Questions 407 Questions denotes answer available in Student Solutions Manual/Study Guide; O denotes objective question 1. O Figure Q14.1 sows aerial views from directly above two dams. Bot dams are equally wide (te vertical dimension in te diagram) and equally ig (into te page in te diagram). Te dam on te left olds back a very large lake, and te dam on te rigt olds back a narrow river. Wic dam as to be built more strongly? (a) te dam on te left (b) te dam on te rigt (c) bot te same (d) cannot be predicted Dam Dam Figure Q14.1 2. Two tin-walled drinking glasses aving equal base areas but different sapes, wit very different cross-sectional areas above te base, are filled to te same level wit water. According to te expression P P 0 rg, te pressure is te same at te bottom of bot glasses. In view of tis equality, wy does one glass weig more tan te oter? 3. Because atmosperic pressure is about 10 5 N/m 2 and te area of a person s cest is about 0.13 m 2, te force of te atmospere on one s cest is around 13 000 N. In view of tis enormous force, wy don t our bodies collapse? 4. A fis rests on te bottom of a bucket of water wile te bucket is being weiged on a scale. Wen te fis begins to swim around, does te scale reading cange? 5. You are a passenger on a spacecraft. For your survival and comfort, te interior contains air just like tat at te surface of te Eart. Te spacecraft is coasting troug a very empty region of space. Tat is, a nearly perfect vacuum exists just outside te wall. Suddenly, a meteoroid pokes a ole, about te size of a large coin, rigt troug te wall next to your seat. Wat appens? Is tere anyting you can or sould do about it? 6. Does a sip float iger in te water of an inland lake or in te ocean? Wy? 7. O An apple is eld completely submerged just below te surface of water in a container. Te apple is ten moved to a deeper point in te water. Compared wit te force needed to old te apple just below te surface, wat is te force needed to old it at te deeper point? (a) larger (b) te same (c) smaller (d) impossible to determine 8. Wen an object is immersed in a liquid at rest, wy is te net force on te object in te orizontal direction equal to zero? 9. A barge is carrying a load of gravel along a river. Te barge approaces a low bridge, and te captain realizes tat te top of te pile of gravel is not going to make it under te bridge. Te captain orders te crew to sovel gravel from te pile into te water. Is tat a good decision? 10. An empty metal soap dis barely floats in water. A bar of Ivory soap floats in water. Wen te soap is stuck in te soap dis, te combination sinks. Explain wy. 11. O A beac ball is made of tin plastic. It as been inflated wit air, but te plastic is not stretced. By swimming wit fins on, you manage to take te ball from te surface of a pool to te bottom. Once te ball is completely submerged, wat appens to te buoyant force exerted on te beac ball as you take it deeper? (a) increases (b) remains constant (c) decreases (d) is impossible to determine 12. If you release a ball wile inside a freely falling elevator, te ball remains in front of you rater tan falling to te floor because te ball, te elevator, and you all experience te same downward gravitational acceleration. Wat appens if you repeat tis experiment wit a elium-filled balloon? (Tis question is tricky.) 13. O A small piece of steel is tied to a block of wood. Wen te wood is placed in a tub of water wit te steel on top, alf of te block is submerged. Now te block is inverted so tat te steel is under water. (i) Does te amount of te block submerged (a) increase, (b) decrease, or (c) remain te same? (ii) Wat appens to te water level in te tub wen te block is inverted? (a) It rises. (b) It falls. (c) It remains te same. 14. How would you determine te density of an irregularly saped rock? 15. O Rank te buoyant forces exerted on te following seven objects, from te largest to te smallest. Assume te objects ave been dropped into a swimming pool and allowed to come to mecanical equilibrium. If any buoyant forces are equal, state tat in your ranking. (a) a block of solid oak (b) an aluminum block of equal volume to te wood (c) a beac ball made of tin plastic and inflated wit air, of equal volume (d) an iron block of equal volume (e) a tin-walled, sealed bottle of water equal in volume to te wood (f) an aluminum block aving te same mass as te wood (g) an iron block of equal mass 16. O A person in a boat floating in a small pond trows an ancor overboard. Wat appens to te level of te pond? (a) It rises. (b) It falls. (c) It remains te same. 17. Is te buoyant force a conservative force? Is a potential energy associated wit it? Explain your answers. 18. An unopened can of diet cola floats wen placed in a tank of water, wereas a can of regular cola of te same brand sinks in te tank. Wat do you suppose could explain tis beavior? 19. O A piece of unpainted porous wood floats in a container partly filled wit water. Te container is sealed and pressurized above atmosperic pressure. Wat appens to te wood? (a) It rises. (b) It falls. (c) It remains at te same level. 20. Te water supply for a city is often provided from reservoirs built on ig ground. Water flows from te reservoir, troug pipes, and into your ome wen you turn
