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CHAPTER 13

Liquids

Summary of TermsPressure The ratio of force to the area over which that force is

distributed: Liquid pressure = weight density × depthBuoyant force The net upward force that a fluid exerts on an immersed

object.Archimedes’ principle

An immersed body is buoyed up by a force equal to the weight of the fluid it displaces.

Principle of flotation

A floating object displaces a weight of fluid equal to its own weight.

Pascal’s principle

The pressure applied to a motionless fluid confined in a container is transmitted undiminished throughout the fluid.

Surface tension The tendency of the surface of a liquid to contract in area and thus to behave like a stretched elastic membrane.

Capillarity The rise of a liquid in a fine, hollow tube or in a narrow space.

Tsing Bardin demonstrates the depth dependence of water pressure.

Molecules that make up a liquid are not confined to fixed positions, as they are in solids, but can flow from position to position by sliding over one another. While a solid maintains a particular shape, a liquid takes the shape of its container. Molecules of a liquid are close together and greatly resist compressive forces. Liquids, like solids, are difficult to compress. Gases, as we shall see in the next chapter, are easily compressed. Both liquids and gases can flow, so both are called fluids.

Pressure

A liquid contained in a vessel exerts forces against the walls of the vessel. To discuss the interaction between the liquid and the walls, it is convenient to introduce the concept of pressure. Pressure is a force divided by the area

over which the force is exerted. 1 pressure = force/area As an illustration of the distinction between pressure and force, consider the two blocks in Figure 13.1. The blocks are identical, but one stands on its end and the other stands on its side. Both blocks are of equal weight and therefore exert the same force on the surface (if you were to put them on a bathroom scale, each would register the same), but the upright block exerts a greater pressure against the surface. If the block were tipped up so that its contact with the table were on a single corner, the pressure would be greater

still.

FIGURE 13.1 Although the weight of both blocks is the same, the upright block exerts greater pressure against the table.

Pressure in a Liquid

When you swim under water, you can feel the water pressure acting against your eardrums. The deeper you swim, the greater the pressure. The pressure you feel is due to the weight of water above you. As you swim deeper, there is more water above you and therefore greater pressure. The pressure a liquid exerts depends on its depth.

FIGURE 13.2 The tower does more than store water. The depth of water above ground level ensures substantial and reliable water pressure to the many homes it serves.

Liquid pressure also depends on the density of the liquid. If you were submerged in a liquid more dense than water, the pressure would be correspondingly greater. The pressure due to a liquid is precisely equal to the product of weight density and depth: 2

Liquid pressure = weight density × depth

Simply put, the pressure a liquid exerts against the sides and bottom of a container depends on the density and the depth of the liquid. If we neglect atmospheric pressure, at twice the depth, the liquid pressure against the bottom is twice as great; at three times the depth, the liquid pressure is threefold; and so on. Or, if the liquid is two or three times as dense, the liquid pressure is correspondingly two or three times as great for any given depth. Liquids are

practically incompressible—that is, their volume can hardly be changed by pressure (water volume decreases by only 50 millionths of its original volume for each atmosphere increase in pressure). So, except for small changes produced by temperature, the density of a particular liquid is practically the same at all depths. 3

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FIGURE 13.3 The dependence of liquid pressure on depth is not a problem for the giraffe because of its large heart and its intricate system of valves and elastic, absorbent blood vessels in the brain. Without these structures, it would faint when suddenly raising its head, and it would be subject to brain hemorrhaging when lowering it.

Earth is the only planet in the solar system with a mostly liquid surface—its oceans. If the Earth were a little closer to the Sun, the oceans would turn to vapor. If the Earth were a little farther away, most of its surface, not just the polar regions, would be solid ice. It’s nice that the Earth is located where it is..If you press your hand against a surface, and somebody else presses against your hand in the same direction, then the pressure against the surface is greater than if you pressed alone. Likewise with the atmospheric pressure that presses on the surface of a liquid. The total pressure of a liquid, then, is weight density × depth plus the pressure of the atmosphere. When this distinction is important, we will use the term total pressure. Otherwise, our discussions of liquid pressure refer to pressure without regard to the normally ever-present atmospheric pressure. (You will learn more about atmospheric pressure in the next chapter.) It is important to recognize that the pressure does not depend on the amount of liquid present. Volume is not the key—depth is. The average water pressure acting against a dam depends on the average depth of the water and not on the volume of water held back. For example, the large, shallow lake in Figure 13.4 exerts only half the average pressure that the small, deep pond exerts.

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FIGURE 13.4 The average water pressure acting against the dam depends on the average depth of the water and not on the volume of water held back. The large, shallow lake exerts only one-half the average pressure that the small, deep pond exerts.

You’ll feel the same pressure whether you dunk your head a meter beneath the surface of the water in a small pool or to the same depth in the middle of a large lake. The same is true for a fish. Refer to the connecting vases in Figure 13.5. If we hold a goldfish by its tail and dunk its head a couple of centimeters under the surface, the water pressure on the fish’s head will be the same in any of the vases. If we release the fish and it swims a few centimeters deeper, the pressure on the fish will increase with depth and be the same no matter which vase the fish is in. If the fish swims to the bottom, the pressure will be greater, but it makes no difference what vase it is in. All vases are filled to equal depths, so the water pressure is the same at the bottom of each vase, regardless of its shape or volume. If water pressure at the

bottom of a vase were greater than water pressure at the bottom of a neighboring narrower vase, the greater pressure would force water sideways and then up the narrower vase to a higher level until the pressures at the bottom were equalized. But this doesn’t happen. Pressure is depth dependent, not volume dependent, so we see that there is a reason why water seeks its own level.

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FIGURE 13.5 Liquid pressure is the same for any given depth below the surface, regardless of the shape of the containing vessel. Liquid pressure = weight density × depth (plus the air pressure at the top).

