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CHAPTER 13 Liquids Summary of Terms Pressure Buoyant force Archimedes’ principle Principle of flotation Pascal’s principle Surface tension Capillarity Review Questions 1. Give two examples of a fluid. Pressure 2. Distinguish between force and pressure. Pressure in a Liquid 3. What is the relationship between liquid pressure and the depth of a liquid? Between liquid pressure and density? 4. If you swim beneath the surface in salt water, will the pressure be greater than in freshwater at the same depth? Why or why not? 5. How does water pressure one meter below the surface of a small pond compare with water pressure one meter below the surface of a huge lake? 6. If you punch a hole in a container filled with water, in what direction does the water initially flow outward from the container? Buoyancy 7. Why does buoyant force act upward on an object submerged in water? 8. Why is there no horizontal buoyant force on a submerged object? 9. How does the volume of a completely submerged object compare with the volume of water displaced? Archimedes’ Principle 10. How does the buoyant force on a submerged object compare with the weight of water displaced? 11. Distinguish between a submerged body and an immersed body. 12. What is the mass of 1 L of water? What is its weight in newtons? 13. If a 1L container is immersed halfway into water, what is the volume of water displaced? What is the buoyant force on the container? What Makes an Object Sink or Float? 14. Is the buoyant force on a submerged object equal to the weight of the object itself or equal to the weight of the fluid displaced by the object? 15. There is a condition in which the buoyant force on an object does equal the weight of the object. What is this condition? 16. Does the buoyant force on a submerged object depend on the volume of the object or the weight of the object? 17. Fill in the blanks: An object denser than water will ______ in water. An object less dense than water will ______ in water. An object with the same density of water will ____________ in water. 18. How is the density of a fish controlled? How is the density of a submarine controlled? Flotation 19. It was emphasized earlier that buoyant force does not equal an object’s weight but does equal the weight of displaced water. Now we say buoyant force equals the object’s weight. Isn’t this a grand contradiction? Explain. 20. What weight of water is displaced by a 100ton ship? What is the buoyant force that acts on a floating 100ton ship? Pascal’s Principle 21. What happens to the pressure in all parts of a confined fluid if the pressure in one part is increased? 22. If the pressure in a hydraulic press is increased by an additional 10 N/cm , how much extra load will the output piston support if its cross sectional area is 50 cm ? Surface Tension 23. What geometrical shape has the least surface area for a given volume? 24. What is the cause of surface tension? Capillarity 25. Distinguish between adhesive and cohesive forces. 26. What determines how high water will climb in a capillary tube? [ previous page ] [ next page ] Projects Place an egg in a pan of tap water. Then dissolve salt in the water until the egg floats. How 1. does the density of an egg compare to that of tap water? To that of salt water? 2.If you punch a couple of holes in the bottom of a waterfilled container, water will spurt out because of water pressure. Now drop the container, and, as it freely falls, note that the water no longer spurts out! If your friends don’t understand this, could you figure it out and then explain it to them? (Click image to enlarge) 3.Float a watersoaked PingPong ball in a can of water held more than a meter above a rigid floor. Then drop the can. Careful inspection will show the ball pulled beneath the surface as both the ball and the can drop. (What does this say about surface tension?). More dramatically, when the can makes impact with the floor, what happens to the ball, and why? Try it and you’ll be astonished! (Caution: Unless you’re wearing safety goggles, keep your head away from above the can when it makes impact.) 4.Soap greatly weakens the cohesive forces between water molecules. You can see this by putting some oil in a bottle of water and shaking it so that the oil and water mix. Notice that the oil and water quickly separate as soon as you stop shaking the bottle. Now add some soap to the mixture. Shake the bottle again and you will see that the soap makes a fine film around each little oil bead and that a longer time is required for the oil to gather after you stop shaking the bottle. This is how soap works in cleaning. It breaks the surface tension around each particle of dirt so that the water can reach the particles and surround them. The dirt is carried away in rinsing. Soap is a good cleaner only in the presence of water. [ previous page ] [ next page ] OneStep Calculations Pressure = weight density × depth (Neglect the pressure due to the atmosphere in the calculations below.) 1. Calculate the water pressure at the bottom of the 100mhigh water tower shown in Figure 13.2. 