408 Capter 14 Fluid Mecanics te tap on your faucet. Wy is te water flow more rapid out of a faucet on te first floor of a building tan in an apartment on a iger floor? 21. If te airstream from a air dryer is directed over a tabletennis ball, te ball can be levitated. Explain. 22. Wen ski jumpers are airborne (Fig. Q14.22), tey bend teir bodies forward and keep teir ands at teir sides. Wy? Pamela Zilly Figure Q14.25 TempSport/CORBIS 26. In Figure Q14.26, an airstream moves from rigt to left troug a tube tat is constricted at te middle. Tree table-tennis balls are levitated in equilibrium above te vertical columns troug wic te air escapes. (a) Wy is te ball at te rigt iger tan te one in te middle? (b) Wy is te ball at te left lower tan te ball at te rigt even toug te orizontal tube as te same dimensions at tese two points? Figure Q14.22 23. Wy do airplane pilots prefer to take off wit te airplane facing into te wind? 24. O A water supply maintains a constant rate of flow for water in a ose. You want to cange te opening of te nozzle so tat water leaving te nozzle reaces a eigt tat is four times te current maximum eigt te water reaces wit te nozzle vertical. To do so, wat sould you do? (a) decrease te area of te opening by a factor of 16 (b) decrease te area by a factor of 8 (c) decrease te area by a factor of 4 (d) decrease te area by a factor of 2 (e) give up because it cannot be done 25. Prairie dogs (Fig. Q14.25) ventilate teir burrows by building a mound around one entrance, wic is open to a stream of air wen wind blows from any direction. A second entrance at ground level is open to almost stagnant air. How does tis construction create an airflow troug te burrow? Figure Q14.26 27. O (i) A glass of water contains floating ice cubes. Wen te ice melts, does te water level in te glass (a) go up, (b) go down, or (c) remain te same? (ii) One of te predicted problems due to global warming is tat ice in te polar ice caps will melt and raise sea level everywere in te world. Is tat more of a worry for ice (a) at te nort pole, were most of te ice floats on water; (b) at te sout pole, were most of te ice sits on land; (c) bot at te nort and te sout poles equally; or (d) at neiter pole? Henry eap and Jim eman Problems Te Problems from tis capter may be assigned online in WebAssign. Sign in at www.tomsonedu.com and go to TomsonNOW to assess your understanding of tis capter s topics wit additional quizzing and conceptual questions. 1, 2, 3 denotes straigtforward, intermediate, callenging; denotes full solution available in Student Solutions Manual/Study Guide; denotes coaced solution wit ints available at www.tomsonedu.com; denotes developing symbolic reasoning; denotes asking for qualitative reasoning; denotes computer useful in solving problem Section 14.1 Pressure 1. Calculate te mass of a solid iron spere tat as a diameter of 3.00 cm. 2. Find te order of magnitude of te density of te nucleus of an atom. Wat does tis result suggest concerning te structure of matter? Model a nucleus as consisting of protons and neutrons closely packed togeter. Eac as mass 1.67 10 27 kg and radius on te order of 10 15 m. 3. A 50.0-kg woman balances on one eel of a pair of igeeled soes. If te eel is circular and as a radius of 0.500 cm, wat pressure does se exert on te floor? 4. Wat is te total mass of te Eart s atmospere? (Te radius of te Eart is 6.37 10 6 m, and atmosperic pressure at te surface is 1.013 10 5 N/m 2.)
Problems 409 Section 14.2 Variation of Pressure wit Dept 5. Te spring of te pressure gauge sown in Figure 14.2 as a force constant of 1 000 N/m, and te piston as a diameter of 2.00 cm. As te gauge is lowered into water, wat cange in dept causes te piston to move in by 0.500 cm? 6. (a) Calculate te absolute pressure at an ocean dept of 1 000 m. Assume te density of seawater is 1 024 kg/m 3 and te air above exerts a pressure of 101.3 kpa. (b) At tis dept, wat force must te frame around a circular submarine portole aving a diameter of 30.0 cm exert to counterbalance te force exerted by te water? 7. Wat must be te contact area between a suction cup (completely exausted) and a ceiling if te cup is to support te weigt of an 80.0-kg student? 8. Te small piston of a ydraulic lift as a cross-sectional area of 3.00 cm 2 and its large piston as a cross-sectional area of 200 cm 2 (Fig. 14.4a). Wat force must be applied to te small piston for te lift to raise a load of 15.0 kn? (In service stations, tis force is usually exerted by compressed air.) 9. For te basement of a new ouse, a ole is dug in te ground, wit vertical sides going down 2.40 m. A concrete foundation wall is built across te 9.60-m widt of te excavation. Tis foundation wall is 0.183 m from te front of te basement ole. During a rainstorm, drainage from te street fills up te space in front of te concrete wall, but not te basement beind te wall. Te water does not soak into te clay soil. Find te force te water causes on te foundation wall. For comparison, te gravitational force exerted on te water is (2.40 m)(9.60 m)(0.183 m)(1 000 kg/m 3 )(9.80 m/s 2 ) 41.3 kn. 10. (a) A powerful vacuum cleaner as a ose 2.86 cm in diameter. Wit no nozzle on te ose, wat is te weigt of te eaviest brick tat te cleaner can lift (Fig. P14.10a)? (b) Wat If? An octopus uses one sucker of diameter 2.86 cm on eac of te two sells of a clam in an attempt to pull te sells apart (Fig. P14.10b). Find te greatest force te octopus can exert in seawater 32.3 m deep. Caution: Experimental verification can be interesting, but do not drop a brick on your foot. Do not overeat te motor of a vacuum cleaner. Do not get an octopus mad at you. wit fres water, wat is te force caused by te water on te bottom? On eac end? On eac side? 12. Te tank in Figure P14.12 is filled wit water 2.00 m deep. At te bottom of one sidewall is a rectangular atc 1.00 m ig and 2.00 m wide tat is inged at te top of te atc. (a) Determine te force te water causes on te atc. (b) Find te torque caused by te water about te inges. 