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Questions denotes answer available in Student Solutions Manual/Study Guide; O denotes objective question 1. O Figure Q14.1 shows aerial views from directly 8. When an object is immersed in a liquid at rest, why above two dams. Both dams are equally wide (the is the net force on the object in the horizontal direction vertical dimension in the diagram) and equally high equal to zero? (into the page in the diagram). The dam on the left 9. A barge is carrying a load of gravel along a river. holds back a very large lake, and the dam on the right The barge approaches a low bridge, and the captain holds back a narrow river. Which dam has to be built realizes that the top of the pile of gravel is not going to more strongly? (a) the dam on the left (b) the dam on make it under the bridge. The captain orders the crew the right (c) both the same (d) cannot be predicted to shovel gravel from the pile into the water. Is that a good decision? 10. An empty metal soap dish barely floats in water. A bar of Ivory soap floats in water. When the soap is stuck in the soap dish, the combination sinks. Explain why. 11. O A beach ball is made of thin plastic. It has been inflated with air, but the plastic is not stretched. By swimming with fins on, you manage to take the ball from the surface of a pool to the bottom. Once the ball is completely submerged, what happens to the buoyant 2. Two thin-walled drinking glasses having equal base force exerted on the beach ball as you take it deeper? areas but different shapes, with very different cross- (a) increases (b) remains constant (c) decreases (d) is sectional areas above the base, are filled to the same impossible to determine level with water. According to the expression P = P0 + gh, the pressure is the same at the bottom of 12. If you release a ball while inside a freely falling both glasses. In view of this equality, why does one elevator, the ball remains in front of you rather than glass weigh more than the other? falling to the floor because the ball, the elevator, and you all experience the same downward gravitational 3. Because atmospheric pressure is about 105 N/m2 acceleration. What happens if you repeat this and the area of a person’s chest is about 0.13 m2, the experiment with a helium-filled balloon? (This force of the atmosphere on one’s chest is around 13 question is tricky.) 000 N. In view of this enormous force, why don’t our bodies collapse? 13. O A small piece of steel is tied to a block of wood. When the wood is placed in a tub of water with the 4. A fish rests on the bottom of a bucket of water steel on top, half of the block is submerged. Now the while the bucket is being weighed on a scale. When block is inverted so that the steel is under water. (i) the fish begins to swim around, does the scale reading Does the amount of the block submerged (a) increase, change? (b) decrease, or (c) remain the same? (ii) What 5. You are a passenger on a spacecraft. For your happens to the water level in the tub when the block is survival and comfort, the interior contains air just like inverted? (a) It rises. (b) It falls. (c) It remains the that at the surface of the Earth. The spacecraft is same. coasting through a very empty region of space. That 14. How would you determine the density of an is, a nearly perfect vacuum exists just outside the wall. irregularly shaped rock? Suddenly, a meteoroid pokes a hole, about the size of a large coin, right through the wall next to your seat. 15. O Rank the buoyant forces exerted on the What happens? Is there anything you can or should do following seven objects, from the largest to the about it? smallest. Assume the objects have been dropped into a swimming pool and allowed to come to mechanical 6. Does a ship float higher in the water of an inland equilibrium. If any buoyant forces are equal, state that lake or in the ocean? Why? in your ranking. (a) a block of solid oak (b) an 7. O An apple is held completely submerged just aluminum block of equal volume to the wood (c) a below the surface of water in a container. The apple is beach ball made of thin plastic and inflated with air, of then moved to a deeper point in the water. Compared equal volume (d) an iron block of equal volume (e) a with the force needed to hold the apple just below the thin-walled, sealed bottle of water equal in volume to surface, what is the force needed to hold it at the the wood (f) an aluminum block having the same mass deeper point? (a) larger (b) the same (c) smaller (d) as the wood (g) an iron block of equal mass impossible to determine 16. O A person in a boat floating in a small pond throws an anchor overboard. What happens to the 2 = intermediate 3 = challenging = SSM/SG = ThomsonNOW = symbolic reasoning =qualitative reasoning level of the pond? (a) It rises. (b) It falls. (c) It remains 25. Prairie dogs (Fig. Q14.25) ventilate their burrows the same. by building a mound around one entrance, which is open to a stream of air when wind blows from any 17. Is the buoyant force a conservative force? Is a direction. A second entrance at ground level is open to potential energy associated with it? Explain your almost stagnant air. How does this construction create answers. an airflow through the burrow? 