Some ancient pipe systems installed in Rome at the time the aqueducts were being built indicate that not all Romans then believed that water couldn’t flow uphill.

The fact that water seeks its own level can be demonstrated by filling a garden hose with water and holding the two ends at the same height. The water levels will be equal. If one end is raised higher than the other, water will flow out of the lower end, even if it has to flow “uphill” part of the way. This fact was not fully understood by some of the early Romans, who built elaborate aqueducts with tall arches and roundabout routes to ensure that water would always flow slightly downward every place along its route from the reservoir to the city. If pipes were laid in the ground and followed the natural contour of the land, in some places the water would have to flow uphill, and the Romans were skeptical of this. Careful experimentation was not yet the mode; so, with plentiful slave labor, the Romans built unnecessarily elaborate aqueducts.

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FIGURE 13.6 Roman aqueducts assured that water flowed slightly downhill from reservoir to city.

An experimentally determined fact about liquid pressure is that it is exerted equally in all directions. For example, if we are submerged in water, no matter which way we tilt our heads we feel the same amount of water pressure on our ears. Because a liquid can flow, the pressure isn’t only downward. We know pressure acts sideways when we see water spurting sideways from a leak in the side of an upright can. We know pressure also acts upward when we try to push a beach ball beneath the surface of the water. The bottom of a boat is certainly pushed upward by water pressure. When liquid presses against a surface, there is a net force that is perpendicular to the surface. Although pressure doesn’t have a specific direction, force does. Consider the triangular block in Figure 13.7. Focus your attention only at the three points midway along each surface. Water is forced against each point from many directions, only a few of which are indicated. Components of the forces that are not perpendicular to the surface cancel each other out, leaving only a net perpendicular force at each point.

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FIGURE 13.7 The forces of a liquid pressing against a surface add up to a net force that is perpendicular to the surface.

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FIGURE 13.8 The force vectors act perpendicular to the inner container surface and increase with increasing depth.

That’s why water spurting from a hole in a bucket initially exits the bucket in a direction at right angles to the surface of the bucket in which the hole is located. Then it curves downward due to gravity. The force exerted by a fluid on a smooth surface is always at right angles to the surface. 4

WATER DOWSING

The practice of water dowsing, which goes back to ancient times in Europe and Africa, was carried across the Atlantic to America by some of the earliest settlers. Dowsing refers to the practice of using a forked stick, or a rod, or some similar device to locate underground water, minerals, or hidden treasure. In the classic method of dowsing, each hand grasps a fork, with palms upward. The pointed end of the stick points skyward at an angle of about 45°. The dowser walks back and forth over the area to be tested, and, when passing over a

source of water (or whatever else is being sought), the stick is supposed to rotate downward. Some dowsers have reported an attraction so great that blisters formed on their hands. Some claim special powers that enable them to “see” through soil and rock, and some are mediums who go into trances when conditions are especially favorable. Although most dowsing is done at the actual site, some dowsers claim to be able to locate water simply by passing the stick over a map.

Since drilling a well is an expensive process, the dowser’s fee is usually seen as reasonable. The practice is widespread, with thousands of dowsers active in the United States. This is because dowsing works. The dowser can hardly miss—not because of special powers, but because groundwater is within 100 meters of the surface at almost every spot on Earth.

If you drill a hole into the ground, you’ll find that the wetness of the soil varies with depth. Near the surface, pores and open spaces are filled mostly with air. Deeper, the pores are saturated with water. The upper boundary of this water-saturated zone is called the water table. It usually rises and falls with the contour of the surface topography. Wherever you see a natural lake or pond, you’re seeing a place where the water table extends above the surface of the land.

The depth, quantity, and quality of water below the water table are studied by hydrologists, who rely on a variety of technological techniques—water dowsing not being one of them. Findings of the U.S. Geological Survey conclude that water dowsing falls into the category of pseudoscience. As mentioned in Chapter 1, the real test of a water dowser would be finding a location in which water can’t be found.

Buoyancy

Anyone who has ever lifted a heavy submerged object out of water is familiar with buoyancy, the apparent loss of weight experienced by objects submerged in a liquid. For example, lifting a large boulder off the bottom of a riverbed is a relatively easy task as long as the boulder is below the surface. When it is lifted above the surface, however, the force required to lift it is increased considerably. This is because, when the boulder is submerged, the water exerts an upward force on it that is exactly opposite to the direction of gravity’s pull. This upward force is called the buoyant force, and it is a consequence of pressure increasing with depth. Figure 13.9 shows why the buoyant force acts upward. Forces due to water pressures are exerted everywhere against the boulder in a direction perpendicular to its surface, as shown by the vectors. Force vectors against the sides at equal depths cancel one another, so there is no horizontal buoyant force. Force vectors in the vertical direction, however, don’t cancel. Pressure is greater against the bottom of the boulder because the bottom is deeper. So upward forces against the bottom are greater than downward forces against the top, producing a net force upward—the buoyant force.

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FIGURE 13.9

The greater pressure against the bottom of a submerged object produces an upward buoyant force.

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Understanding buoyancy requires understanding the expression “volume of water displaced.” If a stone is placed in a container that is brimful of water, some water will overflow (Figure 13.10). Water is displaced by the stone. A little thought will tell us that the volume of the stone—that is, the amount of space it takes up—is equal to the volume of the water displaced. If you place any object in a container partly filled with water, the level of the surface rises (Figure 13.11). By how much? By exactly the same amount as if a volume of water were poured in that equals the volume of the submerged object. This is a good method for determining the volume of irregularly shaped objects: A completely submerged object always displaces a volume of liquid equal to its own volume.

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FIGURE 13.10 When a stone is submerged, it displaces water that has a volume equal to the

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FIGURE 13.11 The increase in water level is the same as if you poured in a volume of water equal to the stone’s volume.

CHECK YOURSELFA recipe calls for a specific amount of butter. How does the displacement method relate to the use of a kitchen measuring cup?