2.00 m 2.00 m Figure P14.12 1.00 m 13. Review problem. Te Abbott of Aberbrotock paid for a bell moored to te Inccape Rock to warn sailors away. Assume te bell was 3.00 m in diameter and cast from brass wit a bulk modulus of 14.0 10 10 N/m 2. Te pirate Ralp te Rover cut loose te bell and trew it into te ocean. By ow muc did te diameter of te bell decrease as it sank to a dept of 10.0 km? Years later, te klutz drowned wen is sip collided wit te rock. Note: Te brass is compressed uniformly, so you may model te bell as a spere of diameter 3.00 m. Section 14.3 Pressure Measurements 14. Figure P14.14 sows Superman attempting to drink water troug a very long straw. Wit is great strengt e acieves maximum possible suction. Te walls of te tubular straw do not collapse. (a) Find te maximum eigt troug wic e can lift te water. (b) Wat If? Still tirsty, te Man of Steel repeats is attempt on te Moon, wic as no atmospere. Find te difference between te water levels inside and outside te straw. (a) Figure P14.10 11. A swimming pool as dimensions 30.0 m 10.0 m and a flat bottom. Wen te pool is filled to a dept of 2.00 m (b) Figure P14.14 15. Blaise Pascal duplicated Torricelli s barometer using a red Bordeaux wine, of density 984 kg/m 3, as te working liquid (Fig. P14.15). Wat was te eigt of te wine column for normal atmosperic pressure? Would
410 Capter 14 Fluid Mecanics you expect te vacuum above te column to be as good as for mercury? P 0 Figure P14.15 16. Mercury is poured into a U-tube as sown in Figure P14.16a. Te left arm of te tube as cross-sectional area A 1 of 10.0 cm 2, and te rigt arm as a cross-sectional area A 2 of 5.00 cm 2. One undred grams of water are ten poured into te rigt arm as sown in Figure P14.16b. (a) Determine te lengt of te water column in te rigt arm of te U-tube. (b) Given tat te density of mercury is 13.6 g/cm 3, wat distance does te mercury rise in te left arm? ave te same density as water. Te pressure of te cerebrospinal fluid can be measured by means of a spinal tap as illustrated in Figure P14.19. A ollow tube is inserted into te spinal column, and te eigt to wic te fluid rises is observed. If te fluid rises to a eigt of 160 mm, we write its gauge pressure as 160 mm H 2 O. (a) Express tis pressure in pascals, in atmosperes, and in millimeters of mercury. (b) Sometimes it is necessary to determine weter an accident victim as suffered a crused vertebra tat is blocking flow of te cerebrospinal fluid in te spinal column. In oter cases, a pysician may suspect tat a tumor or oter growt is blocking te spinal column and inibiting flow of cerebrospinal fluid. Suc conditions can be investigated by means of Queckenstedt s test. In tis procedure, te veins in te patient s neck are compressed to make te blood pressure rise in te brain. Te increase in pressure in te blood vessels is transmitted to te cerebrospinal fluid. Wat sould be te normal effect on te eigt of te fluid in te spinal tap? (c) Suppose compressing te veins ad no effect on te fluid level. Wat migt account for tis result? A 1 A 2 A 1 A Water 2 (a) Mercury Figure P14.16 17. Normal atmosperic pressure is 1.013 10 5 Pa. Te approac of a storm causes te eigt of a mercury barometer to drop by 20.0 mm from te normal eigt. Wat is te atmosperic pressure? (Te density of mercury is 13.59 g/cm 3.) 18. A tank wit a flat bottom of area A and vertical sides is filled to a dept wit water. Te pressure is 1 atm at te top surface. (a) Wat is te absolute pressure at te bottom of te tank? (b) Suppose an object of mass M and density less tan te density of water is placed in te tank and floats. No water overflows. Wat is te resulting increase in pressure at te bottom of te tank? (c) Evaluate your results for a backyard swimming pool wit dept 1.50 m and a circular base wit diameter 6.00 m. Two persons wit combined mass 150 kg enter te pool and float quietly tere. Find te original absolute pressure and te pressure increase at te bottom of te pool. 19. Te uman brain and spinal cord are immersed in te cerebrospinal fluid. Te fluid is normally continuous between te cranial and spinal cavities and exerts a pressure of 100 to 200 mm of H 2 O above te prevailing atmosperic pressure. In medical work, pressures are often measured in units of millimeters of H 2 O because body fluids, including te cerebrospinal fluid, typically (b) Figure P14.19 Section 14.4 Buoyant Forces and Arcimedes s Principle 20. (a) A ligt balloon is filled wit 400 m 3 of elium. At 0 C, te balloon can lift a payload of wat mass? (b) Wat If? In Table 14.1, observe tat te density of ydrogen is nearly one-alf te density of elium. Wat load can te balloon lift if filled wit ydrogen? 21. A table-tennis ball as a diameter of 3.80 cm and average density of 0.084 0 g/cm 3. Wat force is required to old it completely submerged under water? 22. Te gravitational force exerted on a solid object is 5.00 N. Wen te object is suspended from a spring scale and submerged in water, te scale reads 3.50 N (Fig. P14.22). Find te density of te object. T 1 Mg (a) Scale B Mg T 2 (b) Figure P14.22 Problems 22 and 23.