18. An unopened can of diet cola floats when placed in a tank of water, whereas a can of regular cola of the same brand sinks in the tank. What do you suppose could explain this behavior? 19. O A piece of unpainted porous wood floats in a container partly filled with water. The container is sealed and pressurized above atmospheric pressure. What happens to the wood? (a) It rises. (b) It falls. (c) It remains at the same level. 20. The water supply for a city is often provided from reservoirs built on high ground. Water flows from the reservoir, through pipes, and into your home when you turn the tap on your faucet. Why is the water flow 26. In Figure Q14.26, an airstream moves from right more rapid out of a faucet on the first floor of a to left through a tube that is constricted at the middle. building than in an apartment on a higher floor? Three table-tennis balls are levitated in equilibrium 21. If the airstream from a hair dryer is directed over a above the vertical columns through which the air table-tennis ball, the ball can be levitated. Explain. escapes. (a) Why is the ball at the right higher than the 22. When ski jumpers are airborne (Fig. Q14.22), they one in the middle? (b) Why is the ball at the left lower bend their bodies forward and keep their hands at their than the ball at the right even though the horizontal sides. Why? tube has the same dimensions at these two points? 23. Why do airplane pilots prefer to take off with the 27. O (i) A glass of water contains floating ice cubes. airplane facing into the wind? When the ice melts, does the water level in the glass 24. O A water supply maintains a constant rate of flow (a) go up, (b) go down, or (c) remain the same? (ii) for water in a hose. You want to change the opening of One of the predicted problems due to global warming the nozzle so that water leaving the nozzle reaches a is that ice in the polar ice caps will melt and raise sea height that is four times the current maximum height level everywhere in the world. Is that more of a worry the water reaches with the nozzle vertical. To do so, for ice (a) at the north pole, where most of the ice what should you do? (a) decrease the area of the floats on water; (b) at the south pole, where most of opening by a factor of 16 (b) decrease the area by a the ice sits on land; (c) both at the north and the south factor of 8 (c) decrease the area by a factor of 4 (d) poles equally; or (d) at neither pole? decrease the area by a factor of 2 (e) give up because it cannot be done 2 = intermediate 3 = challenging = SSM/SG = ThomsonNOW = symbolic reasoning =qualitative reasoning Problems The Problems from this chapter may be assigned online in WebAssign. Sign in at www.thomsonedu.com and go to ThomsonNOW to assess your understanding of this chapter’s topics with additional quizzing and conceptual questions. 1,2,3 denotes straightforward, intermediate, challenging; denotes full solution available in Student Solutions Manual/Study Guide; denotes coached solution with hints available at www.thomsonedu.com; denotes developing symbolic reasoning; denotes asking for qualitative reasoning; denotes computer useful in solving problem rainstorm, drainage from the street fills up the space in Section 14.1 Pressure front of the concrete wall, but not the basement behind the wall. The water does not soak into the clay soil. 1. Calculate the mass of a solid iron sphere that has a Find the force the water causes on the foundation wall. diameter of 3.00 cm. For comparison, the gravitational force exerted on the 2. Find the order of magnitude of the density of the water is nucleus of an atom. What does this result suggest (2.40 m)(9.60 m)(0.183 m) concerning the structure of matter? Model a nucleus as (1 000 kg/m3)(9.80 m/s2) = 41.3 kN consisting of protons and neutrons closely packed together. Each has mass 1.67 10–27 kg and radius on 10. (a) A powerful vacuum cleaner has a hose 2.86 cm the order of 10–15 m. in diameter. With no nozzle on the hose, what is the weight of the heaviest brick that the cleaner can lift 3. A 50.0-kg woman balances on one heel of a pair of (Fig. P14.10a)? (b) What If? An octopus uses one high-heeled shoes. If the heel is circular and has a sucker of diameter 2.86 cm on each of the two shells radius of 0.500 cm, what pressure does she exert on of a clam in an attempt to pull the shells apart (Fig. the floor? P14.10b). Find the greatest force the octopus can exert 4. What is the total mass of the Earth’s atmosphere? in seawater 32.3 m deep. Caution: Experimental (The radius of the Earth is 6.37 10 m, and 6 verification can be interesting, but do not drop a brick atmospheric pressure at the surface is 1.013 105 on your foot. Do not overheat the motor of a vacuum 2 N/m .) cleaner. Do not get an octopus mad at you. Section 14.2 Variation of Pressure with Depth 5. The spring of the pressure gauge shown in Figure 14.2 has a force