 mm. Water at 200 e flows out of the open end at 0.020 Us. What is the

FIGURE P13.62

gauge pressure at point P, where the diameter is 4.0 mm? 63. II Smoking tobacco is bad for your c irculatory health. In an BIO attempt to maintain the blood's capacity to deliver oxygen, the

body increases its red blood cell production, and thi s increases the viscosity of the blood. In addition, nicotine from tobacco causes arteries to constrict.

For a nonsmoker, with blood viscos ity of 2.S X 10-3 Pa· s, normal blood flow requires a pressure difference of 8.0 mm Hg between the two ends of an artery. If thi s person were to smoke regularly, hi s blood viscos ity would increase to 2.7 X 10- 3 Pa· s, and the arterial diameter would constrict to 90% of it s normal value. What pressure difference would be needed to maintain the same blood flow?

64. III A stitf, IO-cm-Iong tube with an inner diameter of 3.0 mm is auached to a small hole in the side of a tall beaker. The tube sticks out horizon tally. The beaker is fill ed with 200e water to a level 45 cm above the hole, and it is continually topped off to maintain thai level. What is the vo lume flow rate through the tube?

Passage Problems

Blood Pressure and Blood Flow BID

The blood pressure at your heart is approximately 100 mm Hg. As blood is pumped from the left ventri cle of your heart, it flows through the aorta, a single large blood vessel with a diameter of about 2.5 cm. The speed of blood flow in the aorta is about 60 cm/s. Any change in pressure as blood flows in the aorta is due to the change in height: the

Stop to Think 13.1: PI = pz = P3. Density depends only on what the object is made of, not how big the pieces are.

Stop to Think 13.2: C. These are all open tubes, so the liquid ri ses to the same height in all three despite their different shapes.

Stop to Think 13.3: B. The weight of the displaced water equals the weight of the ice cube. When the ice cube melts and turns into water, that amount of water will exactl y filJ the volume that the ice cube is now displacing.

Stop to Think 13.4: 1 cmJjs out. The fluid is incompressible, so the sum of what flow s in must match the sum of what flow s out. 13 cm3/s is known to be flowing in while 12 cm3/s flows out. An additional I c1113/s must now out to achieve balance.

Problems 437

vesse l is large enough that viscoLi s drag is not a major factor. As the blood moves through the circulatory system, it flows into succes sive ly smaller and smaller blood vessels until it reaches the capillaries. Blood flows in the capillari es at the much lower speed of approxi mately 0.7 mm/s. The diameter of capillaries and other small blood vessels is so small that viscous drag is a major factor.

65. I There is a limit to how long yo ur neck can be. If your neck were too long, no blood would reach your brain! What is the maximum height a person ' s brain could be above his heart , given the noted pressure and assuming that there are no valves or supplementary pumping mechanisms in the neck? The den sity of blood is 1060 kg/m). A. 0.97 m B. 1.3 m C. 9.7 m D. 13 III

66. I Because the flow speed in your capillaries is much less than in the aorta, the total cross-section area of the capillaries con sidered together must be much larger than that of the aorta. Given the flow speeds noted , the total area of the capillaries considered together is equi valent to the cross-sec tion area of a si ngle vessel of approximately what diameter? A. 2Scm B. SOcm C. 7Scm D. 100 cm

67. I Suppose that in response to some stimulus a small blood ves sel narrows to 90% its original diameter. [f there is no change in the pressure ac ross the vessel, what is the ratio ofthe new vol ume flow rate to the original now rate? A. 0.66 B. 0.73 C. 0.81 D. 0.90

68. I Sustained exerc ise can increase the blood flow rate of the heart by a factor of 5 with only a modest increase in blood pres sure. This is a large change in fl ow. Although several factors come into play, which of the follo wing physiological changes would most plausibly account for such a large increase in fl ow with a small change in pressure? A. A decrease in the viscos ity of the blood B. Dilation of the smaller blood vessels to larger diameters C. Dilation of the aorta to larger diameter D. An increase in the oxygen carried by the blood

Stop to Think 13.5: 112 > h4 > IlJ > "l' The liquid level is higher where the pressure is lower. The pressure is lower where the flow speed is higher. The flow speed is highest in the narrowest tube, zero in the open air.