Youngbull Science Center - Home - 18science.telosrtc.com/uploads/1/6/5/9/16598904/chapter_18.pdf · 2019. 12. 3. · that hits the crystal of sodium chloride. ... You can see metal

SOLIDSSolids can be described in terms of crystal structure, density, and elasticity.

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How Does Size Affect the Relationship Between Surface Area and Volume?1. Place a single sugar cube on your desk.

2. Using additional sugar cubes, construct the next largest cube possible—that is, a cube with two sugar cubes on a side.

3. Now construct a cube with three sugar cubes on a side.

Analyze and Conclude1. Observing What happened to the surface

area of the cube and the volume of the cube as the length of a side increased?

2. Predicting What would be the surface area and volume of a cube with four sugar cubes on a side? Five sugar cubes on a side?

3. Making Generalizations What happens to the ratio of surface area to volume as an object’s linear dimensions increase?

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THE BIG

IDEA ......

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H umans have been classifying and using solid materials for many thousands of years. The names Stone Age, Bronze Age, and Iron Age tell us the importance of solid

materials in the development of civilization. Wood and clay were perhaps the first materials important to early peoples, and gems were put to use for art and adornment.

Not until recent times has the dis-covery of atoms and their interactions made it possible to understand the structure of materials. We have progressed from being finders and assemblers of materials to actual makers of materials. In today’s laboratories chemists, metallurgists, and materials scientists routinely design and produce new materials to meet specific needs.

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SOLIDSObjectives• Describe the structure of

crystals. (18.1)

• Describe the factors that determine the density of a material. (18.2)

• Explain the property of elasticity. (18.3)

• Explain how a load-carrying beam undergoes compression and tension. (18.4)

• Describe the relationship among linear growth, surface area growth, and volumetric growth. (18.5)

discover!

MATERIALS sugar cubes

EXPECTED OUTCOME As the size of the cube increases, the surface area and volume increase at different rates.

ANALYZE AND CONCLUDE

Both the surface area and volume of the cube increased as the length of a side increased. Surface area grew as the square of the enlargement; volume grew as the cube of the enlargement. When the 1 3 1 3 1 cube was scaled up by a factor of 3, surface area grew from 6 square units to 6 3 (32) 5 54 square units, and the volume grew from 1 cubic unit to 1 3 (33) 5 27 cubic units.

For a 4 3 4 3 4 cube, A 5 6 3 (42) 5 96 square units, and V 5 1 3 (43) 5 64 cubic units. For a 5 3 5 3 5 cube, A 5 6 3 (52) 5 150 square units, and V 5 1 3 (53) 5 125 cubic units.

As linear size of an object increases, the ratio of surface area to volume decreases.

1.

2.

3.

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CHAPTER 18 SOLIDS 345

18.1 Crystal StructureWhen we look at samples of minerals such as quartz, mica, or galena, we see many smooth, flat surfaces at angles to one another within the mineral. The mineral samples are made of crystals, or regular geometric shapes whose component particles are arranged in an orderly, repeating pattern. The shape of a crystal mirrors the geometric arrangement of atoms within the crystal. The mineral samples themselves may have very irregular shapes, as if they were tiny cubes or other small units stuck together to make a free-form solid sculpture.

Not all crystals are evident to the naked eye. Their existence in many solids was not discovered until X-rays became a tool of research early in the twentieth century. The X-ray pattern caused by X-rays passing through the crystal structure of common table salt (sodium chloride) is shown in Figure 18.1. Rays from the X-ray tube are blocked by a lead screen except for a narrow beam that hits the crystal of sodium chloride. The radiation that penetrates the crystal produces the pattern shown on the pho-tographic film beyond the crystal. The white spot in the center is caused by the main unscattered beam of X-rays. The size and arrangement of the other spots indicate the arrangement of sodium and chlorine atoms in the crystal. All crystals of sodium chloride produce this same design.

The patterns made by X-rays on photo-graphic film show that the atoms in a crystal have an orderly arrangement. Every crys-talline structure has its own unique X-ray pattern. For example, in a sodium chloride crystal, the atoms are arranged like a three-dimensional chess board or a child’s jungle gym, as shown in Figure 18.2.

Metals such as iron, copper, and gold have relatively simple crystal structures. Tin and cobalt are only slightly more complex. You can see metal crystals if you look carefully at a metal surface that has been cleaned (etched) with acid.

You can also see them on the surface of galvanized iron that has been exposed to the weather, or on brass doorknobs that have been etched by the perspiration of hands.

CONCEPTCHECK ...

... What determines the shape of a crystal?

FIGURE 18.1 �When X-rays pass through a crystal of common table salt (sodium chloride), they produce a distinctive pat-tern on photographic film.

� FIGURE 18.2

In this model of a sodium chloride crystal, the large spheres represent chloride ions, and the small ones represent sodium ions.

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This chapter may be skipped with no particular consequence. If this chapter is skipped and Chapter 19 is assigned, the concept of density should be introduced then.

18.1 Crystal Structure

Key Termcrystal

� Teaching Tip Crystals are very evident on galvanized (zinc-coated) sheets of iron. You can get a demonstration piece at a local heating and air conditioning duct supplier.

� Teaching Tip Crystal-growing kits are a great way to show the structures of crystals.

The shape of a crystal mirrors the geometric

arrangement of atoms within the crystal.

T e a c h i n g R e s o u r c e s

• Reading and Study Workbook

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• PresentationEXPRESS

• Interactive Textbook

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18.2 DensityOne of the properties of solids, as well as liquids and even gases, is the measure of how tightly the material is packed together: density.Density is a measure of how much matter occupies a given space; it is the amount of mass per unit volume:

densitymass

volume

Density is not mass and it is not volume. Density is a ratio; it is the amount of mass per unit volume. Density is a property of a mate-rial; it doesn’t matter how much you have. A pure iron nail has the same density as a pure iron frying pan. The frying pan may have 100 times as many iron atoms and have 100 times as much mass, but its atoms will take up 100 times as much space. The mass per unit vol-ume for the iron nail and the iron frying pan is the same.