constant of 1 000 N/m, and the piston has a diameter of 2.00 cm. As the gauge is lowered into water, what change in depth causes the piston to move in by 0.500 cm? 6. (a) Calculate the absolute pressure at an ocean depth of 1 000 m. Assume the density of seawater is 1 024 kg/m3 and the air above exerts a pressure of 101.3 kPa. (b) At this depth, what force must the frame around a 11. A swimming pool has dimensions circular submarine porthole having a diameter of 30.0 30.0 m 10.0 m and a flat bottom. When the pool is cm exert to counterbalance the force exerted by the filled to a depth of 2.00 m with fresh water, what is the water? force caused by the water on the bottom? On each 7. What must be the contact area between a suction end? On each side? cup (completely exhausted) and a ceiling if the cup is 12. The tank in Figure P14.12 is filled with water to support the weight of an 80.0-kg student? 2.00 m deep. At the bottom of one sidewall is a 8. The small piston of a hydraulic lift has a cross- rectangular hatch 1.00 m high and 2.00 m wide that is sectional area of 3.00 cm2 and its large piston has a hinged at the top of the hatch. (a) Determine the force cross-sectional area of 200 cm2 (Fig. 14.4a). What the water causes on the hatch. (b) Find the torque force must be applied to the small piston for the lift to caused by the water about the hinges. raise a load of 15.0 kN? (In service stations, this force is usually exerted by compressed air.) 9. For the basement of a new house, a hole is dug in the ground, with vertical sides going down 2.40 m. A concrete foundation wall is built across the 9.60-m width of the excavation. This foundation wall is 0.183 m from the front of the basement hole. During a 2 = intermediate 3 = challenging = SSM/SG = ThomsonNOW = symbolic reasoning =qualitative reasoning 13. Review problem. The Abbott of Aberbrothock 16. Mercury is poured into a U-tube as shown in paid for a bell moored to the Inchcape Rock to warn Figure P14.16a. The left arm of the tube has cross- sailors away. Assume the bell was 3.00 m in diameter sectional area A1 of 10.0 cm2, and the right arm has a and cast from brass with a bulk modulus of 14.0 1010 cross-sectional area A2 of 5.00 cm2. One hundred N/m2. The pirate Ralph the Rover cut loose the bell grams of water are then poured into the right arm as and threw it into the ocean. By how much did the shown in Figure P14.16b. (a) Determine the length of diameter of the bell decrease as it sank to a depth of the water column in the right arm of the U-tube. (b) 10.0 km? Years later, the klutz drowned when his ship Given that the density of mercury is 13.6 g/cm3, what collided with the rock. Note: The brass is compressed distance h does the mercury rise in the left arm? uniformly, so you may model the bell as a sphere of diameter 3.00 m. Section 14.3 Pressure Measurements 14. Figure P14.14 shows Superman attempting to drink water through a very long straw. With his great strength he achieves maximum possible suction. The walls of the tubular straw do not collapse. (a) Find the maximum height through which he can lift the water. (b) What If? Still thirsty, the Man of Steel repeats his attempt on the Moon, which has no atmosphere. Find 17. Normal atmospheric pressure is 1.013 105 Pa. the difference between the water levels inside and The approach of a storm causes the height of a outside the straw. mercury barometer to drop by 20.0 mm from the normal height. What is the atmospheric pressure? (The density of mercury is 13.59 g/cm3.) 18. A tank with a flat bottom of area A and vertical sides is filled to a depth h with water. The pressure is 1 atm at the top surface. (a) What is the absolute pressure at the bottom of the tank? (b) Suppose an object of mass M and density less than the density of water is placed in the tank and floats. No water overflows. What is the resulting increase in pressure at the bottom of the tank? (c) Evaluate your results for a backyard swimming pool with depth 1.50 m and a circular base with diameter 6.00 m. Two persons with combined mass 150 kg enter the pool and float quietly there. Find the original absolute pressure and the pressure increase at the bottom of the pool. 15. Blaise Pascal duplicated Torricelli’s 19. The human brain and spinal cord are immersed barometer using a red Bordeaux wine, of density 984 in the cerebrospinal fluid. The fluid is normally kg/m3, as the working liquid (Fig. P14.15). What was continuous between the cranial and spinal cavities and the height h of the wine column for normal exerts a pressure of 100 to 200 mm of H2O above the atmospheric pressure? Would you expect the vacuum prevailing atmospheric pressure. In medical work, above the column to be as good as for mercury? pressures are often measured in units of millimeters of H2O because body fluids, including the cerebrospinal fluid, typically have the same density as water. The pressure of the cerebrospinal fluid can be measured by means of a spinal tap as illustrated in Figure P14.19. 