96 98 100 102 104 Problems 411 23. A 10.0-kg block of metal measuring 12.0 cm 10.0 cm 10.0 cm is suspended from a scale and immersed in water as sown in Figure P14.22b. Te 12.0-cm dimension is vertical, and te top of te block is 5.00 cm below te surface of te water. (a) Wat are te forces acting on te top and on te bottom of te block? (Take P 0 101.30 kpa.) (b) Wat is te reading of te spring scale? (c) Sow tat te buoyant force equals te difference between te forces at te top and bottom of te block. 24. Te weigt of a rectangular block of low-density material is 15.0 N. Wit a tin string, te center of te orizontal bottom face of te block is tied to te bottom of a beaker partly filled wit water. Wen 25.0% of te block s volume is submerged, te tension in te string is 10.0 N. (a) Sketc a free-body diagram for te block, sowing all forces acting on it. (b) Find te buoyant force on te block. (c) Oil of density 800 kg/m 3 is now steadily added to te beaker, forming a layer above te water and surrounding te block. Te oil exerts forces on eac of te four sidewalls of te block tat te oil touces. Wat are te directions of tese forces? (d) Wat appens to te string tension as te oil is added? Explain ow te oil as tis effect on te string tension. (e) Te string breaks wen its tension reaces 60.0 N. At tis moment, 25.0% of te block s volume is still below te waterline. Wat additional fraction of te block s volume is below te top surface of te oil? (f) After te string breaks, te block comes to a new equilibrium position in te beaker. It is now in contact only wit te oil. Wat fraction of te block s volume is submerged? 25. Preparing to ancor a buoy at te edge of a swimming area, a worker uses a rope to lower a cubical concrete block, 0.250 m on eac edge, into ocean water. Te block moves down at a constant speed of 1.90 m/s. You can accurately model te concrete and te water as incompressible. (a) At wat rate is te force te water exerts on one face of te block increasing? (b) At wat rate is te buoyant force on te block increasing? 26. To an order of magnitude, ow many elium-filled toy balloons would be required to lift you? Because elium is an irreplaceable resource, develop a teoretical answer rater tan an experimental answer. In your solution, state wat pysical quantities you take as data and te values you measure or estimate for tem. 27. A cube of wood aving an edge dimension of 20.0 cm and a density of 650 kg/m 3 floats on water. (a) Wat is te distance from te orizontal top surface of te cube to te water level? (b) Wat mass of lead sould be placed on te cube so tat te top of te cube will be just level wit te water? 28. A sperical aluminum ball of mass 1.26 kg contains an empty sperical cavity tat is concentric wit te ball. Te ball barely floats in water. Calculate (a) te outer radius of te ball and (b) te radius of te cavity. 29. Determination of te density of a fluid as many important applications. A car battery contains sulfuric acid, for wic density is a measure of concentration. For te battery to function properly, te density must be witin a range specified by te manufacturer. Similarly, te effectiveness of antifreeze in your car s engine coolant depends on te density of te mixture (usually etylene glycol and water). Wen you donate blood to a blood bank, its screening includes determination of te density of te blood because iger density correlates wit iger emoglobin content. A ydrometer is an instrument used to determine liquid density. A simple one is sketced in Figure P14.29. Te bulb of a syringe is squeezed and released to let te atmospere lift a sample of te liquid of interest into a tube containing a calibrated rod of known density. Te rod, of lengt and average density r 0, floats partially immersed in te liquid of density r. A lengt of te rod protrudes above te surface of te liquid. Sow tat te density of te liquid is r r 0 96 98 100 102 104 Figure P14.29 Problems 29 and 30. 30. Refer to Problem 29 and Figure P14.29. A ydrometer is to be constructed wit a cylindrical floating rod. Nine fiduciary marks are to be placed along te rod to indicate densities aving values of 0.98 g/cm 3, 1.00 g/cm 3, 1.02 g/cm 3, 1.04 g/cm 3,..., 1.14 g/cm 3. Te row of marks is to start 0.200 cm from te top end of te rod and end 1.80 cm from te top end. (a) Wat is te required lengt of te rod? (b) Wat must be its average density? (c) Sould te marks be equally spaced? Explain your answer. 31. How many cubic meters of elium are required to lift a balloon wit a 400-kg payload to a eigt of 8 000 m? (Take r He 0.180 kg/m 3.) Assume te balloon maintains a constant volume and te density of air decreases wit te altitude z according to te expression r air r 0 e z/8 000, were z is in meters and r 0 1.25 kg/m 3 is te density of air at sea level. 32. A batyspere used for deep-sea exploration as a radius of 1.50 m and a mass of 1.20 10 4 kg. To dive, tis submarine takes on mass in te form of seawater. Determine te amount of mass te submarine must take on if it is to descend at a constant speed of 1.20 m/s, wen te resistive force on it is 1 100 N in te upward direction. Te density of seawater is 1.03 10 3 kg/m 3. 33. A plastic spere floats in water wit 50.0% of its volume submerged. Tis same spere floats in glycerin wit 40.0% of its volume submerged. Determine te densities of te glycerin and te spere.