The density of a material depends upon the masses of the individual atoms that make it up, and the spacing between those atoms. Iridium, a hard, brittle, silvery-white metal in the platinum family, is the densest substance on Earth, even though an individual iridium atom is less massive than individual atoms of gold, mercury, lead, or uranium. The close spacing of iridium atoms in an iridium crystal gives it the greatest density. A cubic centimeter of iridium contains more atoms than a cubic centimeter of gold or uranium.

Table 18.1 lists the densities of a few materials in units of grams per cubic centimeter.18.2.1 Density varies somewhat with temperature and pressure, so, except for water, densities are given at 0°C and atmospheric pressure. Note that water at 4°C has a density of 1.00 g/cm3. The gram was originally defined as the mass of a cubic centimeter of water at a temperature of 4°C. A gold brick, with a density of 19.3 g/cm3, is 19.3 times more massive than an equal volume of water.

Which has greater density—1 kg of water or 10 kg of water? 5 kg of lead or 10 kg of aluminum?Answer: 18.2.1

think!

FIGURE 18.3 �When the loaf of bread is squeezed, its volume decreases and its density increases.

The symbol for density is the Greek letter ®.


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18.2 Density

Key Termsdensity, weight density, specific gravity

Common MisconceptionDensity is the same as mass, but expressed in different units.

FACT Two objects can have the same mass but different densities.

� Teaching Tip Pass around the room a lead block and a wood block that are about the same volume but have vastly different masses. Have students shake each block to dispel the notion that mass depends on volume. Then pass around two objects that have the same mass and different volumes to dispel the notion that volume depends on mass.

Density is introduced in this chapter, but also could have been introduced in Chapter 3. It works nicely here, however, for it plays a central role in the chapters that follow. I like to introduce ideas when they are needed, so they aren’t considered excess baggage.

Pass a small iron ball and a large, more massive Styrofoam ball around the room. Ask which has the greater mass. (Students will likely perceive the iron ball to be more massive—it exerts more pressure on the hand than the Styrofoam ball.) Place both balls on a balance to prove that the Styrofoam ball has more mass even though it is much less dense.

DemonstrationDemonstration


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A quantity known as weight density can be expressed by the amount of weight a body has per unit volume:

weight densityweightvolume

Weight density is commonly used when discussing liquid pressure (see next chapter).18.2.2

A standard measure of density is specific gravity —the ratio of the mass (or weight) of a substance to the mass (or weight) of an equal volume of water. For example, if a substance weighs five times as much as an equal volume of water, its specific gravity is 5. Or put another way, specific gravity is a ratio of the density of a material to the density of water. So specific gravity has no units (density units divided by density units cancel). If you want to know the specific gravity of any material listed in Table 18.1, it’s there. The magnitude of its density is its specific gravity.

CONCEPTCHECK ...... What determines the density of a material?

The density of gold is 19.3 g/cm3. What is its specific gravity?Answer: 18.2.2

think!

For:Visit:Web Code: –

Links on density www.SciLinks.org csn 1802

Density(g/cm3)

Density(g/cm3)

Densities of a Few SubstancesTable 18.1

Solids Liquids

Iridium

Osmium

Platinum

Gold

Uranium

Lead

Silver

Copper

Brass

Iron

Steel

Tin

Diamond

Aluminum

Graphite

Ice

Pine wood

Balsa wood

22.7

22.6

21.4

19.3

19.0

11.3

10.5

8.9

8.6

7.8

7.8

7.3

3.5

2.7

2.25

0.92

0.50

0.12

Mercury

Glycerin

Sea water

Water at 4°C

Benzene

Ethyl alcohol

13.6

1.26

1.03

1.00

0.90

0.81

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� Teaching Tip Discuss the idea of “unit” volume, i.e., gold has a density of 19.3 grams per “unit” of volume—per one cubic centimeter. The numerical value of the density of a substance is equal to the mass of the substance when the volume is one unit.

� Teaching Tip It is interesting to note that the three metals lithium, sodium, and potassium are all less dense than water and so will float in water. Caution: These metals are also highly reactive with water. Do not demonstrate this in your classroom.

� Teaching Tip Tell students that the density of atomic nuclei is about 2 3 1014 g/cm3, and density in the interior of neutron stars is about 1016 g/cm3. That’s dense!

� Teaching Tip Point out that most materials denser than lead are very expensive (platinum, gold, mercury, etc.), so they are generally not used when density is required. A lead fishing weight may not be as dense as a gold fishing weight, but it is certainly more practical. Uranium, on the other hand, is relatively cheap, and is used as a substitute for lead in some cases—like bullets in electronically fired naval Gatling guns, or as low-volume but heavy weights at the bottom of boat keels.

The density of a material depends

upon the masses of the individual atoms that make it up, and the spacing between those atoms.

CONCEPTCHECK ...

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18.3 ElasticityWhen we hang a weight on a spring, the spring stretches. When we add additional weights, the spring stretches still more. When we remove the weights, the spring returns to its original length. A mate-rial that returns to its original shape after it has been stretched or compressed is said to be elastic.

When a batter hits a baseball, the bat temporar-ily changes the ball’s shape. When an archer shoots an arrow, he first bends the bow, which springs back to its original form when the arrow is released. The spring, the baseball, and the bow are examples of elastic objects.

A body’s elasticity describes how much it changes shape when a deforming force acts on it, and how well it returns to its original shape when the deforming force is removed.

Not all materials return to their original shape when a deforming force is applied and then removed. Materials that do not resume their original shape after being dis-torted are said to be inelastic. Clay, putty, and dough are inelastic materials. Lead is also inelastic, since it is easy to distort it permanently.

Suppose you have a gold nugget with a mass of 57.9 g and a volume of 3.00 cm3. How can you determine if the nugget is pure gold?

One of the reasons gold was used as money was that it is one of the densest of all substances and could therefore be easily identified. A merchant suspicious that gold was diluted with a less valuable sub-stance had only to compute its density by measuring its mass and dividing by its volume. The merchant would then compare this value with the density of gold, 19.3 g/cm3.