2 = intermediate 3 = challenging = SSM/SG = ThomsonNOW = symbolic reasoning =qualitative reasoning 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 shown in Figure P14.22b. The 12.0-cm dimension is vertical, and the top of the block is 5.00 cm below the surface of the water. (a) What are the forces acting on the top and on the bottom of the block? (Take P0 = 101.30 kPa.) (b) What is the reading of the spring scale? (c) Show that the buoyant force equals the difference between the forces at the top and bottom of the block. A hollow tube is inserted into the spinal column, and 24. The weight of a rectangular block of low- the height to which the fluid rises is observed. If the density material is 15.0 N. With a thin string, the fluid rises to a height of 160 mm, we write its gauge center of the horizontal bottom face of the block is tied pressure as 160 mm H2O. (a) Express this pressure in to the bottom of a beaker partly filled with water. pascals, in atmospheres, and in millimeters of When 25.0% of the block’s volume is submerged, the mercury. (b) Sometimes it is necessary to determine tension in the string is 10.0 N. (a) Sketch a free-body whether an accident victim has suffered a crushed diagram for the block, showing all forces acting on it. vertebra that is blocking flow of the cerebrospinal (b) Find the buoyant force on the block. (c) Oil of fluid in the spinal column. In other cases, a physician density 800 kg/m3 is now steadily added to the beaker, may suspect that a tumor or other growth is blocking forming a layer above the water and surrounding the the spinal column and inhibiting flow of cerebrospinal block. The oil exerts forces on each of the four fluid. Such conditions can be investigated by means of sidewalls of the block that the oil touches. What are Queckenstedt’s test. In this procedure, the veins in the the directions of these forces? (d) What happens to the patient’s neck are compressed to make the blood string tension as the oil is added? Explain how the oil pressure rise in the brain. The increase in pressure in has this effect on the string tension. (e) The string the blood vessels is transmitted to the cerebrospinal breaks when its tension reaches 60.0 N. At this fluid. What should be the normal effect on the height moment, 25.0% of the block’s volume is still below of the fluid in the spinal tap? (c) Suppose compressing the waterline. What additional fraction of the block’s the veins had no effect on the fluid level. What might volume is below the top surface of the oil? (f) After account for this result? the string breaks, the block comes to a new equilibrium position in the beaker. It is now in contact Section 14.4 Buoyant Forces and Archimedes’s only with the oil. What fraction of the block’s volume Principle is submerged? 20. (a) A light balloon is filled with 400 m3 of helium. 25. Preparing to anchor a buoy at the edge of a At 0°C, the balloon can lift a payload of what mass? swimming area, a worker uses a rope to lower a (b) What If? In Table 14.1, observe that the density of cubical concrete block, 0.250 m on each edge, into hydrogen is nearly one-half the density of helium. ocean water. The block moves down at a constant What load can the balloon lift if filled with hydrogen? speed of 1.90 m/s. You can accurately model the concrete and the water as incompressible. (a) At what 21. A table-tennis ball has a diameter of 3.80 cm and rate is the force the water exerts on one face of the average density of 0.084 0 g/cm3. What force is block increasing? (b) At what rate is the buoyant force required to hold it completely submerged under water? on the block increasing? 22. The gravitational force exerted on a solid object is 26. To an order of magnitude, how many helium-filled 5.00 N. When the object is suspended from a spring toy balloons would be required to lift you? Because scale and submerged in water, the scale reads 3.50 N helium is an irreplaceable resource, develop a (Fig. P14.22). Find the density of the object. theoretical answer rather than an experimental answer. In your solution, state what physical quantities you take as data and the values you measure or estimate for them. 27. A cube of wood having an edge dimension of 20.0 cm and a density of 650 kg/m3 floats on water. (a) What is the distance from the horizontal top surface of the cube to the water level? (b) What mass of lead should be placed on the cube so that the top of the cube will be just level with the water? 