412 Capter 14 Fluid Mecanics 34. Te United States possesses te eigt largest warsips in te world aircraft carriers of te Nimitz class and is building two more. Suppose one of te sips bobs up to float 11.0 cm iger in te water wen 50 figter planes take off from it in 25 minutes, at a location were te free-fall acceleration is 9.78 m/s 2. Bristling wit bombs and missiles, te planes ave an average mass of 29 000 kg. Find te orizontal area enclosed by te waterline of te $4-billion sip. By comparison, its fligt deck as area 18 000 m 2. Below decks are passageways undreds of meters long, so narrow tat two large men cannot pass eac oter. Section 14.5 Fluid Dynamics Section 14.6 Bernoulli s Equation 35. A large storage tank, open at te top and filled wit water, develops a small ole in its side at a point 16.0 m below te water level. Te rate of flow from te leak is 2.50 10 3 m 3 /min. Determine (a) te speed at wic te water leaves te ole and (b) te diameter of te ole. 36. A village maintains a large tank wit an open top, containing water for emergencies. Te water can drain from te tank troug a ose of diameter 6.60 cm. Te ose ends wit a nozzle of diameter 2.20 cm. A rubber stopper is inserted into te nozzle. Te water level in te tank is kept 7.50 m above te nozzle. (a) Calculate te friction force exerted on te stopper by te nozzle. (b) Te stopper is removed. Wat mass of water flows from te nozzle in 2.00? (c) Calculate te gauge pressure of te flowing water in te ose just beind te nozzle. 37. Water flows troug a fire ose of diameter 6.35 cm at a rate of 0.012 0 m 3 /s. Te fire ose ends in a nozzle of inner diameter 2.20 cm. Wat is te speed wit wic te water exits te nozzle? 38. Water moves troug a constricted pipe in steady, ideal flow. At one point as sown in Figure 14.16 were te pressure is 2.50 10 4 Pa, te diameter is 8.00 cm. At anoter point 0.500 m iger, te pressure is equal to 1.50 10 4 Pa and te diameter is 4.00 cm. Find te speed of flow (a) in te lower section and (b) in te upper section. (c) Find te volume flow rate troug te pipe. 39. Figure P14.39 sows a stream of water in steady flow from a kitcen faucet. At te faucet, te diameter of te stream is 0.960 cm. Te stream fills a 125-cm 3 container in 16.3 s. Find te diameter of te stream 13.0 cm below te opening of te faucet. 40. Water falls over a dam of eigt wit a mass flow rate of R, in units of kilograms per second. (a) Sow tat te power available from te water is Rg were g is te free-fall acceleration. (b) Eac ydroelectric unit at te Grand Coulee Dam takes in water at a rate of 8.50 10 5 kg/s from a eigt of 87.0 m. Te power developed by te falling water is converted to electric power wit an efficiency of 85.0%. How muc electric power does eac ydroelectric unit produce? 41. A legendary Dutc boy saved Holland by plugging a 1.20-cm diameter ole in a dike wit is finger. If te ole was 2.00 m below te surface of te Nort Sea (density 1 030 kg/m 3 ), (a) wat was te force on is finger? (b) If e pulled is finger out of te ole, during wat time interval would te released water fill 1 acre of land to a dept of 1 ft? Assume te ole remained constant in size. (A typical U.S. family of four uses 1 acre-foot of water, 1 234 m 3, in 1 year.) 42. In ideal flow, a liquid of density 850 kg/m 3 moves from a orizontal tube of radius 1.00 cm into a second orizontal tube of radius 0.500 cm. A pressure difference P exists between te tubes. (a) Find te volume flow rate as a function of P. Evaluate te volume flow rate (b) for P 6.00 kpa and (c) for P 12.0 kpa. (d) State ow te volume flow rate depends on P. 43. Water is pumped up from te Colorado River to supply Grand Canyon Village, located on te rim of te canyon. Te river is at an elevation of 564 m, and te village is at an elevation of 2 096 m. Imagine tat te water is pumped troug a single long pipe 15.0 cm in diameter, driven by a single pump at te bottom end. (a) Wat is te minimum pressure at wic te water must be pumped if it is to arrive at te village? (b) If 4 500 m 3 of water is pumped per day, wat is te speed of te water in te pipe? (c) Wat additional pressure is necessary to deliver tis flow? Note: Assume te free-fall acceleration and te density of air are constant over tis range of elevations. Te pressures you calculate are too ig for an ordinary pipe. Te water is actually lifted in stages by several pumps troug sorter pipes. 44. Old Faitful Geyser in Yellowstone National Park erupts at approximately 1- intervals, and te eigt of te water column reaces 40.0 m (Fig. P14.44). (a) Model te rising stream as a series of separate drops. Analyze te free-fall motion of one of te drops to determine te speed at wic te water leaves te ground. (b) Wat If? Model te rising stream as an ideal fluid in streamline George Semple Stan Osolinski/Dembinsky Poto Associates Figure P14.39 Figure P14.44