Compute its density as follows:

density 19.3 g/cm3massvolume

57.9 g3.00 cm3

Its density matches that of gold, so the nugget can be presumed to be pure gold. (It is possible to get the same density by mixing gold with a platinum alloy, but this is unlikely since platinum has several times the value of gold.)

do the math!

FIGURE 18.4 �The bow is elastic. When the deforming force is removed, the bow returns to its original shape.


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� Teaching Tip Acknowledge weight density, common in the British system of units—like the density of water, 62.4 lb/ft3.

� Teaching Tip Emphasize that specific gravity has no unit because it is a ratio.

� Teaching Tip Calculate the density of a wooden cube: Define density as mass/volume. Find the mass (in grams) with a balance. Measure the dimensions (in centimeters) and multiply them to find the volume. Divide the mass by the volume and express the answer in g/cm3.

� Teaching Tip Discuss the Do the math! problem. It is the essence of the “Eureka” story of Archimedes and the gold crown.

Ask What happens to the density of each piece of an object when it is cut into pieces? Each piece has the same density as the original object had. Which has the greater density, a kilogram of lead or a kilogram of feathers? Any amount of lead is more dense than any amount of feathers. Which has the greater density, a single uranium atom or Earth? The uranium atom



• Problem Solving Exercises in Physics 10-1

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18.3 Elasticity

Key Termselastic, inelastic, Hooke’s law, elastic limit


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When you hang a weight on a spring, the weight applies a force to the spring. It is found that the stretch is directly proportional to the applied force, as shown in Figure 18.5.

A certain tree branch is found to obey Hooke’s law. When a 20-kg load is hung from the end of it, the branch sags a distance of 10 cm. If, instead, a 40-kg load is hung from the same place, by how much will the branch sag? What would you find if a 60-kg load were hung from the same place? (Assume that none of these loads makes the branch sag beyond its elastic limit.)Answer: 18.3.1

If a force of 10 N stretches a certain spring 4 cm, how much stretch will occur for an applied force of 15 N?Answer: 18.3.2

think!

This relationship was noted by the British physicist Robert Hooke, a contemporary of Isaac Newton, in the mid-seventeenth century. According to Hooke’s law, the amount of stretch (or com-pression), x, is directly proportional to the applied force F. Double the force and you double the stretch; triple the force and you get three times the stretch, and so on. In equation form,

F x

If an elastic material is stretched or compressed more than a certain amount, it will not return to its original state. Instead, it will remain distorted. The distance at which permanent distortion occurs is called the elastic limit. Hooke’s law holds only as long as the force does not stretch or compress the material beyond its elastic limit.

CONCEPTCHECK ...

... What characteristics are described by an object’s elasticity?

In lab, you’ll learn that the ratio of force to stretch for a spring is called the spring constant (k), and Hooke’s law can bewritten as F k x,where k is expressed in N/m.

� FIGURE 18.5 The stretch of the spring is directly proportional to the applied force. When the weight is doubled, the spring stretches twice as much.

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� Teaching Tip Show students that Hooke’s law can be expressed in the form F 5 kDx, where k is the constant of proportionality. Explain that the value of k varies according to the material being stretched or compressed.

In many courses you will find Hooke’s law covered in the mechanics sections. If you wish to teach Hooke’s law earlier, it could go with the material in Chapters 3 or 7.

Compare the elasticities of glass, steel, and rubber by dropping a sphere of each onto a hard surface. Your students will be surprised to see that glass and steel are considerably more elastic than rubber!


Illustrate Hooke’s Law by hanging weights from a spring. Ask the class to predict the elongations before suspending additional masses. You can also place weights on top of a spring and predict the compression. Hooke’s law holds for both stretching and compression.


A body’s elasticity describes how much it

changes shape when a deforming force acts on it, and how well it returns to its original shape when the deforming force is removed.


• Laboratory Manual 49

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18.4 Compression and TensionSteel is an excellent elastic material. It can be stretched and it can be compressed. Because of its strength and elastic properties, it is used to make not only springs but also construction girders. Vertical gird-ers of steel used in the construction of tall buildings undergo only slight compression. A typical 25-meter-long vertical girder used in high-rise construction is compressed about a millimeter when it car-ries a 10-ton load. Most deformation occurs when girders are used horizontally, where the tendency is to sag under heavy loads.

A horizontal beam supported at one or both ends is under stress from the load it supports, including its own weight. It undergoes a stress of both compression and tension (stretching). Consider the beam supported at one end in Figure 18.6. It sags because of its own weight and because of the load it carries at its end.

A beam in the position of the one below in Figure 18.6 is known as a cantilever beam.

Neutral Layer Can you see that the top part of the beam is stretched? Atoms are tugged away from one another. The top part is slightly longer. And can you see that the bottom part of the beam is compressed? Atoms there are pushed toward one another, making the bottom part slightly shorter. So the top part of the beam is stretched, and the bottom part is compressed. A little thought will show that somewhere in between the top and bottom, there will be a region that is neither stretched nor compressed. This is the neutral layer (indi-cated by the red dashed line in the figure).

Civil Engineer Devastating earthquakes strike in many parts of the world. Civil engineers study the collapsed structures left by earthquakes to learn how to reduce damage done by the vibrations and waves of future earthquakes. They also examine the responses of different building materials to the quake. They use this information to build stronger and more resilient bridges, tunnels, and highways. Civil engineers rely heavily on their knowledge of physics principles when designing these structures.

Physics on the Job

FIGURE 18.6 �

The top part of the beam is stretched and the bot-tom part is compressed. The middle portion is neither stretched nor compressed.


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18.4 Compression and Tension

� Teaching Tip Point out the difference between the terms compression and tension: Compression occurs when two parts of an object are pressed together; tension occurs when two parts of an object are pulled away from each other. Ask students to point out examples of compression and tension in structures around them.

It is interesting to note that one cubic inch of human bone can withstand a 2-ton force.


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Consider the beam shown in Figure 18.7. It is supported at both ends, and carries a load in the middle. This time the top of the beam is in compression and the bottom is in tension. Again, there is a neu-tral layer along the middle portion of the length of the beam where neither tension nor compression occurs.