28. A spherical aluminum ball of mass 1.26 kg contains an empty spherical cavity that is concentric with the ball. The ball barely floats in water. Calculate 2 = intermediate 3 = challenging = SSM/SG = ThomsonNOW = symbolic reasoning =qualitative reasoning (a) the outer radius of the ball and (b) the radius of the m/s, when the resistive force on it is 1 100 N in the cavity. upward direction. The density of seawater is 29. Determination of the density of a fluid has many 1.03 103 kg/m3. important applications. A car battery contains sulfuric 33. A plastic sphere floats in water with 50.0% of its acid, for which density is a measure of concentration. volume submerged. This same sphere floats in For the battery to function properly, the density must glycerin with 40.0% of its volume submerged. be within a range specified by the manufacturer. Determine the densities of the glycerin and the sphere. Similarly, the effectiveness of antifreeze in your car’s engine coolant depends on the density of the mixture 34. The United States possesses the eight largest (usually ethylene glycol and water). When you donate warships in the world—aircraft carriers of the Nimitz blood to a blood bank, its screening includes class—and is building two more. Suppose one of the determination of the density of the blood because ships bobs up to float 11.0 cm higher in the water higher density correlates with higher hemoglobin when 50 fighter planes take off from it in 25 minutes, content. A hydrometer is an instrument used to at a location where the free-fall acceleration is 9.78 determine liquid density. A simple one is sketched in m/s2. Bristling with bombs and missiles, the planes Figure P14.29. The bulb of a syringe is squeezed and have an average mass of 29 000 kg. Find the released to let the atmosphere lift a sample of the horizontal area enclosed by the waterline of the $4- liquid of interest into a tube containing a calibrated billion ship. By comparison, its flight deck has area rod of known density. The rod, of length L and 18 000 m2. Below decks are passageways hundreds of meters long, so narrow that two large men cannot pass average density , floats partially immersed in the each other. liquid of density . A length h of the rod protrudes above the surface of the liquid. Show that the density Section 14.5 Fluid Dynamics of the liquid is Section 14.6 Bernoulli’s Equation 0 L 35. A large storage tank, open at the top and filled Lh with water, develops a small hole in its side at a point 16.0 m below the water level. The rate of flow from the leak is 2.50 10–3 m3/min. Determine (a) the speed at which the water leaves the hole and (b) the diameter of the hole. 36. A village maintains a large tank with an open top, containing water for emergencies. The water can drain from the tank through a hose of diameter 6.60 cm. The hose ends with a nozzle of diameter 2.20 cm. A rubber stopper is inserted into the nozzle. The water level in the tank is kept 7.50 m above the nozzle. (a) Calculate the friction force exerted on the stopper by the nozzle. (b) The stopper is removed. What mass of water flows 30. Refer to Problem 29 and Figure P14.29. A from the nozzle in 2.00 h? (c) Calculate the gauge hydrometer is to be constructed with a cylindrical pressure of the flowing water in the hose just behind floating rod. Nine fiduciary marks are to be placed the nozzle. along the rod to indicate densities having values of 0.98 g/cm3, 1.00 g/cm3, 1.02 g/cm3, 1.04 g/cm3, …, 37. Water flows through a fire hose of diameter 6.35 1.14 g/cm3. The row of marks is to start 0.200 cm cm at a rate of 0.012 0 m3/s. The fire hose ends in a from the top end of the rod and end 1.80 cm from the nozzle of inner diameter 2.20 cm. What is the speed top end. (a) What is the required length of the rod? (b) with which the water exits the nozzle? What must be its average density? (c) Should the 38. Water moves through a constricted pipe in steady, marks be equally spaced? Explain your answer. ideal flow. At one point as shown in Figure 14.16 31. How many cubic meters of helium are required to where the pressure is 2.50 104 Pa, the diameter is lift a balloon with a 400-kg payload to a height of 8.00 cm. At another point 0.500 m higher, the pressure 8 000 m? (Take He = 0.180 kg/m3.) Assume the is equal to 1.50 104 Pa and the diameter is 4.00 cm. balloon maintains a constant volume and the density Find the speed of flow (a) in the lower section and (b) of air decreases with the altitude z according to the in the upper section. (c) Find the volume flow rate through the pipe. expression air = 0e–z/8 000, where z is in meters and 0 = 1.25 kg/m3 is the density of air at sea level. 39. Figure P14.39 shows a stream of water in steady flow from a kitchen faucet. At the faucet, the diameter 32. A bathysphere used for deep-sea exploration has a of the stream is 0.960 cm. The stream fills a 125-cm3 radius of 1.50 m and a mass of 1.20 104 kg. To dive, container in 16.3 s. Find the diameter of the stream this submarine takes on mass in the form of seawater. 13.0 cm below the opening of the faucet. Determine the amount of mass the submarine must take on if it is to descend at a constant speed of 1.20 2 = intermediate 3 = challenging = SSM/SG = ThomsonNOW = symbolic reasoning =qualitative reasoning 44. Old Faithful Geyser in Yellowstone National Park erupts at approximately 1-h intervals, and the height of the water column reaches 40.0 m (Fig. P14.44). (a) Model the rising stream as a series of separate drops. Analyze the free-fall motion of one of the drops to determine the speed at which the water leaves the ground. (b) What If? Model the rising stream as an ideal fluid in streamline flow. Use Bernoulli’s equation to determine the speed of the water as it leaves ground level. (c) How does the answer from part (a) compare with the answer from part (b)? (d) What is the pressure (above atmospheric) 40. Water falls over a dam of height h with a mass in the heated underground chamber if its depth is 175 flow rate of R, in units of kilograms per second. (a) m? Assume the chamber is large compared with the Show that the power available from the water is geyser’s vent. = Rgh where g is the free-fall acceleration. (b) Each hydroelectric unit at the Grand Coulee Dam takes in water at a rate of 8.50 105 kg/s from a height of 87.0 m. The power developed by the falling water is converted to electric power with an efficiency of 85.0%. How much electric power does each hydroelectric unit produce? 