CHECK YOUR ANSWERPut some water in the cup before you add the butter. Note the water-level reading on the side of the cup. Then add the butter and watch the water level rise. Because butter floats, poke it beneath the surface. When you subtract the lower-level reading from the higher-level reading, you know the volume of the butter.

© 2010 Pearson Education, Inc.

Archimedes’ Principle

The relationship between buoyancy and displaced liquid was first discovered in the third century BC by the Greek scientist Archimedes. It is stated as follows:An immersed object is buoyed up by a force equal to the weight of the fluid it displaces.If you stick your foot in water, it’s immersed. If you jump in and sink and immersion is total, you’re submerged.This relationship is called Archimedes’ principle. It is true of liquids and gases, both of which are fluids. If an immersed object displaces 1 kilogram of fluid, the buoyant force acting on it is equal to the weight of 1 kilogram. 5 By immersed, we mean either completely or partially submerged. If we immerse a sealed 1-liter container halfway into the water, it will displace a half-liter of water and be buoyed up by a force equal to the weight of a half-liter of water—no matter what is in the container. If we immerse it completely (submerge

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it), it will be buoyed up by a force equivalent to the weight of a full liter of water (1 kilogram of mass). If the container is fully submerged and doesn’t compress, the buoyant force will equal the weight of 1 kilogram of water at any depth. This is because, at any depth, the container can displace no greater volume of water than its own volume. And the weight of this displaced water (not the weight of the submerged object!) is equal to the buoyant force.

A liter of water occupies a volume of 1000 cm3, has a mass of 1 kg, and weighs 9.8 N. Its density may therefore be expressed as 1 kg/L and its weight density as 9.8 N/L. (Seawater is slightly denser, about 10.0 N/L).

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If a 30-kilogram object displaces 20 kilograms of fluid upon immersion, its apparent weight will be equal to the weight of 10 kilograms (98 newtons). Note that, in Figure 13.13, the 3-kilogram block has an apparent weight equal to the weight of 1 kilogram when submerged. The apparent weight of a submerged object is its usual weight in air minus the buoyant force.

CHECK YOURSELF

1.Does Archimedes’ principle tell us that, if an immersed object displaces liquid weighing 10 N, the buoyant force on the object is 10 N?

2. A 1-liter container completely filled with lead has a mass of 11.3 kg and is submerged in water. What is the buoyant force acting on it?

3. A boulder is thrown into a deep lake. As it sinks deeper and deeper into the water, does the buoyant force upon it increase or decrease?

4. Since buoyant force is the net force that a fluid exerts on a body and, as we learned in Chapter 4, that net force produces acceleration, why doesn’t a submerged body accelerate?

CHECK YOUR ANSWERS

1.Yes. Looking at it another way, the immersed object pushes 10 N of fluid aside. The displaced fluid reacts by pushing back on the immersed object with 10 N.

2. The buoyant force is equal to the weight of the liter of water displaced. One L of water has a mass of 1 kg, and a weight of 9.8 N. So the buoyant force on it is 9.8 N. (The 11.3 kg of the lead is irrelevant; 1 L of anything submerged in water will displace 1 L and be buoyed upward with a force of 9.8 N, the weight of 1 kg.)

3. Buoyant force remains unchanged as the boulder sinks, because the boulder displaces the same volume of water at any depth.

4. A submerged body does accelerate if the buoyant force is not balanced by other forces acting on it—the force of gravity and fluid resistance. The net force on a submerged body is the result of the net force the fluid exerts (buoyant force), the weight of the body, and, if the submerged body is moving, the force of fluid friction.

Perhaps your instructor will summarize Archimedes’ principle by way of a numerical example to show that the difference between the upward-acting and the downward-acting forces due to the similar pressure differences on a submerged cube is numerically identical to the weight of fluid displaced. It makes no difference how deep the cube is placed because, although the pressures are greater with increasing depths, the difference between the pressure up against the bottom of the cube and the pressure down against the top of the cube is the same at any depth (Figure 13.14). Whatever the shape of the submerged body, the buoyant force is equal to the weight of fluid displaced.

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FIGURE 13.14

The difference in the upward and downward force acting on the submerged block is the same at any depth.

What Makes an Object Sink or Float?

It’s important to remember that the buoyant force acting on a submerged object depends on the volume of the object. Small objects displace small amounts of water and are acted on by small buoyant forces. Large objects displace large amounts of water and are acted on by larger buoyant forces. It is the volume of the submerged object—not its weight—that determines the buoyant force. The buoyant force is equal to the weight of the volume of fluid displaced. (Misunderstanding this idea is the root of much confusion that people have about buoyancy.) The weight of an object does play a role, however, in floating. Whether an object will sink or float in a liquid depends on how the buoyant force compares with the object’s weight. This, in turn, depends on the object’s density. Consider these three simple rules: 1. An object more dense than the fluid in which it is immersed will sink. 2. An object less dense than the fluid in which it is immersed will float. 3. An object having a density equal to the density of the fluid in which it is

immersed will neither sink nor float.

People who can’t float are, nine times out of ten, males. Most males are


more muscular and slightly denser than females. Also, cans of diet soda float, whereas cans of regular soda sink in water. What does this tell you about their relative densities?

Rule 1 seems reasonable enough, for objects denser than water sink to the bottom, regardless of the water’s depth. Scuba divers near the bottoms of deep bodies of water may sometimes encounter a waterlogged piece of wood hovering above the ocean floor (with a density equal to that of water at that depth), but never do they encounter hovering rocks! From Rules 1 and 2, what can you say about people who, try as they may, cannot float? They’re simply too dense! To float more easily, you must reduce your density. The formula weight density = weight/volume says you must either reduce your weight or increase your volume. Wearing a life jacket increases volume while correspondingly adding very little to your weight. It reduces your overall density. Rule 3 applies to fish, which neither sink nor float. A fish normally has the same density as water. A fish can regulate its density by expanding and contracting an air sac in its body that changes its volume. The fish can move upward by increasing its volume (which decreases its density) and downward by contracting its volume (which increases its density). For a submarine, weight, not volume, is varied to achieve the desired density. Water is taken into or blown out of its ballast tanks. Similarly, the overall density of a crocodile increases when it swallows stones. From 4 to 5 kilograms of stones have been found in the stomachs of large crocodiles. Because of this increased density, the crocodile swims lower in the water, thus exposing itself less to its prey. (Figure 13.15).