Problems 413 flow. Use Bernoulli s equation to determine te speed of te water as it leaves ground level. (c) How does te answer from part (a) compare wit te answer from part (b)? (d) Wat is te pressure (above atmosperic) in te eated underground camber if its dept is 175 m? Assume te camber is large compared wit te geyser s vent. 45. A Venturi tube may be used as a fluid flowmeter (see Fig. 14.19). Taking te difference in pressure as P 1 P 2 21.0 kpa, find te fluid flow rate in cubic meters per second given tat te radius of te outlet tube is 1.00 cm, te radius of te inlet tube is 2.00 cm, and te fluid is gasoline (r 700 kg/m 3 ). Section 14.7 Oter Applications of Fluid Dynamics 46. An airplane as a mass of 1.60 10 4 kg, and eac wing as an area of 40.0 m 2. During level fligt, te pressure on te lower wing surface is 7.00 10 4 Pa. Determine te pressure on te upper wing surface. 47. A sipon of uniform diameter is used to drain water from a tank as illustrated in Figure P14.47. Assume steady flow witout friction. (a) If 1.00 m, find te speed of outflow at te end of te sipon. (b) Wat If? Wat is te limitation on te eigt of te top of te sipon above te water surface? (For te flow of te liquid to be continuous, te pressure must not drop below te vapor pressure of te liquid.) F S were is 1 atm. A force of magnitude 2.00 N acts on te plunger, making medicine squirt orizontally from te needle. Determine te speed of te medicine as it leaves te needle s tip. 50. Te Bernoulli effect can ave important consequences for te design of buildings. For example, wind can blow around a skyscraper at remarkably ig speed, creating low pressure. Te iger atmosperic pressure in te still air inside te buildings can cause windows to pop out. As originally constructed, te Jon Hancock Building in Boston popped windowpanes tat fell many stories to te sidewalk below. (a) Suppose a orizontal wind blows wit a speed of 11.2 m/s outside a large pane of plate glass wit dimensions 4.00 m 1.50 m. Assume te density of te air to be 1.30 kg/m 3. Te air inside te building is at atmosperic pressure. Wat is te total force exerted by air on te windowpane? (b) Wat If? If a second skyscraper is built nearby, te airspeed can be especially ig were wind passes troug te narrow separation between te buildings. Solve part (a) again wit a wind speed of 22.4 m/s, twice as ig. Additional Problems 51. A elium-filled balloon is tied to a 2.00-m-long, 0.050 0-kg uniform string. Te balloon is sperical wit a radius of 0.400 m. Wen released, it lifts a lengt of string and ten remains in equilibrium as sown in Figure P14.51. Determine te value of. Te envelope of te balloon as a mass of 0.250 kg. y v He Figure P14.47 48. An airplane is cruising at altitude 10 km. Te pressure outside te craft is 0.287 atm; witin te passenger compartment, te pressure is 1.00 atm and te temperature is 20 C. A small leak occurs in one of te window seals in te passenger compartment. Model te air as an ideal fluid to find te speed of te stream of air flowing troug te leak. 49. A ypodermic syringe contains a medicine aving te density of water (Fig. P14.49). Te barrel of te syringe as a cross-sectional area A 2.50 10 5 m 2, and te needle as a cross-sectional area a 1.00 10 8 m 2. In te absence of a force on te plunger, te pressure every- Figure P14.51 52. Figure P14.52 sows a water tank wit a valve at te bottom. If tis valve is opened, wat is te maximum eigt attained by te water stream coming out of te rigt side of te tank? Assume 10.0 m, 2.00 m, and u 30.0 and assume te cross-sectional area at A is very large compared wit tat at B. A F A v Valve B u a Figure P14.49 Figure P14.52
414 Capter 14 Fluid Mecanics 53. Te true weigt of an object can be measured in a vacuum, were buoyant forces are absent. An object of volume V is weiged in air on an equal-arm balance wit te use of counterweigts of density r. Representing te density of air as r air and te balance reading as F g, sow tat te true weigt F g is F g F g a V F g rg b r airg 54. Water is forced out of a fire extinguiser by air pressure as sown in Figure P14.54. How muc gauge air pressure in te tank (above atmosperic) is required for te water jet to ave a speed of 30.0 m/s wen te water level is 0.500 m below te nozzle? 