41. A legendary Dutch boy saved Holland by plugging a 1.20-cm diameter hole in a dike with his finger. If the hole was 2.00 m below the surface of the North 45. A Venturi tube may be used as a fluid flowmeter Sea (density 1 030 kg/m3), (a) what was the force on (see Fig. 14.19). Taking the difference in pressure as his finger? (b) If he pulled his finger out of the hole, P1 – P2 = 21.0 kPa, find the fluid flow rate in cubic during what time interval would the released water fill meters per second given that the radius of the outlet 1 acre of land to a depth of 1 ft? Assume the hole tube is 1.00 cm, the radius of the inlet tube is 2.00 cm, remained constant in size. (A typical U.S. family of four uses 1 acre-foot of water, 1 234 m3, in 1 year.) and the fluid is gasoline ( = 700 kg/m3). 42. In ideal flow, a liquid of density 850 kg/m3 Section 14.7 Other Applications of Fluid Dynamics moves from a horizontal tube of radius 1.00 cm into a second horizontal tube of radius 0.500 cm. A pressure 46. An airplane has a mass of 1.60 104 kg, and each wing has an area of 40.0 m2. During level flight, the difference P exists between the tubes. (a) Find the pressure on the lower wing surface is 7.00 104 Pa. volume flow rate as a function of P. Evaluate the Determine the pressure on the upper wing surface. volume flow rate (b) for P = 6.00 kPa and (c) for P = 12.0 kPa. (d) State how the volume flow rate 47. A siphon of uniform diameter is used to drain depends on P. water from a tank as illustrated in Figure P14.47. Assume steady flow without friction. (a) If h = 1.00 m, 43. Water is pumped up from the Colorado River to find the speed of outflow at the end of the siphon. (b) supply Grand Canyon Village, located on the rim of What If? What is the limitation on the height of the the canyon. The river is at an elevation of 564 m, and top of the siphon above the water surface? (For the the village is at an elevation of 2 096 m. Imagine that flow of the liquid to be continuous, the pressure must the water is pumped through a single long pipe 15.0 not drop below the vapor pressure of the liquid.) cm in diameter, driven by a single pump at the bottom end. (a) What is the minimum pressure at which the water must be pumped if it is to arrive at the village? (b) If 4 500 m3 of water is pumped per day, what is the speed of the water in the pipe? (c) What additional pressure is necessary to deliver this flow? Note: Assume the free-fall acceleration and the density of air are constant over this range of elevations. The pressures you calculate are too high for an ordinary pipe. The water is actually lifted in stages by several pumps through shorter pipes. 2 = intermediate 3 = challenging = SSM/SG = ThomsonNOW = symbolic reasoning =qualitative reasoning 48. An airplane is cruising at altitude 10 km. The 52. Figure P14.52 shows a water tank with a valve at pressure outside the craft is 0.287 atm; within the the bottom. passenger compartment, the pressure is 1.00 atm and the temperature is 20°C. A small leak occurs in one of the window seals in the passenger compartment. Model the air as an ideal fluid to find the speed of the stream of air flowing through the leak. 49. A hypodermic syringe contains a medicine having the density of water (Fig. P14.49). If this valve is opened, what is the maximum height attained by the water stream coming out of the right side of the tank? Assume h = 10.0 m, L = 2.00 m, and The barrel of the syringe has a cross-sectional area = 30.0° and assume the cross-sectional area at A is A = 2.50 10–5 m2, and the needle has a cross- very large compared with that at B. sectional area a= 1.00 10–8 m2. In the absence of a force on the plunger, the pressure everywhere is 1 atm. 53. The true weight of an object can be measured in a A force of F magnitude 2.00 N acts on the plunger, vacuum, where buoyant forces are absent. An object making medicine squirt horizontally from the needle. of volume V is weighed in air on an equal-arm balance Determine the speed of the medicine as it leaves the with the use of counterweights of density . needle’s tip. Representing the density of air as air and the 50. The Bernoulli effect can have important balance reading as F’g, show that the true weight consequences for the design of buildings. For Fg is example, wind can blow around a skyscraper at F remarkably high speed, creating low pressure. The Fg Fg V g air g