I-Beams Have you ever wondered why the cross section of many steel girders has the form of the letter I, as shown in Figure 18.8? Most of the material in these I-beams is concentrated in the top and bottom parts, called the flanges. The piece joining the bars, called the web, is thinner. Why is it shaped like this?

The answer is that the stress is predominantly in the top and bot-tom flanges when the beam is used horizontally in construction. One flange tends to be stretched while the other tends to be compressed. The web between the top and bottom flanges is a region of low stress that acts principally to hold the top and bottom flanges apart. Heavier loads are supported by farther-apart flanges. For this pur-pose, comparatively little material is needed. An I-beam is nearly as strong as a solid bar, and its weight is considerably less.

CONCEPTCHECK ...

... How is a horizontal beam affected by the load it supports?

If you had to make a hole horizontally through the tree branch shown, in a location that would weaken it the least, would you bore it through the top, the middle, or the bottom?

Answer: 18.4

think!

� FIGURE 18.7The top part of the beam is compressed and the bottom part is stretched. The dashed line indicates the neutral layer, which is not under stress.

FIGURE 18.8 �An I-beam is like a solid bar with some of the steel scooped from its middle where it is needed least. The beam is therefore lighter for nearly the same strength.

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� Teaching Tip An interesting follow-up to the material discussed in the text is the ease with which tiny cracks can lead to big breaks in strong structures. Ships can break in two or wings can fall off airplanes. Bonds between atoms are broken in a crack or scratch, which places greater stress upon neighboring bonds. A concentration of stress occurs at the tip of a crack, so that a relatively small force can cause the overstrained bond to break. The next bond in turn breaks, and like a zipper, the whole structure separates. The initial dislocation need not be a large one. A glass cutter need only make a shallow scratch on the surface of a pane of glass, and the glass will break easily along the line of the scratch. Similarly, a piece of cloth is easily ripped if a small nick is first made in the material.

A horizontal beam supported at one or

both ends is under stress from the load it supports, including its own weight. It undergoes a stress of both compression and tension (stretching).





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Hold an egg verti-cally and dangle a small chain beside it by both ends. Note that the chain follows the contour of the egg—a shallow sag for the rounded end and deeper sag for the more pointed end. Nature has not over-looked the catenary!

Why construct with triangles?You have probably noticed triangular shapes in steel bridges and sports domes. You can verify for yourself the structural merits of the triangle.

1. Nail or bolt three sticks together as shown below.

2. Nail or bolt four sticks together.

3. Think How well do the two shapes resist collapse when you apply pressure on them?

discover!

Link to

The Catenary Tension in a stretched rope lies along the direction of the rope, and likewise for a taut chain. Tension between chain links lie along the direction of the chain, even when the chain sags. The curved shape of a rope or chain that sags under its own weight is a catenary. In the photo at the right, both the curve of the sagging chain and the Gateway Arch in the background are catenaries.

In a free-standing stone arch, stones press against one another, producing compression between them. An arch that takes the shape of an inverted catenary is extremely stable because compression within it occurs exactly along the curve. Even a catenary arch made of slippery blocks of ice is stable, for compression only

presses the blocks firmly together with no side components of force. The same is true for the catenary arch that graces the city of St. Louis. If you twirl an arch through a complete circle, you have a dome. The weight of the dome, like that of the arch, produces compression. The catenary shape applied to domes was not appreciated by those who built early domes such as the Notre Dame Cathedral in Paris, which required elaborate buttressing. One of the first successful domes without buttressing was St. Paul’s Cathedral in London, designed by Christopher Wren. Its curved

shape, a catenary, was suggested by Robert Hooke. Modern domes since then, such as the Astrodome in Houston, employ the catenary shape.

Link to ARCHITECTURELink to ARCHITECTURE


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discover!

MATERIALS nails or bolts, seven sticks

EXPECTED OUTCOME Students will find that the shape made from the four sticks will collapse easily under a small pressure while the shape made from three sticks is quite rigid.

Show how a paper drinking straw reacts when a force is applied perpendicular to its length. A small load buckles the straw. Then show how along its length, just like the vertical girders used in construction, the straw has great strength. Show this by grasping a straw firmly at its end with forefinger and thumb and jabbing it endwise into a potato held in your other hand. With very little practice you can pierce the potato.


Show how a paper drinking straw reacts when a force is applied perpendicular to its length. A small load buckles the straw. Then show how along its length, just like the vertical girders used in construction, the straw has great strength. Show this by grasping a straw firmly at its end with forefinger and thumb and jabbing it endwise into a potato held in your other hand. With very little practice you can pierce the potato.


Link to ARCHITECTURE

Have your students compare the contour of a sagging jewelry chain with the shape of an egg. They’ll see it forms a catenary, as the mouse suggests. Note a different catenary for each end of the egg. The strength of the catenary is illustated by the very large force required to crush the egg when the forces are directed along the long axis of the egg.


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18.5 ScalingDid you ever notice how strong an ant is for its size? An ant can carry the weight of several ants on its back, whereas a strong elephant could not even carry one elephant on its back. How strong would an ant be if it were scaled up to the size of an elephant? Would this “super ant” be several times stronger than an elephant? Surprisingly, the answer is no. Such an ant would not be able to lift its own weight off the ground. Its legs would be too thin for its greater weight and would likely break.

Ants have thin legs and elephants have thick legs for a reason. The proportions of things in nature are in accord with their size. The study of how size affects the relationship between weight, strength, and surface area is known as scaling. As the size of a thing increases, it grows heavier much faster than it grows stronger. You can support a toothpick horizontally at its ends, and you’ll notice no sag. But support a tree of the same kind of wood horizontally at its ends and you’ll see a noticeable sag. The tree is not as strong per unit mass as the toothpick is.

Galileo studied scalingand described the dif-ferent bone sizes of various creatures.

FIGURE 18.9 �If the linear dimensions of an object are multiplied by some number, then the area will grow by the square of the number, and the volume (and mass and weight) will grow by the cube of the number. If the linear dimensions of the cube grow by 2, the area will grow by 22 = 4, and the volume will grow by 23 = 8. If the linear dimensions grow by 3, the area will grow by 32 = 9, and the volume will grow by 33 = 27.