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FIGURE 13.15 (left) A crocodile coming toward you in the water. (right) A stoned crocodile coming toward you in the water.

CHECK YOURSELF

1.Two solid blocks of identical size are submerged in water. One block is lead and the other is aluminum. Upon which is the buoyant force greater?

2. If a fish makes itself denser, it will sink; if it makes itself less dense, it will rise. In terms of buoyant force, why is this so?

CHECK YOUR ANSWERS

1.The buoyant force is the same on each, for both blocks displace the same volume of water. For submerged objects, only the volume of water displaced, not the object’s weight, determines the buoyant force.

2. When the fish increases its density by decreasing its volume, it displaces less water, so the buoyant force decreases. When the fish decreases its density by expanding, it displaces a greater volume of water, and the buoyant force increases.

Flotation

Primitive peoples made their boats of wood. Could they have conceived of an iron ship? We don’t know. The idea of floating iron might have seemed strange. Today it is easy for us to understand how a ship made of iron can float. Consider a 1-ton block of solid iron. Because iron is nearly eight times denser than water, it displaces only 1/8 ton of water when submerged, which is not enough to keep it afloat. Suppose we reshape the same iron block into a bowl (Figure 13.16). It still weighs 1 ton. But when we put it in water, it displaces a greater volume of water than when it was a block. The deeper the iron bowl is immersed, the more water it displaces and the greater the buoyant force acting on it. When the buoyant force equals 1 ton, it will sink no farther.

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FIGURE 13.16 An iron block sinks, while the same quantity of iron shaped like a bowl floats.


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Only in the special case of floating does the buoyant force acting on an object equal the object’s weight When any boat displaces a weight of water equal to its own weight, it floats. This is sometimes called the principle of flotation: A floating object displaces a weight of fluid equal to its own weight.

Every ship, every submarine, and every dirigible must be designed to displace a weight of fluid equal to its own weight. Thus, a 10,000-ton ship must be built wide enough to displace 10,000 tons of water before it sinks too deep in the water. The same holds true for vessels in air. A dirigible that weighs 100 tons displaces at least 100 tons of air. If it displaces more, it rises; if it displaces less, it falls. If it displaces exactly its weight, it hovers at constant altitude.

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FIGURE 13.17 The weight of a floating object equals the weight of the water displaced by the submerged part.

For a given volume of displaced fluid, a denser fluid exerts a greater buoyant force than a less dense fluid. A ship, therefore, floats higher in salt water than in freshwater because salt water is slightly denser. Similarly, a solid chunk of iron will float in mercury, even though it will sink in water.

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FIGURE 13.18 A floating object displaces a weight of fluid equal to its own weight.

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FIGURE 13.19 The same ship empty and loaded. How does the weight of its load compare with the weight of extra water displaced?

CHECK YOURSELF

1. Why is it easier for you to float in salt water than in freshwater?

2. On a boat ride, the skipper gives you a life preserver filled with lead pellets. When he sees the skeptical look on your face, he says that you’ll experience a greater buoyant force if you fall overboard than your friends who wear Styrofoam-filled life preservers. Is he being truthful?

CHECK YOUR ANSWERS

1.It’s easier because a lesser amount of your body is immersed when displacing your weight—you don’t “sink” as far. You’d float even higher in mercury (density 13.6 g/cm3), and you’d sink completely in alcohol (density 0.8 g/cm3).

2. He’s truthful. But what he doesn’t tell you is that you’ll drown! Your life preserver will submerge and displace more water than those of your friends who float at the surface. Although the buoyant force on you will be greater, your increased weight is greater still! Whether you float or sink depends on whether or not the buoyant force equals your weight.

FLOATING MOUNTAINS

The tip of a floating iceberg above the ocean’s surface is approximately 10% of the whole iceberg. That’s because ice is 0.9 times the density of water, so 90% of it submerges in water. Similarly, a mountain floats on the Earth’s semiliquid mantle with only its tip showing. That’s because Earth’s continental crust is about 0.85 times the density of the mantle it floats upon; thus, about 85% of a mountain extends beneath the Earth’s surface. So, like floating icebergs, mountains are appreciably deeper than they are high.

There is an interesting gravitational sidelight to this: Recall, from Chapter 9, that the gravitational field at the Earth’s surface varies slightly with varying densities of underlying rock (which is valuable information to geologists and oil prospectors), and that gravitation is less at the top of a mountain because of the greater distance to Earth’s center. Combining these ideas, we see that, because the bottom of a mountain extends deep into the Earth’s mantle, there is increased distance between a mountaintop and the denser mantle. This increased “gap” further reduces gravitation at the top of a mountain.

Another interesting fact about mountains: If you could shave off the top of an iceberg, the iceberg would be lighter and would be buoyed up to nearly its original height before being shaved. Similarly, when mountains erode, they are lighter, and they are pushed up from below to float to nearly their original heights. So, when a kilometer of mountain erodes away, some 85% of a kilometer of mountain thrusts up from below. That’s why it takes so long for mountains to weather away.

CHECK YOURSELFA river barge loaded with gravel approaches a low bridge that it cannot quite pass under. Should gravel be removed from or added to the barge?

DO YOU HAVE AN ANSWER?Ho, ho, ho! Do you think ol’ Hewitt is going to give all the answers to Check Yourself questions? Good teaching is asking good questions, not providing all answers. You’re on your own with this one!