higher atmospheric pressure in the still air inside the g buildings can cause windows to pop out. As originally constructed, the John Hancock Building in Boston 54. Water is forced out of a fire extinguisher by air popped windowpanes that fell many stories to the pressure as shown in Figure P14.54. How much gauge sidewalk below. (a) Suppose a horizontal wind blows air pressure in the tank (above atmospheric) is with a speed of 11.2 m/s outside a large pane of plate required for the water jet to have a speed of 30.0 m/s glass with dimensions 4.00 m 1.50 m. Assume the when the water level is 0.500 m below the nozzle? density of the air to be 1.30 kg/m3. The air inside the building is at atmospheric pressure. What is the total force exerted by air on the windowpane? (b) What If? If a second skyscraper is built nearby, the airspeed can be especially high where wind passes through the narrow separation between the buildings. Solve part (a) again with a wind speed of 22.4 m/s, twice as high. Additional Problems 51. A helium-filled balloon is tied to a 2.00-m-long, 55. A light spring of constant k = 90.0 N/m is attached 0.050 0-kg uniform string. The balloon is spherical vertically to a table (Fig. P14.55a). A 2.00-g balloon is with a radius of 0.400 m. When released, it lifts a filled with helium (density = 0.180 kg/m3) to a volume length h of string and then remains in equilibrium as of 5.00 m3 and is then connected to the spring, causing shown in Figure P14.51. Determine the value of h. the spring to stretch as shown in Figure P14.55b. The envelope of the balloon has a mass of 0.250 kg. Determine the extension distance L when the balloon is in equilibrium. 2 = intermediate 3 = challenging = SSM/SG = ThomsonNOW = symbolic reasoning =qualitative reasoning hemispheres. Two teams of eight horses each could pull the hemispheres apart only on some trials and then ―with greatest difficulty,‖ with the resulting sound likened to a cannon firing (Fig. P14.60). (a) Show that the force F required to pull the thin-walled evacuated hemispheres apart is R2(P0 – P), where R is the radius of the hemispheres and P is the pressure inside the hemispheres, which is much less than P0. (b) Determine the force for P = 0.100P0 and R = 0.300 m. 56. We can’t call it Flubber. Assume a certain liquid, with density 1 230 kg/m3, exerts no friction force on spherical objects. A ball of mass 2.10 kg and radius 9.00 cm is dropped from rest into a deep tank of this liquid from a height of 3.30 m above the surface. (a) Find the speed at which the ball enters the liquid. (b) What two forces are exerted on the ball as it moves through the liquid? (c) Explain why the ball moves down only a limited distance into the liquid and calculate this distance. (d) With what speed does the ball pop up out of the liquid? (e) How does the time interval tdown, during which the ball moves from the surface down to its lowest point, compare with the time interval tup for the return trip between the same two points? (f) What If? Now modify the model to suppose the liquid exerts a small friction force on the ball, opposite in direction to its motion. In this case, how do the time intervals tdownand tup compare? Explain your answer with a conceptual argument rather than a numerical calculation. 57. As a 950-kg helicopter hovers, its horizontal rotor pushes a column of air downward at 40.0 m/s. What can you say about the quantity of this air? Explain your answer. You may model the air motion Figure P14.60 The colored engraving, dated 1672, illustrates Otto as ideal flow. von Guericke’s demonstration of the force due to air pressure as it 58. Evangelista Torricelli was the first person to might have been performed before Emperor Ferdinand III. realize that we live at the bottom of an ocean of air. He correctly surmised that the pressure of our 61. A 1.00-kg beaker containing 2.00 kg of oil atmosphere is attributable to the weight of the air. The (density = 916.0 kg/m3) rests on a scale. A 2.00-kg density of air at 0°C at the Earth’s surface is 1.29 block of iron suspended from a spring scale is kg/m3. The density decreases with increasing altitude completely submerged in the oil as shown in Figure (as the atmosphere thins). On the other hand, if we P14.61. Determine the equilibrium readings of both assume the density is constant at 1.29 kg/m3 up to scales. some altitude h and is zero above that altitude, then h would represent the depth of the ocean of air. Use this model to determine the value of h that gives a pressure of 1.00 atm at the surface of the Earth. Would the peak of Mount Everest rise above the surface of such an atmosphere? 59. Review problem. With reference to Figure 14.5, show that the total torque exerted by the water behind the dam about a horizontal axis through O is 1 6 gwH 3 . Show that the effective line of action of the total force exerted by the water is at a distance 1 H above O. 62. A beaker of mass mb containing oil of mass mo and 3 density o rests on a scale. A block of iron of mass mFe 60. In about 1657, Otto von Guericke, inventor of the suspended from a spring scale is completely air pump, evacuated a sphere made of two brass 2 = intermediate 3 = challenging = SSM/SG = ThomsonNOW = symbolic reasoning =qualitative reasoning submerged in the oil as shown in Figure P14.61. Determine the equilibrium readings of both scales. 