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18.5 Scaling

Key Termscaling

Common MisconceptionWhen a structure is scaled up or down in size, its properties go up or down in direct proportion.

FACT Some properties, such as weight, strength, surface area, and volume do not increase in direct proportion to an increase in linear dimensions.

Scaling is becoming enormously important as more and more devices are being miniaturized. Researchers are finding that when something shrinks enough, whether an electronic circuit, a motor, a film of lubricant, or an individual crystal of metal or ceramic, it stops acting like a miniature version of its larger self and starts behaving in new and different ways. Palladium metal, for example, which is normally composed of grains about 1000 nanometers in size, is found to be five times as strong when formed from 5-nanometer grains.


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How Scaling Affects Strength Weight depends on volume, and strength comes from the area of the cross section of limbs—tree limbs or animal limbs. To understand this weight-strength relation-ship, let’s consider the simple case of a solid cube of matter, 1 centi-meter on a side.

A 1-cubic-centimeter cube has a cross section of 1 square cen-timeter. That is, if we sliced through the cube parallel to one of its faces, the sliced area would be 1 square centimeter. Compare this to a cube made of the same material that has double the linear dimen-sions, a cube 2 centimeters on each side. Its cross-sectional area will be 2 × 2 (or 4) square centimeters, and its volume will be 2 × 2 × 2 (or 8) cubic centimeters. It will be eight times more massive.

When linear dimensions are enlarged, the cross-sectional area (as well as the total surface area) grows as the square of the enlarge-ment, whereas volume and weight grow as the cube of the enlarge-ment. These relationships are illustrated in Figure 18.9.

The volume (and weight) increases much faster than the cor-responding enlargement of cross-sectional area. Although the figure demonstrates the simple example of a cube, the principle applies to an object of any shape. Consider an athlete who can lift his weight with one arm. Suppose he could somehow be scaled up to twice his size—that is, twice as tall, twice as broad, his bones twice as thick, and every linear dimension enlarged by a factor of 2. Would he be twice as strong? Would he be able to lift himself with twice the ease? The answer to both questions is no. Since his twice-as-thick arms would have four times the cross-sectional area, he would be four times as strong. At the same time, his volume would be eight times as great, so he would be eight times as heavy. So, for comparable effort, he could lift only half his weight. In relation to his weight, he would be weaker than before.

The fact that volume (and weight) grows as the cube of linear enlargement, while strength (and surface area) grows as the square of linear enlargement is evident in the disproportionately thick legs of large animals compared with those of small animals. Consider the different legs of an elephant and a deer, or a tarantula and a daddy longlegs.

So the great strengths attributed to King Kong and other fictional giants cannot be taken seriously. The fact that the consequences of scaling are conveniently omitted is one of the differences between science and science fiction.

CONCEPTCHECK ...

... If the linear dimensions of an object double, by how much will the cross-sectional area grow?

Suppose a cube 1 cm long on each side were scaled up to a cube 10 cm long on each edge. What would be the volume of the scaled-up cube? What would be its cross-sectional surface area? Its total surface area?Answer: 18.5

think!

Muscular strength depends on the num-ber of fibers in a partic-ular muscle. Hence the strength of a muscle is proportional to its cross-sectional area.


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� Teaching Tip To explain the above results, call attention to the fact that the area of the spherical flask is considerably smaller than the surface area of the cylinder. We see the greater area of the cylindrical surface and we tend to think that the volume of the cylinder should be greater.

� Teaching Tip Display a selection of toy cubical blocks, or a box of sugar cubes. Be sure to actually show the cubes, and even pass sets of eight or more to groups of students. Playing with cubes will remove any last remnants of confusion between area and volume.

Introduce the relationship between area and volume using two differently shaped flasks having the same volume. Fill a 500-ml or 1000-ml spherical flask with colored water. Then produce a tall cylindrical flask of the same volume (unknown to your students). Ask for speculations as to how high the water level will be when all the water from the spherical flask is poured into it. Ask for a show of hands from those who think that the water will reach more than half the height, for those who think it will fill to less than half the height, and for those who guess it will fill to exactly half the height. Your students will be amazed when they see that the seemingly smaller spherical flask has the same volume as the tall cylinder.



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How Scaling Affects Surface Area vs. Volume Important also is the comparison of total surface area with volume. Look at Figure 18.10. As the linear size of an object increases, the volume grows faster than the total surface area. (Vol ume grows as the cube of the enlargement, and both cross-sectional area and total surface area grow as the square of the enlargement.) So as an object grows, the surface area to volume ratio decreases. The following examples may be helpful.

An experienced cook knows that more skin results when peel-ing 5 kg of small potatoes than when peeling 5 kg of large potatoes. Smaller objects have more surface area per kilogram. Since cooling occurs at the surfaces of objects, crushed ice will cool a drink much faster than a single ice cube of the same mass. This is because crushed ice presents more surface area to the beverage.

The rusting of iron is also a surface phenomenon. The greater the amount of surface exposed to the air, the faster rusting takes place. That’s why small filings and iron in the form of “steel wool,” which have large surfaces compared with their volumes, are soon eaten away. The same mass of iron packed in a solid cube or sphere would undergo little rusting in comparison.

Chunks of coal burn, while coal dust explodes when ignited. Thin French fries cook faster in oil than fat fries. Flat hamburgers cook faster than meatballs of the same mass. Large raindrops fall faster than small raindrops, and large fish move faster than small fish. These are all consequences of the fact that volume and area are not in direct proportion to each other.

A sphere has less sur-face area per volume of material than any other shape. When a fat ball-shaped burger is flattened, its surface area increases—which allows greater heat transfer from the grill to the burger.

FIGURE 18.10 �As an object grows proportionally in all directions, there is a greater increase in volume than in surface area. As a result, the ratio of surface area to volume decreases.

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355

TE bubble text

Ask Explain why small cars are more affected by wind. (They have more cross-sectional area compared to weight than larger cars.)