Pascal’s Principle

One of the most important facts about fluid pressure is that a change in pressure at one part of the fluid will be transmitted undiminished to other parts. For example, if the pressure of city water is increased at the pumping station by 10 units of pressure, the pressure everywhere in the pipes of the connected system will be increased by 10 units of pressure (provided that the water is at rest). This rule is called Pascal’s principle: A change in pressure at any point in an enclosed fluid at rest is transmitted undiminished to all points in the fluid.Pascal’s principle was discovered in the seventeenth century by Blaise Pascal (who was an invalid at the age of 18 and remained so until his death at the age of 39), for whom the SI unit of pressure, the pascal (1 Pa = 1 N/m2), was named. Fill a U-tube with water and place pistons at each end, as shown in Figure 13.21. Pressure exerted against the left piston will be transmitted throughout the liquid and against the bottom of the right piston. (The pistons are simply “plugs” that can slide freely but snugly inside the tube.) The pressure that the left piston exerts against the water will be exactly equal to the pressure the water exerts against the right piston. This is nothing to write home about. But suppose you make the tube on the right side wider and use a piston of larger area: Then the result will be impressive. In Figure 13.22 the piston on the right has 50 times the area of the piston on the left (let’s say that the left piston has a cross-sectional area of 100 square centimeters and that the right piston has a cross-sectional area of 5000 square centimeters). Suppose a 10-kg load is placed on the left piston. Then an additional pressure (nearly 1 N/cm2) due to the weight of the load is transmitted throughout the liquid and up against the larger piston. Here is where the difference between force and pressure comes in. The additional pressure is exerted against every square centimeter of the larger piston. Since there is 50 times the area, 50 times as much force is exerted on the larger piston. Thus, the larger piston will support a 500-kg load—fifty times the load on the smaller piston!

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FIGURE 13.21 The force exerted on the left piston increases the pressure in the liquid and is transmitted to the right piston.


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FIGURE 13.22 A 10-kg load on the left piston will support 500 kg on the right piston.

This is something to write home about, for we can multiply forces using such a device. One newton input produces 50 newtons output. By further increasing the area of the larger piston (or reducing the area of the smaller piston), we can multiply force, in principle, by any amount. Pascal’s principle underlies the operation of the hydraulic press.

Seventeenth-century theologian and scientist Blaise Pascal is remembered scientifically for hydraulics, which changed the technological landscape more than he imagined. He is remembered theologically for his many assertions, one of which relates to centuries of human landscape: “Men never do evil so cheerfully and completely as when they do so from religious conviction.”

The hydraulic press does not violate energy conservation, because a decrease in distance moved compensates for the increase in force. When the small piston in Figure 13.22 is moved downward 10 centimeters, the large piston will be raised only one-fiftieth of this, or 0.2 centimeter. The input force multiplied by the distance moved by the smaller piston is equal to the output force multiplied by the distance moved by the larger piston; this is one more example of a simple machine operating on the same principle as a mechanical lever. Pascal’s principle applies to all fluids, whether gases or liquids. A typical application of Pascal’s principle for gases and liquids is the automobile lift seen in many service stations (Figure 13.23). Increased air pressure produced by an air compressor is transmitted through the air to the surface of oil in an underground reservoir. The oil, in turn, transmits the pressure to a

piston, which lifts the automobile. The relatively low pressure that exerts the lifting force against the piston is about the same as the air pressure in automobile tires.

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FIGURE 13.23 Pascal’s principle in a service station.

Hydraulics is employed by modern devices ranging from very small to enormous. Note the hydraulic pistons in almost all construction machines where heavy loads are involved (Figure 13.24)

CHECK YOURSELF

1.As the automobile in Figure 13.23 is being lifted, how does the change in oil level in the reservoir compare with the distance the automobile moves?

2. If a friend commented that a hydraulic device is a common way of multiplying energy, what would you say?

CHECK YOUR ANSWERS

1.The car moves up a greater distance than the oil level drops, since the area of the piston is smaller than the surface area of the oil in the reservoir.

2. No, no, no! Although a hydraulic device, like a mechanical lever, can multiply force, it always does so at the expense of distance. Energy is the product of force and distance. If you increase one, you decrease the other. No device has ever been found that can multiply energy!


Surface Tension

Suppose that you suspend a bent piece of clean wire from a sensitive spiral spring (Figure 13.25), lower the wire into water, and then raise it. As you attempt to free the wire from the water surface, you see from the stretched spring that the water surface exerts an appreciable force on the wire. The water surface resists being stretched, for it has a tendency to contract. You can also see this when a fine-haired paintbrush is wet. When the brush is under water, the hairs are fluffed pretty much as they are when the brush is dry, but when the brush is lifted out, the surface film of water contracts and pulls the hairs together (Figure 13.26). This contractive tendency of the surface of liquids is called surface tension.

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FIGURE 13.25 When the bent wire is lowered into the water and then raised, the spring will stretch because of surface tension.

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FIGURE 13.26 When the brush is taken out of the water, the hairs are held together by surface tension.

Surface tension accounts for the spherical shape of liquid drops. Raindrops, drops of oil, and falling drops of molten metal are all spherical because their surfaces tend to contract and force each drop into the shape having the least surface area. This is a sphere, the geometrical figure that has the least surface area for a given volume. For this reason, the mist and dewdrops on spider webs or on the downy leaves of plants are nearly spherical blobs. (The larger they are, the more that gravity flattens them.)

(Click image

to enlarge)

FIGURE 13.27 Small blobs of water are drawn by surface tension into spherelike shapes.