0.500 m Figure P14.54 55. A ligt spring of constant k 90.0 N/m is attaced vertically to a table (Fig. P14.55a). A 2.00-g balloon is filled wit elium (density 0.180 kg/m 3 ) to a volume of 5.00 m 3 and is ten connected to te spring, causing te spring to stretc as sown in Figure P14.55b. Determine te extension distance wen te balloon is in equilibrium. v 57. As a 950-kg elicopter overs, its orizontal rotor puses a column of air downward at 40.0 m/s. Wat can you say about te quantity of tis air? Explain your answer. You may model te air motion as ideal flow. 58. Evangelista Torricelli was te first person to realize tat we live at te bottom of an ocean of air. He correctly surmised tat te pressure of our atmospere is attributable to te weigt of te air. Te density of air at 0 C at te Eart s surface is 1.29 kg/m 3. Te density decreases wit increasing altitude (as te atmospere tins). On te oter and, if we assume te density is constant at 1.29 kg/m 3 up to some altitude and is zero above tat altitude, ten would represent te dept of te ocean of air. Use tis model to determine te value of tat gives a pressure of 1.00 atm at te surface of te Eart. Would te peak of Mount Everest rise above te surface of suc an atmospere? 59. Review problem. Wit reference to Figure 14.5, sow tat te total torque exerted by te water beind te dam 1 about a orizontal axis troug O is 6rgwH 3. Sow tat te effective line of action of te total force exerted by 1 te water is at a distance 3H above O. 60. In about 1657, Otto von Guericke, inventor of te air pump, evacuated a spere made of two brass emisperes. Two teams of eigt orses eac could pull te emisperes apart only on some trials and ten wit greatest difficulty, wit te resulting sound likened to a cannon firing (Fig. P14.60). (a) Sow tat te force F required to pull te tin-walled evacuated emisperes apart is pr 2 (P 0 P), were R is te radius of te emisperes and P is te pressure inside te emisperes, wic is muc less tan P 0. (b) Determine te force for P 0.100P 0 and R 0.300 m. k k F R F (a) (b) P P 0 Figure P14.55 56. We can t call it Flubber. Assume a certain liquid, wit density 1 230 kg/m 3, exerts no friction force on sperical objects. A ball of mass 2.10 kg and radius 9.00 cm is dropped from rest into a deep tank of tis liquid from a eigt of 3.30 m above te surface. (a) Find te speed at wic te ball enters te liquid. (b) Wat two forces are exerted on te ball as it moves troug te liquid? (c) Explain wy te ball moves down only a limited distance into te liquid and calculate tis distance. (d) Wit wat speed does te ball pop up out of te liquid? (e) How does te time interval t down, during wic te ball moves from te surface down to its lowest point, compare wit te time interval t up for te return trip between te same two points? (f) Wat If? Now modify te model to suppose te liquid exerts a small friction force on te ball, opposite in direction to its motion. In tis case, ow do te time intervals t down and t up compare? Explain your answer wit a conceptual argument rater tan a numerical calculation. Figure P14.60 Te colored engraving, dated 1672, illustrates Otto von Guericke s demonstration of te force due to air pressure as it migt ave been performed before Emperor Ferdinand III. Te Granger Collection
Problems 415 61. A 1.00-kg beaker containing 2.00 kg of oil (density 916.0 kg/m 3 ) rests on a scale. A 2.00-kg block of iron suspended from a spring scale is completely submerged in te oil as sown in Figure P14.61. Determine te equilibrium readings of bot scales. 67. An incompressible, nonviscous fluid is initially at rest in te vertical portion of te pipe sown in Figure P14.67a, were 2.00 m. Wen te valve is opened, te fluid flows into te orizontal section of te pipe. Wat is te speed of te fluid wen it is all in te orizontal section as sown in Figure P14.67b? Assume te cross-sectional area of te entire pipe is constant. Figure P14.61 Problems 61 and 62. 62. A beaker of mass m b containing oil of mass m o and density r o rests on a scale. A block of iron of mass m Fe suspended from a spring scale is completely submerged in te oil as sown in Figure P14.61. Determine te equilibrium readings of bot scales. 63. In 1983, te United States began coining te cent piece out of copper-clad zinc rater tan pure copper. Te mass of te old copper penny is 3.083 g and tat of te new cent is 2.517 g. Calculate te percent of zinc (by volume) in te new cent. Te density of copper is 8.960 g/cm 3 and tat of zinc is 7.133 g/cm 3. Te new and old coins ave te same volume. 64. Sow tat te variation of atmosperic pressure wit altitude is given by P P 0 e ay, were a r 0 g/p 0, P 0 is atmosperic pressure at some reference level y 0, and r 0 is te atmosperic density at tis level. Assume te decrease in atmosperic pressure over an infinitesimal cange in altitude (so tat te density is approximately uniform) is given by dp rg dy and tat te density of air is proportional to te pressure. 