Ask In terms of surface to volume, why should parents take extra care that a baby is warm enough in a cold environment? The baby has more surface per bodyweight, and will therefore cool more rapidly than a larger person. Why do cooks chop food in such small pieces for cooking quickly in a wok? Since cooking occurs from the surface inward, the greater area per volume speeds cooking.

� Teaching Tip State that a sphere contains the largest possible volume for a given surface area.

� Teaching Tip The size to which insects grow is limited by how much oxygen they can route to their body tissues. Researchers have long thought that the big bugs of the Paleozoic period could grow large because each milliliter of atmosphere then contained nearly twice as much oxygen as it does today.

Our technology is changing from “top down” to “bottom up.” In the top-down method, relatively large pieces of material are carved into smaller pieces. Such has been the era of milling machines and lathes. In the bottom-up method, matter is assembled an atom at a time—interestingly enough, nature’s way. Trees grow an atom at a time. Will tomorrow’s human-made devices in the era of nanotechnology do the same?


356

How Scaling Affects Living Organisms The big ears of elephants are not for better hearing, but for better cooling. They are nature’s way of making up for the small ratio of surface area to volume for these large animals. The heat that an animal generates is proportional to its mass (or volume), but the heat that it can dissipate is proportional to its surface area. If an elephant did not have large ears, it would not have enough surface area to cool its huge mass. The large ears of the African elephant greatly increase its overall surface area, and enable it to cool off in hot climates.

At the biological level, living cells must contend with the fact that the growth of volume is faster than the growth of surface area. A cell obtains nourishment by diffusion through its surface. As it grows, its surface area enlarges, but not fast enough to keep up with the cell’s volume. For example, when the diameter of the cell doubles, its surface area increases four times, while its volume increases eight times. Eight times the mass must be sustained by only four times the access to nour-ishment. This puts a limit on the growth of a living cell. So cells divide, and there is life as we know it. That’s nice.

Not so nice is the fate of large animals when they fall. The state-ment “the bigger they are, the harder they fall” holds true and is a consequence of the small ratio of surface area to weight. Air resis-tance to movement through the air depends on the surface area of the moving object. If you fell off a cliff, even with air resistance, for a short time your speed would increase at the rate of very nearly 1 g.You would have too little surface area relative to your weight—unless you wore a parachute. Small animals need no parachute. They have plenty of surface area relative to their small weights. An insect can fall from the top of a tree to the ground below without harm. The surface-area-to- weight ratio is in the insect’s favor—in a sense, the insect is its own parachute.

It is interesting to note that the rate of heartbeat in a mammal is related to the size of the mammal. The heart of a tiny shrew beats about twenty times as fast as the heart of an elephant. In general, small mammals live fast and die young; larger animals live at a lei-surely pace and live longer. Don’t feel bad about a pet hamster that doesn’t live as long as a dog. All warm-blooded animals have about the same life span—not in terms of years, but in the average number of heartbeats (about 800 million). Humans are the exception: we live two to three times longer than other mammals of our size.

CONCEPTCHECK ...

... If the linear dimensions of an object double, by how much will the volume grow?

FIGURE 18.11 �The African elephant has less surface area compared with its weight than other animals. It compensates for this with its large ears, which significantly increase the surface area through which heat is dissipated, and promote cooling.


356

� Teaching Tip When discussing Figure 18.11, point out that the span from ear tip to ear tip is almost the same as the height of the elephant. The dense packing of veins and arteries in the elephant’s ears gives a five-degree difference in temperature between blood entering and blood leaving the ears.

Ask Which has more surface area, an elephant or a mouse? An elephant Which has more surface area, 2000 kg of elephant or 2000 kg of mice? 2000 kg of mice Distinguish carefully between these different questions.

Did you know that the surface area of human lungs is about 20 times greater than the surface skin area? And the digestive tract of an adult human is about 10 m long!

When linear dimensions are

enlarged, the cross-sectional area grows as the square of the enlargement, whereas volume and weight grow as the cube of the enlargement. As the linear size of an object increases, the volume grows faster than the total surface area.



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Concept Summary ••••••

• The shape of a crystal mirrors the geometric arrangement of atoms within the crystal.

• The density of a material depends upon the masses of its individual atoms and the spacing between those atoms.

• A body’s elasticity describes how much it changes shape when a deforming force acts on it, and how well it returns to its original shape when the deforming force is removed.

• A horizontal beam supported at one or both ends is under stress from the load it supports, including its own weight. It undergoes stresses of both compression and tension (stretching).

• When linear dimensions are enlarged, the cross-sectional area (as well as the total surface area) grows as the square of the enlargement, whereas volume and weight grow as the cube of the enlarge-ment. As the linear size of an object increases, the volume grows faster than the total surface area.

crystal (p. 345)

density (p. 346)

weight density(p. 347)

specific gravity(p. 347)

elastic (p. 348)

inelastic (p. 348)

Hooke’s law (p. 349)

elastic limit (p. 349)

scaling (p. 353)

18.2.1 The density of any amount of water (at 4°C) is 1.00 g/cm3. Any amount of lead always has a greater density than any amount of aluminum; the amount of material is irrelevant.

18.2.2 19.3density of golddensity of water

19.3 g/cm3

1.0 g/cm3

18.3.1 A 40-kg load has twice the weight of a 20-kg load. In accord with Hooke’s law, F x, two times the applied force will result in two times the stretch, so the branch should sag 20 cm. The weight of the 60-kg load will make the branch sag three times as much, or 30 cm.

18.3.2 The spring will stretch 6 cm. By ratio and proportion:

Then x (15 N) (4 cm)/(10 N) 6 cm.

10 N4 cm

15 Nx

18.4 Drill the hole through the middle. Wood fibers in the top part of the branch are stretched, and if you drill the hole there, tension in that part may pull the branch apart. Fibers in the lower part are com-pressed, and a hole there might crush under compression. In between, in the neutral layer, the hole will not affect the strength of the branch because fibers there are neither stretched nor compressed.