2. Calculate the water pressure at the base of a dam when the depth of water behind the dam is 100 m. 3. The top floor of a building is 50 m above the basement. Calculate how much greater the water pressure is in the basement compared with the pressure at the top floor. 4. Water pressure at the bottom of a 1mtall closed barrel is 98 kPa. What is the pressure at the bottom of the barrel when a 5m pipe filled with water is inserted into the top of the barrel? Exercises 1. What common liquid covers more than twothirds of our planet, makes up 60% of our bodies, and sustains our lives and lifestyles in countless ways? 2. Which is more likely to hurt—being stepped on by a 200lb man wearing loafers or being stepped on by a 100lb woman wearing high heels? 3. Which do you suppose exerts more pressure on the ground—an elephant or a lady standing on spike heels? (Which will be more likely to make dents in a linoleum floor?) Approximate a rough calculation for each. 4. Stand on a bathroom scale and read your weight. When you lift one foot up so that you’re standing on one foot, does the reading change? Does a scale read force or pressure? 5. Why are persons who are confined to bed less likely to develop bedsores on their bodies if they use a waterbed rather than an ordinary mattress? 6. You know that a sharp knife cuts better than a dull knife. Do you know why this is so? Defend your answer. 7. If water faucets upstairs and downstairs are turned fully on, will more water per second flow out of the upstairs faucets or the downstairs faucets? 8. The photo shows physics instructor Marshall Ellenstein walking barefoot on broken glass bottles in his class. What physics concept is Marshall demonstrating, and why is he careful that the broken pieces are small and numerous? (The BandAids on his feet are for humor!) (Click image to enlarge) 9. Why does your body get more rest when you’re lying down than it does when you’re sitting? And why is blood pressure measured in the upper arm, at the elevation of your heart? Is blood pressure in your legs greater? 10. When standing, blood pressure in your legs is greater than in your upper body. Would this be true for an astronaut in orbit? Defend your answer. 11. How does water pressure 1 meter beneath the surface of a lake compare with water pressure 1 meter beneath the surface of a swimming pool? 12. Which teapot holds more liquid? (Click image to enlarge) 13. The sketch shows a reservoir that supplies water to a farm. It is made of wood and is reinforced with metal hoops. (a) Why is it elevated? (b) Why are the hoops closer together near the bottom part of the tank? (Click image to enlarge) 14. A block of aluminum with a volume of 10 cm is placed in a beaker of water filled to the b3im. Water overflows. The same is done in another beaker with a 10cm block of lead. Does the lead displace more, less, or the same amount of water? 15. A block of aluminum with a mass of 1 kg is placed in a beaker of water filled to the brim. Water overflows. The same is done in another beaker with a 1kg block of lead. Does the lead displace more, less, or the same amount of water? 16. A block of aluminum with a weight of 10 N is placed in a beaker of water filled to the brim. Water overflows. The same is done in another beaker with a 10N block of lead. Does the lead displace more, less, or the same amount of water? (Why are your answers to this exercise and to Exercise 15 different from your answer to Exercise 14?) 17. In 1960, the U.S. Navy’s bathyscaphe Trieste (a submersible) descended to a depth of nearly 11 kilometers in the Marianas Trench near the Philippines in the Pacific Ocean. Instead of a large viewing window, it was a small circular window 15 centimeters in diameter. What is your explanation for so small a window? 18. There is a story about Pascal in which it is said that he climbed a ladder and poured a small container of water into a tall, thin, vertical pipe inserted into a wooden barrel full of water below. The barrel burst when the water in the pipe reached about 12 m. This was all the more intriguing because the weight of added water in the tube was very small. What two physical principles was Pascal demonstrating? 19. There is a legend of a Dutch boy who bravely held back the whole North Sea by plugging a hole in a dike with his finger. Is this possible and reasonable? (See also Problem 4.) 20. If you’ve wondered about the flushing of toilets on the upper floors of city skyscrapers, how do you suppose the plumbing is designed so that there is not an enormous impact of sewage arriving at the basement level? (Check your speculations with someone who is knowledgeable about architecture.) 21. Why does water “seek its own level”? 22. Suppose that you wish to lay a level foundation for a home on hilly and bushy terrain. How can you use a garden hose filled with water to determine equal elevations for distant points? 23. When you are bathing on a stony beach, why do the stones hurt your feet less when you’re standing in deep water? 24. If liquid pressure were the same at all depths, would there be a buoyant force on an object submerged in the liquid? Explain. 25. A can of diet soda floats in water, whereas a can of regular soda sinks. Explain this phenomenon first in terms of density, then in terms of weight versus buoyant force. 