Solutions: Chapter 5 Exercises
1. 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.
2. 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.
3. Like the loaf of bread in Figure 5.1, its volume is decreased. Its mass stays the same so the density increases. A whale is denser when it swims deeper in the ocean.
4. This is similar to Exercise 3. Whereas a rock is not compressible, you are. When you’re submerged, water pressure squeezes in on you and reduces 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.)
5. A person lying on a waterbed experiences less bodyweight pressure because more of the body is in contact with the supporting surface. The greater area reduces the support pressure.
6. 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.
7. A woman with spike heels exerts considerably more pressure on the ground than an elephant! Example: A 500-N woman with 1-cm2 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 cm2) = 125 N/cm2. A 20,000-N elephant with 1000 cm2 feet exerting 1⁄4 its weight on each foot produces (5000 N/1000 cm2) = 5 N/cm2; about 25 times less pressure. (So a woman with spike heels will make greater dents in a new linoleum floor than an elephant will.)
8. 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.
9. 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.
10. 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.
11. 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.
12. 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.
13. 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 when 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.
14. 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.)
15. 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.)
16. 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.
17. 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.
18. 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 lead-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.
19. 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.
20. 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 14.
21. 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.
22. The weight of a truck is distributed over the part of the tires that make contact with the road. Weight/surface area = pressure, so the greater the surface area, or equivalently, the greater the number of tires, the greater the weight of the truck can be for a given pressure. What pressure? The pressure exerted by the tires on the road, which is determined by (but is somewhat greater than) the air pressure in its tires. Can you see how this relates to Activity 5?
23. The density of air in a deep mine is greater than at the surface. The air filling up the mine adds weight and pressure at the bottom of the mine, and according to Boyle’s law, greater pressure in a gas means greater density.
24. To begin with, the two teams of horses used in the Magdeburg hemispheres demonstration were for showmanship and effect, for a single team and a strong tree would have provided the same force on the hemispheres. So if two teams of nine horses each could pull the hemispheres apart, a single team of nine horses could also, if a tree or some other strong object were used to hold the other end of the rope.
25. If the item is sealed in an air-tight package at sea level, then the pressure in the package is about 1 atmosphere. Cabin pressure is reduced somewhat for high altitude flying, so the pressure in the package is greater than the surrounding pressure and the package therefore puffs outwards.
26. Airplane windows are small because the pressure difference between the inside and outside surfaces result in large net forces that are directly proportional to the window’s surface area. (Larger windows would have to be proportionately thicker to withstand the greater net force—windows on underwater research vessels are similarly small.)
27. Unlike water, air is easily compressed—and also, foam rubber. In fact, air density is proportional to its pressure. So, near Earth’s surface, where the pressure is greater, the air’s density is greater (and a foam brick is more squashed), and at high altitude, where the pressure is less, the air’s density is less.
28. A vacuum cleaner wouldn’t work on the Moon. A vacuum cleaner operates on Earth because the atmospheric pressure pushes dust into the machine’s region of reduced pressure. On the Moon there is no atmospheric pressure to push the dust anywhere.
29. The height would be less. The weight of the column balances the weight of an equal-area column of air. The denser liquid would need less height to have the same weight as the mercury column.
30. Drinking through a straw is slightly more difficult atop a mountain. This is because the reduced atmospheric pressure is less effective in pushing soda up into the straw.
31. You agree with your friend, for the elephant displaces far more air than a small helium-filled balloon, or small anything. The effects of the buoyant forces, however, are a different story. The large buoyant force on the elephant is insignificant relative to its enormous weight. The tiny buoyant force acting on the balloon of tiny weight, however, is significant.
32. One’s lungs, like an inflated balloon, are compressed when submerged in water, and the air within is compressed. Air will not of itself flow from a region of low pressure into a region of higher pressure. The diaphragm in one’s body reduces lung pressure to permit breathing, but this limit is strained when nearly 1 m below the water surface. The limit is exceeded at more than a 1-m depth.
33. The buoyant force does not change, because the volume of the balloon does not change. The buoyant force is the weight of air displaced, and doesn’t depend on what is doing the displacing.
34. An object rises in air only when buoyant force exceeds its weight. A steel tank of anything weighing more than the air it displaces, so it won’t rise. A helium-filled balloon weighs less than the air it displaces and rises.
35. The end supporting the punctured balloon tips upwards as it is lightened by the amount of air that escapes. There is also a loss of buoyant force on the punctured balloon, but that loss of upward force is less than the loss of downward force, since the density of air in the balloon before puncturing was greater than the density of surrounding air.
36. The force of the atmosphere is on both sides of the window; the net force is zero, so windows don’t normally break under the weight of the atmosphere. In a strong wind, however, pressure will be reduced on the windward side (Bernoulli’s Principle) and the forces no longer cancel to zero. Many windows are blown outward in strong winds.
37. The strong man will be unsuccessful. He will have to push with 50 times the weight of the 10 kilograms. That’s 5000 N, more than his weight. The hydraulic arrangement is arranged to his disadvantage. Ordinarily, the input force is applied to the smaller piston and the output force to the large piston. This arrangement is just the opposite.
38. (a) Speed increases (so that the same quantity of gas can move through the pipe in the same time).
(b) Pressure decreases (Bernoulli’s principle).
(c) The spacing between the streamlines decreases, because the same number of streamlines fit in a smaller area.
39. Bernoulli’s Principle. For the moving car the pressure will be less on the side of the car where the air is moving fastest—the side nearest the truck, resulting in the car’s being pushed by the atmosphere towards the truck. Inside the convertible, atmospheric pressure is greater than outside, and the canvas roof top is pushed upwards towards the region of lesser pressure. Similarly for the train windows, where the interior air is at rest relative to the window and the air outside is in motion. Air pressure against the inner surface of the window is greater than the atmospheric pressure outside. When the difference in pressures is significant enough, the window is blown out.
40. An airplane flies upside down by tilting its fuselage so that there is an angle of attack of the wing with oncoming air. (It does the same when flying right side up, but then, because the wings are designed for right-side-up flight, the tilt of the fuselage may not need to be as great.)
Solutions: Chapter 5 Problems