Surface tension is caused by molecular attractions. Beneath the surface, each molecule is attracted in every direction by neighboring molecules, resulting in no tendency to be pulled in any specific direction. A molecule on the surface of a liquid, however, is pulled only by neighbors on each side and downward from below; there is no pull upward (Figure 13.28). These molecular attractions thus tend to pull the molecule from the surface into the liquid, and this tendency minimizes the surface area. The surface behaves as if it were tightened into an elastic film. This is evident when dry steel needles or razor blades seem to float on water. They don’t float in the usual sense, but they are supported by the surface molecules opposing an increase in surface area. The water surface sags like a piece of plastic wrap, which allows certain insects, such as water striders, to run across the surface of a pond.

(Click image

to enlarge)

FIGURE 13.28 A molecule at the surface is pulled only sideways and downward by neighboring molecules. A molecule beneath the surface is pulled equally in all directions.

The surface tension of water is greater than that of other common liquids, and pure water has a stronger surface tension than soapy water. We can see this when a little soap film on the surface of water is effectively pulled out over the entire surface. This minimizes the surface area of the water. The same thing happens for oil or grease floating on water. Oil has less surface tension than cold water, and it is drawn out into a film covering the whole surface. But hot water has less surface tension than cold water because the faster-moving molecules are not bonded as tightly. This allows the grease or oil in hot soups to float in little bubbles on the surface of the soup. When the soup cools and the surface tension of the water increases, the grease or oil is dragged out over the surface of the soup. The soup becomes “greasy.” Hot soup tastes different from cold soup primarily because the surface tension of water in the soup changes with temperature.

Capillarity

When the end of a thoroughly clean glass tube with a small inside diameter is dipped into water, the water wets the inside of the tube and rises in it. In a

tube with a bore of about millimeter in diameter, for example, the water rises slightly higher than 5 centimeters. With a still smaller bore, the water rises much higher (Figure 13.30). This rise of a liquid in a fine, hollow tube or in a narrow space is capillarity.


(Click image

to enlarge)

FIGURE 13.29 Bubble Master Tom Noddy blows bubbles within bubbles. The large bubble is elongated due to blowing, but it will quickly settle to a spherical shape due to surface tension.

FIGURE 13.30

Capillary tubes.

When thinking of capillarity, think of molecules as sticky balls. Water molecules stick to glass more than to each other. The attraction between unlike substances such as water and glass is called adhesion. The attraction between like substances, molecular stickiness, is called cohesion. When a glass tube is dipped into water, the adhesion between the glass and water causes a thin film of water to be drawn up over the inner and outer surfaces of the tube (Figure 13.31a). Surface tension causes this film to contract (Figure 13.31b). The film on the outer surface contracts enough to make a rounded edge. The film on the inner surface contracts more and raises water with it until the adhesive force is balanced by the weight of the water lifted (Figure 13.31c). In a narrower tube, the weight of the water in the tube is small and the water is lifted higher than it would be if the tube were wider.

(Click image

to enlarge)

FIGURE 13.31 Hypothetical stages of capillary action, as seen in a cross-sectional view of a capillary tube.

If a paintbrush is dipped part way into water, the water will rise up into the narrow spaces between the bristles by capillary action. If your hair is long, let it hang into the sink or bathtub, and water will seep up to your scalp in the same way. This is how oil soaks upward in a lamp wick and water soaks into a bath towel when one end hangs in water. Dip one end of a lump of sugar in coffee, and the entire lump is quickly wet. Capillary action is essential for plant growth. It brings water to the roots of plants and carries sap and nourishment to high branches of trees. Just about everywhere we look, we can see capillary action at work. That’s nice. But from the point of view of an insect, capillarity is not so nice. Recall from the previous chapter that, because of an insect’s relatively large surface area, it falls slowly in air. Gravity poses almost no risk at all—but not so with capillarity. Being in the grip of water may be fatal to an insect—unless it is equipped for water like a water strider.


Review Questions1. Give two examples of a fluid.

Pressure2. Distinguish between force and pressure.

Pressure in a Liquid3. 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?

Buoyancy7. 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’ Principle10. 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 1-L 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?

Flotation19. 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 100-ton ship? What is the buoyant force that acts on a floating 100-ton ship?

Pascal’s Principle21. 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/cm2, how much extra load will the output piston support if its cross-sectional area is 50 cm2?

Surface Tension23. What geometrical shape has the least surface area for a given volume?24. What is the cause of surface tension?

Capillarity25. Distinguish between adhesive and cohesive forces.26. What determines how high water will climb in a capillary tube?

[ previous page ] [ next page ]Projects

1.Place an egg in a pan of tap water. Then dissolve salt in the water until the egg floats. How 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 water-filled 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 water-soaked Ping-Pong 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 ]One-Step 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 100-m-high 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 1-m-tall closed barrel is 98 kPa.

What is the pressure at the bottom of the barrel when a 5-m pipe filled with water is inserted into the top of the barrel?

Exercises

1. What common liquid covers more than two-thirds 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 200-lb man wearing loafers or being stepped on by a 100-lb 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 Band-Aids 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?


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?


14. A block of aluminum with a volume of 10 cm3 is placed in a beaker of water filled to the brim. Water overflows. The same is done in another beaker with a 10-cm3 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 1-kg 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 10-N 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 1-L 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.


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?)


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.


50. A half-filled 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?


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 Ping-Pong 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.


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?



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 6-kg 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 400-kg horse. (b) If

the barge can only be pushed 15 cm deeper into the water before water overflows to sink it, how many 400-kg horses can it carry?

4. A dike in Holland springs a leak through a hole of area 1 cm2 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 solid-gold 1-kg 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 2.0-kg 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 1-cm 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 m3? (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/m3. 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?

13 LiquidsConceptual Physics Instructor’s Manual, 10th Edition

Pressure

Pressure in a Liquid

Water Dowsing

Buoyancy

Archimedes’ Principle

What Makes an Object Sink or Float

Floatation

Floating Mountains

Pascal’s Principle

Surface Tension

Capillarity

The depths of the ocean as well as the expanse of outer space are of current interest, yet liquids are seldom studied in introductory physics classes anymore. Perhaps this is because Archimedes’ Principle and the like are too far from the frontiers of present research. But because much of the physics in this chapter is more than 2000 years old is no reason that it should not be in your physics course. Liquids are a very real part of your students’ everyday world.