63. In 1983, the United States began coining the cent piece out of copper-clad zinc rather than pure copper. The mass of the old copper penny is 3.083 g and that of the new cent is 2.517 g. Calculate the percent of zinc (by volume) in the new cent. The density of copper is 8.960 g/cm3 and that of zinc is 7.133 g/cm3. The new and old coins have the same volume. 64. Show that the variation of atmospheric pressure with altitude is given by P = P0 e–y, where = 0g/P0, P0 is atmospheric pressure at some 68. The water supply of a building is fed through a reference level y = 0, and 0 is the atmospheric density main pipe 6.00 cm in diameter. A 2.00-cm-diameter at this level. Assume the decrease in atmospheric faucet tap, located 2.00 m above the main pipe, is pressure over an infinitesimal change in altitude (so observed to fill a 25.0-L container in 30.0 s. (a) What that the density is approximately uniform) is given by is the speed at which the water leaves the faucet? (b) dP = –g dy and that the density of air is proportional What is the gauge pressure in the 6-cm main pipe? to the pressure. (Assume the faucet is the only ―leak‖ in the building.) 65. Review problem. A uniform disk of mass 10.0 kg 69. A U-tube open at both ends is partially filled with and radius 0.250 m spins at 300 rev/min on a low- water (Fig. P14.69a). Oil having a density 750 kg/m3 friction axle. It must be brought to a stop in 1.00 min is then poured into the right arm and forms a column by a brake pad that makes contact with the disk at an L = 5.00 cm high (Fig. P14.69b). (a) Determine the average distance of 0.220 m from the axis. The difference h in the heights of the two liquid surfaces. coefficient of friction between the pad and the disk is (b) The right arm is then shielded from any air motion 0.500. A piston in a cylinder of diameter 5.00 cm while air is blown across the top of the left arm until presses the brake pad against the disk. Find the the surfaces of the two liquids are at the same height pressure required for the brake fluid in the cylinder. (Fig. P14.69c). Determine the speed of the air being 66. A cube of ice whose edges measure 20.0 mm is blown across the left arm. Take the density of air as floating in a glass of ice-cold water, and one of the ice 1.29 kg/m3. cube’s faces is parallel to the water’s surface. (a) How far below the water surface is the bottom face of the ice cube? (b) Ice-cold ethyl alcohol is gently poured onto the water surface to form a layer 5.00 mm thick above the water. The alcohol does not mix with the water. When the ice cube again attains hydrostatic equilibrium, what is the distance from the top of the water to the bottom face of the block? (c) Additional cold ethyl alcohol is poured onto the water’s surface until the top surface of the alcohol coincides with the top surface of the ice cube (in hydrostatic equilibrium). How thick is the required layer of ethyl alcohol? 70. A woman is draining her fish tank by siphoning 67. An incompressible, nonviscous fluid is initially at the water into an outdoor drain as shown in Figure rest in the vertical portion of the pipe shown in Figure P14.70. The rectangular tank has footprint area A and P14.67a, where L = 2.00 m. When the valve is opened, depth h. The drain is located a distance d below the the fluid flows into the horizontal section of the pipe. surface of the water in the tank, where What is the speed of the fluid when it is all in the d >> h. The cross-sectional area of the siphon tube is horizontal section as shown in Figure P14.67b? A. Model the water as flowing without friction. (a) Assume the cross-sectional area of the entire pipe is Show that the time interval required to empty the tank constant. is Ah t A 2 gd (b) Evaluate the time interval required to empty the tank if it is a cube 0.500 m on each edge, taking A = 2.00 cm2 and d = 10.0 m. 2 = intermediate 3 = challenging = SSM/SG = ThomsonNOW = symbolic reasoning =qualitative reasoning that the upward lift force exerted by the water on the hydrofoil has a magnitude F 1 (n2 1) v2 A 2 b (b) The boat has mass M. Show that the liftoff speed is 2Mg v (n2 1) A (c) Assume an 800-kg boat is to lift off at 9.50 m/s. Evaluate the area A required for the hydrofoil if its design yields n = 1.05. 71. The hull of an experimental boat is to be lifted above the water by a hydrofoil mounted below its keel as shown in Figure P14.71. The hydrofoil is shaped like an airplane wing. Its area projected onto a horizontal surface is A. When the boat is towed at sufficiently high speed, water of density moves in streamline flow so that its average speed at the top of the hydrofoil is n times larger than its speed vb below the hydrofoil. (a) Ignoring the buoyant force, show 2 = intermediate 3 = challenging = SSM/SG = ThomsonNOW = symbolic reasoning =qualitative reasoning