26. Why will a block of iron float in mercury but sink in water? 27. The mountains of the Himalayas are slightly less dense than the mantle material upon which they “float.” Do you suppose that, like floating icebergs, they are deeper than they are high? 28. Why is a high mountain composed mostly of lead an impossibility on the planet Earth? 29. How much force is needed to push a nearly weightless but rigid 1L carton beneath a surface of water? 30. Why will a volleyball held beneath the surface of water have more buoyant force than if it is floating? 31. Why does an inflated beach ball pushed beneath the surface of water swiftly shoot above the water surface when released? 32. Why is it inaccurate to say that heavy objects sink and that light objects float? Give exaggerated examples to support your answer. 33. Why is the buoyant force on a submerged submarine appreciably greater than the buoyant force on it while it is floating? 34. A piece of iron placed on a block of wood makes it float lower in the water. If the iron were instead suspended beneath the wood, would it float as low, lower, or higher? Defend your answer. (Click image to enlarge) 35. Compared with an empty ship, would a ship loaded with a cargo of Styrofoam sink deeper into the water or rise in the water? Defend your answer. 36. If a submarine starts to sink, will it continue to sink to the bottom if no changes are made? Explain. 37. A barge filled with scrap iron is in a canal lock. If the iron is thrown overboard, does the water level at the side of the lock rise, fall, or remain unchanged? Explain. 38. Would the water level in a canal lock go up or down if a battleship in the lock sank? 39. Will a rock gain or lose buoyant force as it sinks deeper in water? Or will the buoyant force remain the same at greater depths? Defend your answer. 40. Will a swimmer gain or lose buoyant force as she swims deeper in the water? Or will her buoyant force remain the same at greater depths? Defend your answer, and contrast it with your answer to Exercise 39. 41. A balloon is weighted so that it is barely able to float in water. If it is pushed beneath the surface, will it return to the surface, stay at the depth to which it is pushed, or sink? Explain. (Hint: Does the balloon’s density change?) (Click image to enlarge) 42. The density of a rock doesn’t change when it is submerged in water, but your density changes when you are submerged. Explain. 43. In answering the question of why bodies float higher in salt water than in freshwater, your friend replies that the reason is that salt water is denser than freshwater. (Does your friend often answer questions by reciting only factual statements that relate to the answers but don’t provide any concrete reasons?) How would you answer the same question? 44. A ship sailing from the ocean into a freshwater harbor sinks slightly deeper into the water. Does the buoyant force on the ship change? If so, does it increase or decrease? 45. Suppose that you are given the choice between two life preservers that are identical in size, the first a light one filled with Styrofoam and the second a very heavy one filled with gravel. If you submerge these life preservers in the water, upon which will the buoyant force be greater? Upon which will the buoyant force be ineffective? Why are your answers different? 46. The weight of the human brain is about 15 N. The buoyant force supplied by fluid around the brain is about 14.5 N. Does this mean that the weight of fluid surrounding the brain is at least 14.5 N? Defend your answer. 47. The relative densities of water, ice, and alcohol are 1.0, 0.9, and 0.8, respectively. Do ice cubes float higher or lower in a mixed alcoholic drink? What comment can you make about a cocktail in which the ice cubes lie submerged at the bottom of the glass? 48. When an ice cube in a glass of water melts, does the water level in the glass rise, fall, or remain unchanged? Does your answer change if the ice cube has many air bubbles? How about if the ice cube contains many grains of heavy sand? 49. When the wooden block is placed in the beaker, what happens to the scale reading? Answer the same question for an iron block. (Click image to enlarge) 50. A halffilled bucket of water is on a spring scale. Will the reading of the scale increase or remain the same if a fish is placed in the bucket? (Will your answer be different if the bucket is initially filled to the brim?) 51. The weight of the container of water, as shown in a, is equal to the weight of the stand and the suspended solid iron ball. When the suspended ball is lowered into the water, as shown in b, the balance is upset. Will the additional weight needed on the right side to restore balance be greater than, equal to, or less than the weight of the solid iron ball? (Click image to enlarge) 52. If the gravitational field of the Earth were to increase, would a fish float to the surface, sink, or stay at the same depth? 53. What would you experience when swimming in water in an orbiting space habitat where simulated gravity is g? Would you float in the water as you do on Earth? 