It is well known that falling from great heights into water has much the same effect as falling to solid ground. Less well known are the new “water saws,” with pressures of about 5500 lbin2 used for cutting through armor-plate steel.

Regarding Figure 13.3, you may point out that the average mass of a giraffe’s heart is about 40 kilograms. That’s quite a pump.

The dedicated teacher walking on broken glass with bare feet in his classroom with Exercise 8 is Marshall Ellenstein. Marshall has been a contributor to this book for years and is the editor of the video and DVD series Conceptual Physics Alive! And the newly released 3-DVD set of The San Francisco Years.

In student laboratory exercises, it is more common to work with mass density than with weight density, and floating or submerged materials are more often described in units of

mass rather than weight. Displaced liquid is also described in units of mass rather than weight. This is why buoyant force in this chapter is treated as “the weight of so many kilograms,” rather than “so many newtons.” The expression of buoyancy in terms of mass units should be more in keeping with what goes on in lab.

An impressive buoyancy DEMONSTRATION: Place about 8 grams of dry ice in a large (several cm) uninflated balloon. Tie the balloon. Immediately set it on a digital balance reading to the nearest milligram. As the balloon inflates (over a few minutes) the balance readout plummets at a rate of about 2 mg sec. The scale will finally read about 2.4 grams less, assuming the balloon inflates to about 2 liters (density of air is about 1.2 gL). I learned of this demo from my nephew John Suchocki.

Oceans tidbit: The Atlantic is getting wider, the Pacific narrower.

There is only 1 OHT for this chapter, Figure 13.22.

In the Practicing Physics book:

• Archimedes’ Principle I

• Archimedes’ Principle II

There are problems for this chapter in the Problem Solving in Conceptual Physics book.

Three activities and 1 experiment on chapter material are in the Laboratory Manual.

In the Next-Time Questions book:

• Styrofoam Cargo • Ice Cube on the Moon

• Water’s Own Level • Deuterium Ice Cube

• Fire Truck • Balance Stand

• Boat with Scrap Iron • Submerged Cube and Sphere

• Wood and Rock Float • Submerged Teabag

• Floating Block

Prerequisite to this chapter is understanding of density, covered in the previous chapter. So if you skipped Chapter 12, discuss density here. This chapter is prerequisite to the following chapter but is not prerequisite to the remaining chapters of the text.

SUGGESTED LECTURE PRESENTATION

Force versus Pressure: Begin by distinguishing between force and pressure. Illustrate with examples: Somebody pushing on your back with a force of only 1 N—with a pin! As you’re lying on the floor, a 400-N lady stands on your stomach—perched atop spike heels! Indian master lying on a bed of 1000 nails—apprentice being advised to start with one nail! The rounded corners on tables, sharp blades on cutting knives, and the absurdity of standing tall while pointing your toes downward when caught in quicksand.

Have students compare in their hands the weights of a small steel ball and a large Styrofoam ball, and after agreeing that the little ball is heavier (since density was treated in the previous chapter), weigh them and show the Styrofoam ball is heavier! Another example of pressure (on the nerve endings).

Liquid Pressure: Liquid pressure = density depth: After a few words about density, you may want to derive or call attention to the derivation of this relationship in the second footnote in the chapter.

DEMONSTRATION: Pascal’s Vases (similar to Figure 13.5) and shown by Tsing Bardin in the chapter opener photo. (Dr. Bardin is a first-rate physicist who has done extensive research for IBM and now donates a day each week teaching at City

College of San Francisco—a fascinating woman.) Rationalize your results in terms of the supporting forces exerted by the sloping sides of the vases. [That is, in the wide sloping vase, the water pushes against the glass, and the glass reacts by pushing against the water. So the glass supports the extra water without the pressure below increasing. For the narrow vase that slopes outward near the bottom, the water pushes up against the sloping glass. By reaction, the glass pushes down on the water, so the pressure at the bottom is as if water were present all the way to the surface.]

Discuss Figure 13.3 and why your heart gets more rest if you sleep in a prone position versus sitting up. Call attention to the fact that when swimming, the pressure one feels against the eardrums is a function of only depth—that swimming 3 meters deep in a small pool has the same effect as swimming 3 meters deep in the middle of a huge lake.

CHECK QUESTION: Would the pressure be greater swimming 3 m deep in the middle of the ocean? (Then compare the densities of fresh and salt water.)

Ask why dams are built thicker at the bottom, and after discussing Figure 13.4 sketch the top view of a couple of dams on the board and ask which design is best (Exercise 22, previous chapter). Then relate this to the shape of stone bridges (which actually need no mortar), and the arched shape of doorways in old stone structure (Figure 12.12, previous chapter), and the aqueducts shown in Figure 13.6. Another illustration is the concave ends of large wine barrels (Exercise 23, previous chapter).

Buoyant Force: Show that a consequence of pressure being depth dependent is the phenomenon of buoyancy. Sketch Figure 13.9 on the board. Follow this up with a sketch and explanation of Figure 13.14.

DEMONSTRATION: Show how an overflow can enables the measure of an object’s volume. Ask how one could measure a quarter of a cup of butter in a liquid measuring cup using this method.

DEMONSTRATION: Archimedes’ Principle, as shown in Figure 13.15.