65. Review problem. A uniform disk of mass 10.0 kg and radius 0.250 m spins at 300 rev/min on a low-friction axle. It must be brougt to a stop in 1.00 min by a brake pad tat makes contact wit te disk at an average distance of 0.220 m from te axis. Te coefficient of friction between te pad and te disk is 0.500. A piston in a cylinder of diameter 5.00 cm presses te brake pad against te disk. Find te pressure required for te brake fluid in te cylinder. 66. A cube of ice wose edges measure 20.0 mm is floating in a glass of ice-cold water, and one of te ice cube s faces is parallel to te water s surface. (a) How far below te water surface is te bottom face of te ice cube? (b) Icecold etyl alcool is gently poured onto te water surface to form a layer 5.00 mm tick above te water. Te alcool does not mix wit te water. Wen te ice cube again attains ydrostatic equilibrium, wat is te distance from te top of te water to te bottom face of te block? (c) Additional cold etyl alcool is poured onto te water s surface until te top surface of te alcool coincides wit te top surface of te ice cube (in ydrostatic equilibrium). How tick is te required layer of etyl alcool? Valve closed (a) Figure P14.67 Valve opened 68. Te water supply of a building is fed troug a main pipe 6.00 cm in diameter. A 2.00-cm-diameter faucet tap, located 2.00 m above te main pipe, is observed to fill a 25.0- container in 30.0 s. (a) Wat is te speed at wic te water leaves te faucet? (b) Wat is te gauge pressure in te 6-cm main pipe? (Assume te faucet is te only leak in te building.) 69. A U-tube open at bot ends is partially filled wit water (Fig. P14.69a). Oil aving a density 750 kg/m 3 is ten poured into te rigt arm and forms a column 5.00 cm ig (Fig. P14.69b). (a) Determine te difference in te eigts of te two liquid surfaces. (b) Te rigt arm is ten sielded from any air motion wile air is blown across te top of te left arm until te surfaces of te two liquids are at te same eigt (Fig. P14.69c). Determine te speed of te air being blown across te left arm. Take te density of air as 1.29 kg/m 3. Water P 0 Oil (b) (a) (b) (c) Figure P14.69 v v Sield 70. A woman is draining er fis tank by siponing te water into an outdoor drain as sown in Figure P14.70 (page 416). Te rectangular tank as footprint area A and dept. Te drain is located a distance d below te surface of te water in te tank, were d. Te crosssectional area of te sipon tube is A. Model te water as
416 Capter 14 Fluid Mecanics flowing witout friction. (a) Sow tat te time interval required to empty te tank is t A A 22gd (b) Evaluate te time interval required to empty te tank if it is a cube 0.500 m on eac edge, taking A 2.00 cm 2 and d 10.0 m. d wing. Its area projected onto a orizontal surface is A. Wen te boat is towed at sufficiently ig speed, water of density r moves in streamline flow so tat its average speed at te top of te ydrofoil is n times larger tan its speed v b below te ydrofoil. (a) Ignoring te buoyant force, sow tat te upward lift force exerted by te water on te ydrofoil as a magnitude F 1 2 1n 2 12rv b 2 A (b) Te boat as mass M. Sow tat te liftoff speed is 2Mg v B 1n 2 12Ar (c) Assume an 800-kg boat is to lift off at 9.50 m/s. Evaluate te area A required for te ydrofoil if its design yields n 1.05. Figure P14.70 71. Te ull of an experimental boat is to be lifted above te water by a ydrofoil mounted below its keel as sown in Figure P14.71. Te ydrofoil is saped like an airplane Figure P14.71 Answers to Quick Quizzes 14.1 (a). Because te basketball player s weigt is distributed over te larger surface area of te soe, te pressure (F/A) e applies is relatively small. Te woman s lesser weigt is distributed over te very small cross-sectional area of te spiked eel, so te pressure is ig. 14.2 (a). Because bot fluids ave te same dept, te one wit te smaller density (alcool) will exert te smaller pressure. 14.3 (c). All barometers will ave te same pressure at te bottom of te column of fluid: atmosperic pressure. Terefore, te barometer wit te igest column will be te one wit te fluid of lowest density. 14.4 (b) or (c). In all tree cases, te weigt of te treasure cest causes a downward force on te raft tat makes te raft sink into te water. In (b) and (c), owever, te treasure cest also displaces water, wic provides a buoyant force in te upward direction, reducing te effect of te cest s weigt. 14.5 (a). Te ig-speed air between te balloons results in low pressure in tis region. Te iger pressure on te outer surfaces of te balloons puses tem toward eac oter.