54. We say that the shape of a liquid is that of its container. But, with no container and no gravity, what is the natural shape of a blob of water? Why? 55. If you release a PingPong ball beneath the surface of water, it will rise to the surface. Would it do the same if it were inside a big blob of water floating weightless in an orbiting spacecraft? 56. So you’re on a run of bad luck, and you slip quietly into a small, quiet pool as hungry crocodiles lurking at the bottom are relying on Pascal’s principle to help them to detect a tender morsel. What does Pascal’s principle have to do with their delight at your arrival? 57. In the hydraulic arrangement shown, the larger piston has an area that is fifty times that of the smaller piston. The strong man hopes to exert enough force on the large piston to raise the 10 kg that rest on the small piston. Do you think he will be successful? Defend your answer. (Click image to enlarge) 58. In the hydraulic arrangement shown in Figure 13.22, the multiplication of force is equal to the ratio of the areas of the large and small pistons. Some people are surprised to learn that the area of the liquid surface in the reservoir of the arrangement shown in Figure 13.23 is immaterial. What is your explanation to clear up this confusion? 59. Why will hot water leak more readily than cold water through small leaks in a car radiator? 60. On the surface of a pond, it is common to see water striders, insects that can “walk” on the surface of water without sinking. What physics concept explains their ability? [ previous page ] [ next page ] 2010 Pearson Education, Inc. Problems 1. The depth of water behind the Hoover Dam in Nevada is 220 m. What is the water pressure at the base of this dam? (Neglect the pressure due to the atmosphere.) 2. A 6kg piece of metal displaces 1 liter of water when submerged. What is its density? 3. A rectangular barge, 5 m long and 2 m wide, floats in freshwater. (a) Find how much deeper it floats when its load is a 400kg horse. (b) If the barge can only be pushed 15 cm deeper into the water before water overflows to sink it, how many 400kg horses can it carry? 4. A dike in Holland springs a leak through a hole of area 1 cm at a depth of 2 m below the water surface. How much force must a boy apply to the hole with his thumb to stop the leak? Could he do it? 5. A merchant in Kathmandu sells you a solidgold 1kg statue for a very reasonable price. When you return home, you wonder whether or not you got a bargain, so you lower the statue into a container of water and measure the volume of displaced water. What volume will verify that it’s pure gold? 6. When a 2.0kg object is suspended in water, it “masses” 1.5 kg. What is the density of the object? 7. An ice cube measures 10 cm on a side and floats in water. One cm extends above water level. If you shaved off the 1cm part, how many cm of the remaining ice would extend above water level? 8. A swimmer wears a heavy belt to make her average density exactly equal to the density of water. Her mass, including the belt, is 60 kg. (a) What is the swimmer’s weight in newtons? (b) What is the swimmer’s volume in m ? (c) At a depth of 2 m below the surface of a pond, what buoyant force acts on the swimmer? What net force acts on her? 9. A vacationer floats lazily in the ocean with 90% of his body below the surface. The density of the ocean water is 1,025 kg/m . What is the vacationer’s average density? 10. In the hydraulic pistons shown in the sketch, the small piston has a diameter of 2 cm. The larger piston has a diameter of 6 cm. How much more force can the larger piston exert compared with the force applied to the smaller piston? Solutions to Chapter 13 Exercises 1. Water. 2. Pressure would be appreciably greater by the woman, which would hurt you more. 3. A woman with spike heels exerts considerably more pressure on the ground than an elephant! Example: A 500-N woman with 1-cm spike heels puts half her weight on each foot, distributed (let’s say) half on her heel and half on her sole. So the pressure exerted by each heel will be (125 N/1 cm ) = 125 N/cm . A 20,000-N elephant with 1000 cm feet exerting / 14 its weight on each foot 2 2 produces (5000N/1000 cm ) = 5N/cm ; about 25 times less pressure. (So a woman with spike heels will make greater dents in a new linoleum floor than an elephant will.) 4. The scale measures force, not pressure, and is calibrated to read your weight. That’s why your weight on the scale is the same whether you stand on one foot or both. 5. There is less pressure with a waterbed due to the greater contact area. 6. A sharp knife cuts better than a dull knife because it has a thinner cutting area which results in more cutting pressure for a given force. 7. More water will flow from a downstairs open faucet because of the greater pressure. Since pressure depends on depth, the downstairs faucet is effectively “deeper” than the upstairs faucet. The pressure downstairs is greater by an amount = weight density depth, where the depth is the vertical distance between faucets. 