18.5 Volume of the scaled-up cube is (10 cm)3,or 1000 cm3. Its cross-sectional surface area is (10 cm)2, or 100 cm2. Its total surface area � 6 � 100 cm2 � 600 cm2.

think! Answers

18

Key Terms ••••••

Self-Assessment PHSchool.com csa 1800

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358

Check Concepts ••••••

Section 18.1 1. How does the arrangement of atoms

differ in a crystalline and a noncrystalline substance?

2. What evidence do we have for the microscopic crystal nature of some solids?

3. What evidence do we have for the visible crystal nature of some solids?

Section 18.2 4. What happens to the density of a uni-

form piece of wood when we cut it in half?

5. Uranium is the heaviest atom found in nature. Why isn’t uranium metal the most dense material?

6. Which has the greater density—a heavy bar of pure gold or a pure gold ring?

7. a. Does the mass of a loaf of bread change when you squeeze it?

b. Does its volume change? c. Does its density change?

8. What is the difference between mass density and weight density?

Section 18.3 9. a. What is the evidence for the claim that

steel is elastic? b. That putty is inelastic?

10. What is Hooke’s law?

11. What is an elastic limit?

12. A 2-kg mass stretches a spring 3 cm. How far does the spring stretch when it supports 6 kg? (Assume the spring has not reached its elastic limit.)

Section 18.4 13. Is a steel beam slightly shorter when it

stands vertically? Explain.

14. Where is the neutral layer in a horizon-tal beam that supports a load?

15. Why is the cross section of a metal beam I-shaped and not rectangular?

Section 18.5 16. What is the weight–strength relationship

in scaling?

17. a. If the linear dimensions of an object are doubled, how much does the total area increase?

b. How much does the volume increase?

18. True or false: As the volume of an object increases, its surface area also increases, but the ratio of surface area to volume decreases. Explain.

19. Which will cool a drink faster—a 10-gram ice cube or 10 grams of crushed ice?

ASSESS


358

ASSESS

Check Concepts 1. Crystalline—regular geometric

shapes; noncrystalline—nonregular shapes

2. X-ray patterns (Figure 18.1)

3. Crystals in etched metals

4. no change

5. The atoms are not the most closely packed.

6. same

7. no; yes, decreases; yes, increases

8. Mass density—mass/volume; weight density—weight/volume

9. a. It returns to its original shape when bent.

b. It doesn’t return to its shape.

10. F ~ Dx (or F 5 kDx), where Dx is the amount of stretch or compression.

11. Stretch limit beyond which the substance is permanently deformed.

12. 9 cm

13. Yes, it is compressed by its own weight.

14. Usually along the middle axis, where there is no tension or compression

15. It is much lighter but has nearly the same strength.

16. With growth, weight increases faster than strength.

17. a. Multiplied by four b. Multiplied by eight

18. True, both increase, but volume increases at a faster rate.

19. Crushed ice because it has more surface area


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CHAPTER 18 SOLIDS 359CHAPTER 18 SOLIDS 359




18 20. a. Which has more skin—an elephant or a

mouse? b. Which has more skin per unit of body

weight—an elephant or a mouse?

Think and Explain ••••••

21. You take 1000 milligrams of a vitamin. Your friend takes 1 gram of the same vita-min. Who takes more?

22. Your friend says that the primary differ-ence between a solid and a liquid is the kind of atoms in the material. Do you agree or disagree, and why?

23. How does the density of a 100-kg iron block compare with the density of an iron filing?

24. Which has more volume—a kilogram of lead or a kilogram of aluminum?

25. Which has more weight—a liter of ice or a liter of water?

26. A certain spring stretches 1 cm for each kilogram it supports.

a. If the elastic limit is not reached, how far will it stretch when it supports a load of 8 kg?

b. Suppose the spring is placed next to an identical spring so the two side-by-side springs equally share the 8-kg load. How far will each spring stretch?

27. A thick rope is stronger than a thin rope of the same material. Is a long rope stronger than a short rope?

28. When you bend a meterstick, one side is under tension, and the other is under com-pression. Which side is which?

29. Compression and tension stress occurs in a beam that supports a load (even when the load is its own weight). Show by means of a simple sketch an example where a horizontal load-carrying beam is in tension at the top and compression at the bottom. Then show a case where the opposite occurs: compression at the top and tension at the bottom.

30. Consider a model steel bridge that is 1/100 the exact scale of the real bridge that is to be built.

a. If the model bridge weighs 50 N, what will the real bridge weigh?

b. If the model bridge doesn’t appear to sag under its own weight, is this evidence that the real bridge, built exactly to scale, will not appear to sag either? Explain.


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359

20. a. elephant b. mouse

Think and Explain21. Both the same, for

1000 mg 5 1 g.

22. Disagree; it is the arrangement of atoms and molecules that distinguishes solid from liquid.

23. The densities are the same—both samples are iron.

24. Aluminum; it is less dense.

25. Water; it is more dense.

26. a. 8 cm b. Each spring supports half

the load (4 kg), so each stretch is half as much (4 cm).

27. No; strength is with thickness, not length.

28. Concave side under compression; convex side under tension.

29. See Figure 18.7; see Figure 18.8.

30. a. 50 N 3 100 3 100 3 100 5 50,000,000 N

b. No; the real bridge will have a much lower strength-to-volume ratio than the model and will very likely sag much more than the model.

31. The symmetrical arches of the egg are catenaries; pressing them together (when you squeeze along the long axis) strengthens the egg. The asymmetrical arches are not catenaries, and they splay outward when you squeeze the egg sideways.

32. In the hanging chain, each chain link is pulled by neighboring links; the tension is parallel to the chain at every point. Similarly, in the arch, the compression is parallel to the arch at every point.



Self-Assessment PHSchool.com csa 1800ASSESS (continued)

360

31. Only with great difficulty can you crush an egg when squeezing it along its long axis, but it breaks easily if you squeeze it side-ways. Why?

32. Archie designs an arch to serve as an outdoor sculpture in a park. The arch is to be a certain width and a certain height. To achieve the size and shape for the strongest arch, Archie suspends a chain from two sup-ports of equal heights that are as far apart as the arch is wide. Archie allows the chain to hang as low as the arch is high. He builds the arch to have exactly the shape of the hanging chain, but inverted. Explain why.