You may find that many students who have trouble with conceptualizing buoyant force are confused about the distinction between area and volume. Be sure to make this distinction clear, (as elementary as it seems). (If you didn’t pour the contents of the spherical flask into the tall cylindrical flask of the same volume as described in the suggested lecture of the previous chapter, be sure to do so here.) Also, point out that because a liquid is incompressible (practically incompressible, as the volume of water decreases by only 50 one-millionths of its original volume for each atmosphere increase in pressure, or equivalently, for each addition 10.3 m in depth) its density is not depth dependent. The density of water near the surface is practically the same as the density far beneath the surface. You may wish to acknowledge that some variation occurs due to temperature differences. Usually a student will inquire about waterlogged objects which lie submerged yet off the bottom of the body of water. Such objects are slightly denser than the warmer surface water and not quite as dense as the cooler water at the bottom. Stress that this is unusual and that objects appreciably denser than water always sink to the bottom, regardless of the depth of the water. Scuba divers do not encounter “floating” rocks near the bottoms of deep bodies of water!

CHECK QUESTION: Two solid blocks of identical size are submerged in water. One block is lead and the other is aluminum. Upon which is the buoyant force greater? [Same, since volumes of water displaced are the same.]

After discussion, try this one:

CHECK QUESTION: Two solid blocks of identical size, one of lead and the other of wood, are put in the same water. Upon which is the buoyant force greater? [This time the BF is greater on the lead because it displaces more water than the wood that floats!]

CHECK QUESTIONS: What is the buoyant force on a ten-ton ship floating in fresh water? In salt water? In a lake of mercury? [Same BF, but different volumes displaced.]

The unit “ton” is used in several places in this text. It may be taken to mean a metric tonne, the weight of 1000 kg, or the British ton, 2000 pounds. Either interpretation is sufficient in treating the concept involved.

Flotation: Discuss boats and rafts and the change of water lines when loaded.

CHECK QUESTIONS: What is the approximate density of a fish? Of a person? What can you say of people who can’t float?

DEMONSTRATION: Cartesian diver (inverted partially filled small bottle submerged in a larger flexible plastic bottle that you squeeze to increase and decrease the weight of water in the small bottle to make it rise and fall).

Discuss the compressibility of the human body in swimming—how the density of most people a meter or two below the surface of the water is still less than the density of water, and that one need only relax and be buoyed to the surface. But that at greater depths, the greater pressure compresses one to densities greater than the density of water, and one must swim to the surface. Simply relaxing, one would sink to the bottom! Relate this to the Cartesian diver demonstration. Also state why one cannot snorkel with a tube that goes deeper than a half-meter or so.

Side point: Contrary to those old Tarzan movies, you cannot sink in quicksand. Quicksand is the name given to a mass of sand particles that are supported by circulating water rather than by each other. Its density is greater than the density of human bodies, so you can float on it. If you struggle, you’ll unfortunately succeed in digging yourself deeper in. So if you’re ever stuck in it, keep yourself still until you stop sinking (you will), and then use slow swimming motions to get yourself into a horizontal position and then roll onto the ground.

Pascal’s Principle: Begin by pushing against the wall with a meterstick and state that the stick affords a means of applying pressure to the wall—then state that the same can be done with a confined fluid. Explain how any external pressure applied to a liquid that tightly fills a volume is transmitted to all parts of the liquid equally. Discuss Figures 13.21 and 13.22. If a hydraulic press is available, crush a block of wood with it. Point out that the pressure transmitted throughout a confined liquid is pressure over and above that already in the liquid. For example, the pressure in a hydraulic system at any point is equal to the applied pressure plus the density depth.

Surface Tension: In the next chapter on gases we’ll think of atoms as rigid balls that ricochet off one another. Here we think of atoms as sticky balls—capillarity. An interesting example of capillarity (not in the text) involves the tallness of trees. The cohesive forces of water explains the transport of water from roots to the top of tall trees. When a single water molecule evaporates from the cell membrane inside a leaf, it is replaced by the one immediately next to it due to the cohesive forces between water molecules. A pull is created on the column of water which is continuous from leaves to roots. Water can be lifted far higher than the 10.2 m that atmospheric pressure would serve—even to 100 m in this way, the height of the largest trees.

NEXT-TIME QUESTION: Show your class an ordinary rectangular chunk of wood with square cross-section, and ask how it will float if placed in water. [Unless it is very low density, like Styrofoam, it will float as in position b. Why?]

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-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 (5000N/1000 cm2) = 5N/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.)

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.

26. Mercury is more dense (13.6 g/cm3) 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 vwave ~ √g).

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/m3 220 m = 2,200,000 N/m2 = 2200 kPa (or for density = 9800 N/m3, pressure = 2160 kPa.

2. Density = m/V = 6 kg/1 liter = 6 kg/liter. (Since there are 1000 liters in 1 cubic meter, density may be expressed in units kg/m3. Density = 6 kg/1 liter 1000 liter/m3 = 6000 kg/m3, 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 m2; 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

density = . Or from this, V = = = 0.4 m3.

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.

4. First you must find the pressure. It is weight density depth = (10,000 N/m3)(2 m) = 20,000 N/m2, or 20,000 Pa. Force is pressure area, and 1 cm2 = 10-4 m2, so F = (20,000 N/m2)(10-4 m2) = 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/cm3. Your gold has a mass of 1000 grams, so = 19.3 g/cm3. Solving for V,

V = = 51.8 cm3.

6. Density = = =

= 4 kg/liter. And since 1 liter = 103 cm3 = 10-3 m3, density = 4,000 kg/m3.

(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 mass/density, and displaced volume = (0.5 kg)/(1000 kg/m3) = 5 10-4 m3. The object’s volume is the same as the volume it displaces, so the object’s density is mass/volume =

(2 kg)/(5 10-4 m3) = 4000 kg/m3, 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/s2) = 600 N (or 588 N if 9.8 m/s2 is used).

(b) She has the same density as water, 1000 kg/m3. Since density = mass/volume, volume = mass/density, so volume = (60 kg)/(1000 kg/m3) = 0.06 m3.

(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/m3) = 923 kg/m3.

10. The relative areas are as the squares of the diameters; 62/22 = 36/4 = 9. The larger piston can lift 9 times the input force to the smaller piston.

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