8. The concept of pressure is being demonstrated. He is careful that the pieces are small and numerous so that his weight is applied over a large area of contact. Then the sharp glass provides insufficient pressure to cut the feet. 9. Your body gets more rest when lying than when sitting or standing because when lying, the heart does not have to pump blood to the heights that correspond to standing or sitting. Blood pressure is normally greater in the lower parts of your body simply because the blood is “deeper” there. Since your upper arms are at the same level as your heart, the blood pressure in your upper arms will be the same as the blood pressure in your heart. 10. No, in orbit there are no pressure differences due to gravity. 11. Both are the same, for pressure depends on depth. 12. The water can be no deeper than the spouts, which are at the same height, so both teapots hold the same amount of liquid. 13. (a) The reservoir is elevated so as to produce suitable water pressure in the faucets that it serves. (b) The hoops are closer together at the bottom because the water pressure is greater at the bottom. Closer to the top, the water pressure is not as great, so less reinforcement is needed there. 14. Both blocks have the same volume and therefore displace the same amount of water. 15. A one-kilogram block of aluminum is larger than a one-kilogram block of lead. The aluminum therefore displaces more water. 16. A 10-N block of aluminum is larger than a 10-N block of lead. The aluminum therefore displaces more water. Only in Exercise 10 were the volumes of the block equal. In this and the preceding exercise, the aluminum block is larger. (These exercises serve only to emphasize the distinctions between volume, mass, and weight.) 17. The smaller the window area, the smaller the crushing force of water on it. 18. One, that water pressure depends on depth. The other, that the pressure due to the column of water is transmitted to all parts of the barrel. 19. From a physics point of view, the event was quite reasonable, for the force of the ocean on his finger would have been quite small. This is because the pressure on his finger has only to do with the depth of the water, specifically the distance of the leak below the sea level—not the weight of the ocean. For a numerical example, see Problem 4. 20. A typical plumbing design involves short sections of pipe bent at 45-degree angles between vertical sections two-stories long. The sewage therefore undergoes a succession of two-story falls which results in a moderate momentum upon reaching the basement level. 21. Water seeking its own level is a consequence of pressure depending on depth. In a bent U-tube full of water, for example, the water in one side of the tube tends to push water up the other side until the pressures at the same depth in each tube are equal. If the water levels were not the same, there would be more pressure at a given level in the fuller tube, which would move the water until the levels were equal. 22. The use of a water-filled garden hose as an elevation indicator is a practical example of water seeking its own level. The water surface at one end of the hose will be at the same elevation above sea level as the water surface at the other end of the hose. 23. In deep water, you are buoyed up by the water displaced and as a result, you don’t exert as much pressure against the stones on the bottom. When you are up to your neck in water, you hardly feel the bottom at all. 24. Buoyant force is the result of differences in pressure; if there are no pressure differences, there is no buoyant force. This can be illustrated by the following example: A Ping-Pong ball pushed beneath the surface of water will normally float back to the surface when released. If the container of water is in free fall, however, a submerged Ping-Pong ball will fall with the container and make no attempt to reach the surface. In this case there is no buoyant force acting on the ball because there are no pressure differences—the local effects of gravity are absent. 25. The diet drink is less dense than water, whereas the regular drink is denser than water. (Water with dissolved sugar is denser than pure water.) Also, the weight of the can is less than the buoyant force that would act on it if totally submerged. So it floats, where buoyant force equals the weight of the can. 3 26. Mercury is more dense (13.6 g/cm ) than iron. A block of iron will displace its weight and still be partially above the mercury surface. Hence it floats in mercury. In water it sinks because it cannot displace its weight. 27. Mountain ranges are very similar to icebergs: Both float in a denser medium, and extend farther down into that medium than they extend above it. Mountains, like icebergs, are bigger than they appear to be. The concept of floating mountains is isostacy—Archimedes’ principle for rocks. 28. A mostly-lead mountain would be more dense than the mantle and would sink in it. Guess where most of the iron in the world is. In the Earth’s center! 29. The force needed will be the weight of 1 L of water, which is 9.8 N. If the weight of the carton is not negligible, then the force needed would be 9.8 N minus the carton’s weight, for then the carton would be “helping” to push itself down. 30. When the ball is held beneath the surface, it displaces a greater weight of water. 