33. Why is cement not needed between the stone blocks of an arch that has the shape of an inverted catenary?

34. If you use a batch of cake batter for cup-cakes instead of a cake and bake them for the time suggested for baking a cake, what will be the result?

35. If you were trapped on a cold mountain, why would it make sense for you to sit in a crouched position and grab your knees? (Hint: A piece of wire will cool faster when stretched out than when rolled up into a ball.)

36. Animals lose heat through the surface areas of their skin. A small animal, such as a mouse, uses a much larger proportion of its energy to keep warm than does a large animal, such as an elephant. Why is the rate of heat loss per unit area greater in a small animal than a large one?

37. Why is heating more efficient in large apartment buildings than in single-family dwellings?

38. Some environmentally conscious people build their homes in the shape of domes. Why is less heat lost in a dome-shaped dwelling?

39. Why does crushed ice melt faster than the same mass of ice cubes?

40. Which fall faster, large or small raindrops?

Think and Solve ••••••

41. A one-cubic-centimeter cube has sides 1 cm in length. What is the length of the sides of a cube of volume two cubic centimeters?

42. A solid cube has sides 4.0 cm long and a mass of 672.0 g.

a. What is the volume of the cube? b. What is the total surface area of the cube? c. What is the density of the cube?

18


360

33. The cement is not needed because compression between the blocks is parallel to the arch everywhere.

34. They will be overcooked because they have more area per volume than the larger cake and will cook faster.

35. In a crouched position, you expose less surface area to the cold.

36. Given the surface-to-volume ratio of each animal, a small animal loses a relatively greater amount of heat than does a large one.

37. An apartment building has less area per dwelling unit exposed to the weather.

38. A sphere has less surface area per unit volume than any other geometrical shape. Similarly, a dome-shaped structure has less surface area per unit volume than conventional structures. Less surface exposed to the climate means less heat loss.

39. Crushed ice has a greater surface area, which means more melting surface is exposed to the surroundings.

40. Large drops fall faster. Large drops have less surface area per unit weight, and therefore less air resistance per unit weight.

Think and Solve41. �3

2� cm, or 1.26 cm

42. a. V 5 L3 5 (4.0 cm)3 5 64.0 cm3

b. Surface area of a cube equals six times the area of one side; A 5 6(4.0 cm)2 5 96 cm2

c. r 5 m/V 5 672.0 g/64.0 cm 5 10.5 g/cm3


11/19/07 12:07:25 PM

CHAPTER 18 SOLIDS 361CHAPTER 18 SOLIDS 361




43. A solid sphere has a radius of 2.0 cm and a mass of 352.0 g.

a. What is the volume of the sphere? b. What is the surface area of the sphere?

(A useful way to remember the formula for the area of a sphere is that it is 4 times the area of a circle of the same radius: 4pr2.)

c. What is the density of the sphere? d. What can you say about the material

used to make this sphere and the cube of the previous problem?

44. A solid 5.0-kg cylinder is 10 cm tall with a radius of 3.0 cm. Show that its density is 18 g/cm3.

45. A cube of metal 0.30 m to a side is said to be pure gold. Gold has a density of 1.93 × 104 kg/m3.

a. What is the volume of the gold cube? b. Calculate the mass of the cube. c. Calculate the weight of the cube in

newtons, and then in pounds (recall that 1 N = 0.22 lb). Could you lift it?

46. What is the weight of a cubic meter of cork? Could you lift it? (For the density of cork, use 400 kg/m3.)

47. A certain spring stretches 3 cm when a load of 15 N is suspended from it. How much will the spring stretch if 45 N is suspended from it (and the spring doesn’t reach its elastic limit)?

48. If a certain spring stretches 4 cm when a load of 10 N is suspended from it, how much will the spring stretch if it is cut in half and 10 N is suspended from it?

49. Consider eight one-cubic-centimeter sugar cubes stacked two-by-two to form a single bigger cube. What will be the volume of the combined cube? How does its surface area compare to the total surface area of the eight separate cubes?

50. Consider eight little spheres of mercury, each with a diameter of 1 millimeter. When they coalesce to form a single sphere, how big will it be? How does its surface area compare to the total surface area of the pre-vious eight little spheres?

More Problem-Solving PracticeAppendix F

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361

43. a. V 5 4/3pr3 5 4/3p(2.0 cm)3 5 33.5 cm3

b. A 5 4pr2 5 4p(2.0 cm)2 5 50 cm2

c. r 5 m/V 5 352.0 g/33.5 cm3 5 10.5 g/cm3

d. Since both are solids with the same density, they are likely composed of the same material. Table 18.1 suggests silver.

44. r 5 m/V 5 5.0 kg/V. The volume of a cylinder is pr2h. So r 5 (5000 g)/[p(3.0 cm)2

� (10 cm)] 5 18 g/cm3

45. a. V 5L35 (0.30 m)350.027 m3

b. m 5 rV 5 (1.93 3 104 kg/m3) 3 (0.027 m3) = 520 kg

c. W 5 mg 5 (520 kg)(10 m/s2) 5 5200 N 3 0.22 lb/N 5 1100 lb. Too much to lift.

46. A cubic meter of cork has a mass of 400 kg and a weight of 4000 N. Its weight in pounds is 400 kg 3 2.2 lb/kg 5 880 lb, much too heavy to lift.

47. The spring constant k 5 15 N/3 cm, or 5 N/cm. From Hooke’s law, F 5 kDx, so Dx 5 F/k 5 45 N/(5 N/cm) 5 9 cm. (Or, 45 N is three times 15 N, so the spring will stretch three times as far, 9 cm.)

48. A half spring means half as much to stretch. (Tension throughout the spring stays 10 N.) So a 10 N load stretches it 2 cm. (Cutting the spring in half doubles the spring constant k. Initially k 5 10 N/4 cm 5 2.5 N/cm. When cut in half, k 5 10 N/2 cm 5 5 N/cm.)

49. 8 cm3 (same as before); half the surface area

50. Twice the diameter of the little spheres; half the total surface area


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