31. The buoyant force on the ball beneath the surface is much greater than the force of gravity on the ball, producing a large net force and large acceleration. 32. Heavy objects may or may not sink, depending on their densities (a heavy log floats while a small rock sinks, or a boat floats while a paper clip sinks, for example). People who say that heavy objects sink really mean that dense objects sink. Be careful to distinguish between how heavy an object is and how dense it is. 33. While floating, BF equals the weight of the submarine. When submerged, BF equals the submarine’s weight plus the weight of water taken into its ballast tanks. Looked at another way, the submerged submarine displaces a greater weight of water than the same submarine floating. 34. The block of wood would float higher if the piece of iron is suspended below it rather than on top of it. By the law of flotation: The iron-and-wood unit displaces its combined weight and the same volume of water whether the iron is on top or the bottom. When the iron is on the top, more wood is in the water; when the iron is on the bottom, less wood is in the water. Or another explanation is that when the iron is below—submerged—buoyancy on it reduces its weight and less of the wood is pulled beneath the water line. 35. When a ship is empty its weight is least and it displaces the least water and floats highest. Carrying a load of anything increases its weight and makes it float lower. It will float as low carrying a few tons of Styrofoam as it will carrying the same number of tons of iron ore. So the ship floats lower in the water when loaded with Styrofoam than when empty. If the Styrofoam were outside the ship, below water line, then the ship would float higher as a person would with a life preserver. 36. A sinking submarine will continue to sink to the bottom so long as the density of the submarine is greater than the density of the surrounding water. If nothing is done to change the density of the submarine, it will continue to sink because the density of water is practically constant. In practice, water is sucked into or blown out of a submarine’s tanks to adjust its density to match the density of the surrounding water. 37. The water level will fall. This is because the iron will displace a greater amount of water while being supported than when submerged. A floating object displaces its weight of water, which is more than its own volume, while a submerged object displaces only its volume. (This may be illustrated in the kitchen sink with a dish floating in a dishpan full of water. Silverware in the dish takes the place of the scrap iron. Note the level of water at the side of the dishpan, and then throw the silverware overboard. The floating dish will float higher and the water level at the side of the dishpan will fall. Will the volume of the silverware displace enough water to bring the level to its starting point? No, not as long as it is denser than water.) 38. For the same reason as in the previous exercise, the water level will fall. (Try this one in your kitchen sink also. Note the water level at the side of the dishpan when a bowl floats in it. Tip the bowl so it fills and submerges, and you’ll see the water level at the side of the dishpan fall.) 39. Bouyant force will remain unchanged on the sinking rock because it displaces the same weight of water at any depth. 40. Bouyant force on a sinking swimmer will decrease as she sinks. This is because her body, unlike the rock in the previous exercise, will be compressed by the greater pressure of greater depths. 41. The balloon will sink to the bottom because its density increases with depth. The balloon is compressible, so the increase in water pressure beneath the surface compresses it and reduces its volume, thereby increasing its density. Density is further increased as it sinks to regions of greater pressure and compression. This sinking is understood also from a buoyant force point of view. As its volume is reduced by increasing pressure as it descends, the amount of water it displaces becomes less. The result is a decrease in the buoyant force that initially was sufficient to barely keep it afloat. 42. You are compressible, whereas a rock is not, so when you are submerged, the water pressure tends to squeeze in on you and reduce your volume. This increases your density. (Be careful when swimming—at shallow depths you may still be less dense than water and be buoyed to the surface without effort, but at greater depths you may be pressed to a density greater than water and you’ll have to swim to the surface.) 43. A body floats higher in denser fluid because it does not have to sink as far to displace a weight of fluid equal to its own weight. A smaller volume of the displaced denser fluid is able to match the weight of the floating body. 44. The buoyant force does not change. The buoyant force on a floating object is always equal to that object’s weight, no matter what the fluid. 45. Since both preservers are the same size, they will displace the same amount of water when submerged and be buoyed up with equal forces. Effectiveness is another story. The amount of buoyant force exerted on the heavy gravel-filled preserver is much less than its weight. If you wear it, you’ll sink. The same amount of buoyant force exerted on the lighter Styrofoam preserver is greater than its weight and it will keep you afloat. The amount of the force and the effectiveness of the force are two different things. 46. No, there does not have to actually be 14.5 N of fluid in the skull to supply a buoyant force of 14.5 N on the brain. To say that the buoyant force is 14.5 N is to say that the brain is taking up the space that 14.5 N of fluid would occupy if fluid instead of the brain were there. The amount of fluid in excess of the fluid that immediately surrounds the brain does not contribute to the buoyancy on the brain. (A ship floats the same in the middle of the ocean as it would if it were floating in a small lock just barely larger than the ship itself. As long as there is enough water to press against the hull of the ship, it will float. It is not important that the amount of water in this tight-fitting lock weigh as much as the ship—think about that, and don’t let a literal word explanation “a floating object displaces a weight of fluid equal to its own weight” and the idea it represents confuse you.) 47. Ice cubes will float lower in a mixed drink because the mixture of alcohol and water is less dense than water. In a less dense liquid a greater volume of liquid must be displaced to equal the weight of the floating ice. In pure alcohol, the volume of alcohol equal to that of the ice cubes weighs less than the ice cubes, and buoyancy is less than weight and ice cubes will sink. Submerged ice cubes in a cocktail indicate that it is predominantly alcohol. 48. When the ice cube melts the water level at the side of the glass is unchanged (neglecting temperature effects). To see this, suppose the ice cube to be a 5 gram cube; then while floating it will displace 5 grams of water. But when melted it becomes the same 5 grams of water. Hence the water level is unchanged. The same occurs when the ice cube with the air bubbles melts. Whether the ice cube is hollow or solid, it will displace as much water floating as it will melted. If the ice cube contains grains of heavy sand, however, upon melting, the water level at the edge of the glass will drop. This is similar to the case of the scrap iron of Exercise 38. 49. The total weight on the scale is the same either way, so the scale reading will be the same whether or not the wooden block is outside or floating in the beaker. Likewise for an iron block, where the scale reading shows the total weight of the system. 50. If water doesn’t overflow, the reading on the scale will increase by the ordinary weight of the fish. However, if the bucket is brim filled so a volume of water equal to the volume of the fish overflows, then the reading will not change. We assume here that the fish and water have the same density. 51. When the ball is submerged (but not touching the bottom of the container), it is supported partly by the buoyant force on the left and partly by the string connected to the right side. So the left pan must increase its upward force to provide the buoyant force in addition to whatever force it provided before, and the right pan’s upward force decreases by the same amount, since it now supports a ball lighter by the amount of the buoyant force. To bring the scale back to balance, the additional weight that must be put on the right side will equal twice the weight of water displaced by the submerged ball. Why twice? Half of the added weight makes up for the loss of upward force on the right, and the other half for the equal gain in upward force on the left. (If each side initially weighs 10 N and the left side gains 2 N to become 12 N, the right side loses 2 N to become 8 N. So an additional weight of 4 N, not 2 N, is required on the right side to restore balance.) Because the density of water is less than half the density of the iron ball, the restoring weight, equal to twice the buoyant force, will still be less than the weight of the ball. 52. If the gravitational field of the Earth increased, both water and fish would increase in weight and weight density by the same factor, so the fish would stay at its prior level in water. 53. Both you and the water would have half the weight density as on Earth, and you would float with the same proportion of your body above the water as on Earth. Water splashed upward with a certain initial speed would rise twice as high, since it would be experiencing only half the “gravity force.” Waves on the water surface would move more slowly than on Earth (at about 70% as fast since v √g). wave~ 54. Because of surface tension, which tends to minimize the surface of a blob of water, its shape without gravity and other distorting forces will be a sphere—the shape with the least surface area for a given volume. 55. A Ping-Pong ball in water in a zero-g environment would experience no buoyant force. This is because buoyancy depends on a pressure difference on different sides of a submerged body. In this weightless state, no pressure difference would exist because no water pressure exists. (See the answer to Exercise 20, and Home Project 2.) 56. Part of whatever pressure you add to the water is transmitted to the hungry crocodiles, via Pascal’s principle. If the water were confined, that is, not open to the atmosphere, the crocs would receive every bit of pressure you exert. But even if you were able to slip into the pool to quietly float without exerting pressure via swimming strokes, your displacement of water raises the water level in the pool. This ever-so-slight rise, and accompanying ever-so-slight increase in pressure at the bottom of the pool, is an ever-so-welcome signal to the hungry crocodiles. 57. The strong man will be unsuccessful. He will have to push with 50 times the weight of the 10 kilograms. The hydraulic arrangement is arranged to his disadvantage. Ordinarily, the input force is applied against the smaller piston and the output force is exerted by the large piston—this arrangement is just the opposite. 58. In Figure 13.21, the increased pressure in the reservoir is a result of the applied force distributed over the input piston area. This increase in pressure is transmitted to the output piston. In Figure 13.23, however, the pressure increase is supplied by the mechanical pump, which has nothing to do with the area of fluid interface between the compressed air and the liquid. 59. When water is hot, the molecules are moving more rapidly and do not cling to one another as well as when they are slower moving, so the surface tension is less. The lesser surface tension of hot water allows it to pass more readily through small openings. 60. Surface tension accounts for the walking of water striders, needles that appear to float, and even razor blades that also appear to float. In these cases the weights of the objects are less than the restoring forces in the water surface that tends to resist stretching. Chapter 13 Problem Solutions 1. Pressure = weight density depth = 10,000 N/m 220 m = 2,200,000 N/m = 2200 kPa (or for 3 density = 9800 N/m , pressure = 2160 kPa. 2. Density = m/V = 6 kg/1 liter = 6 kg/liter. (Since there are 1000 liters in 1 cubic meter, density 3 3 3 may be expressed in units kg/m . Density = 6 kg/1 liter 1000 liter/m = 6000 kg/m , six times the density of water.) 3. (a) The volume of the extra water displaced will weigh as much as the 400-kg horse. And the volume of extra water displaced will also equal the area of the barge times the extra depth. That is, V = Ah, where A is the horizontal area of the barge; Then h = . Now A = 5m 2m = 10 m ; to find the volume V of barge pushed into the water by the horse’s weight, which equals the volume of water displaced, we know that 3 density = . Or from this, V = = = 0.4 m . So h = = = 0.04 m, which is 4 cm deeper. (b) If each horse will push the barge 4 cm deeper, the question becomes: How many 4-cm increments will make 15 cm? 15/4 = 3.75, so 3 horses can be carried without sinking. 4 horses will sink the barge. 3 4. First you must find the pressure. It is weight density depth = (10,000 N/m )(2 m) = 20,000 N/m , or 20,000 Pa. Force is pressure area, and 1 cm = 10 2 -4 m , so F = (20,000 N/m ) 2 (10 -4m ) = 2 N. It would be easy for the boy to exert this force. It is about the weight of a notebook or a small box of cereal. (Note: Air pressure is not figured into this calculation because its effect in pushing down on the water from above is canceled by its effect in pushing from outside the hole against the leaking water.) 5. From Table 12.1 the density of gold is 19.3 g/cm . Your gold has a mass of 1000 grams, so = 19.3 g/cm . Solving for V, 3 V = = 51.8 cm . 6. Density = = = 3 3 -3 3 3 = 4 kg/liter. And since 1 liter = 10 cm = 10 m , density = 4,000 kg/m . (Or this can be reasoned as follows: The buoyant force on the object is the force needed to support 0.5 kg, so 0.5 kg of water is displaced. Since density is mass/volume, volume is 3 -4 3 mass/density, and displaced volume = (0.5 kg)/(1000 kg/m ) = 5 10 m . The object’s volume is the same as the volume it displaces, so the object’s density is mass/volume = (2 kg)/(5 10 m ) = 4000 kg/m , four times the density of water.) 7. 10% of ice extends above water. So 10% of the 9-cm thick ice would float above the water line; 0.9 cm. So the ice pops up. Interestingly, when mountains erode they become lighter and similarly pop up! Hence it takes a long time for mountains to wear away. 8. (a) Weight = mg = (60 kg)(10 m/s ) = 600 N (or 588 N if 9.8 m/s is used). 2 (b) She has the same density as water, 1000 kg/m . Since density = mass/volume, volume = mass/density, so volume = (60 kg)/(1000 kg/m ) = 0.06 m . 3 (c) Buoyant force = weight of water displaced = 600 N (or 588 N). Her weight balances the buoyant force, so net force = 0. 9. The displaced water, with a volume 90 percent of the vacationer’s volume, weighs the same as the vacationer (to provide a buoyant force equal to his weight). Therefore his density is 90 percent of the water’s density. Vacationer’s density = (0.90)(1,025 kg/m ) = 923 kg/m . 3 2 2 10. The relative areas are as the squares of the diameters; 6 /2 = 36/4 = 9. The larger piston can lift 9 times the input force to the smaller piston.
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