Objectives
• Describe the structure of
crystals. (18.1)
• Describe the factors that
determine the density of a
material. (18.2)
18 SOLIDS
• Explain the property of
elasticity. (18.3)
THE BIG
IDEA
• 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
1. 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.
2. 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.
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 discovery 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.
discover!
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.
3. As linear size of an object
increases, the ratio of surface
area to volume decreases.
344
Solids can be described in terms
of crystal structure, density,
and elasticity.
..........
SOLIDS
344
Analyze and Conclude
1. 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?
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
......
CONCEPT
CHECK
18.1 Crystal
Structure
Key Term
crystal
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.
FIGURE 18.1 When X-rays pass through
a crystal of common table
salt (sodium chloride), they
produce a distinctive pattern on photographic film.
Teaching Tip Crystal-growing
kits are a great way to show the
structures of crystals.
FIGURE 18.2
In this model of a sodium
chloride crystal, the large
spheres represent chloride
ions, and the small ones
represent sodium ions.
The shape of a crystal
mirrors the geometric
arrangement of atoms within the
crystal.
......
When 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 photographic 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 photographic film show that the atoms in a crystal
have an orderly arrangement. Every crystalline structure has its own unique X-ray
pattern. For example, in a sodium chloride
crystal, the atoms are arranged like a threedimensional 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.
CONCEPT
CHECK
Teaching Resources
• Reading and Study
Workbook
What determines the shape of a crystal?
• Transparency 29
• PresentationEXPRESS
• Interactive Textbook
CHAPTER 18
SOLIDS
345
345
18.2 Density
Key Terms
density, weight density,
specific gravity
FIGURE 18.3 When the loaf of bread
is squeezed, its volume
decreases and its
density increases.
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.
18.2 Density
One 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:
Common Misconception
Density 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 The symbol for density
is the Greek letter ®.
Demonstration
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.
346
think!
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
346
mass
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 material; 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 volume 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.
Table 18.1
Densities of a Few Substances
Solids
Density
(g/cm3)
Liquids
Density
(g/cm3)
Iridium
22.7
Mercury
13.6
Osmium
22.6
Glycerin
1.26
Platinum
21.4
Sea water
1.03
Gold
19.3
Water at 4°C
1.00
Uranium
19.0
Benzene
0.90
Lead
11.3
Ethyl alcohol
0.81
Silver
10.5
Copper
8.9
Brass
8.6
Iron
7.8
Steel
7.8
Tin
7.3
Diamond
3.5
Aluminum
2.7
Graphite
2.25
Ice
0.92
Pine wood
0.50
Balsa wood
0.12
For: Links on density
Visit: www.SciLinks.org
Web Code: csn – 1802
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 lowvolume but heavy weights at the
bottom of boat keels.
A quantity known as weight density can be expressed by the
amount of weight a body has per unit volume:
weight
weight density volume
......
CONCEPT
CHECK
think!
The density of gold is
19.3 g/cm3. What is its
specific gravity?
Answer: 18.2.2
The density of a
material depends
upon the masses of the individual
atoms that make it up, and the
spacing between those atoms.
......
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.
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.
CONCEPT
CHECK
What determines the density of a material?
CHAPTER 18
SOLIDS
347
347
Teaching Tip Acknowledge
weight density, common in the
British system of units—like the
density of water, 62.4 lb/ft3.
do the math!
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 substance 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:
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.
density 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
Teaching Resources
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.)
18.3 Elasticity
FIGURE 18.4 The bow is elastic.
When the deforming
force is removed,
the bow returns to
its original shape.
• Reading and Study
Workbook
• Problem Solving Exercises in
Physics 10-1
• Transparency 30
• PresentationEXPRESS
• Interactive Textbook
18.3 Elasticity
Key Terms
elastic, inelastic, Hooke’s law,
elastic limit
348
57.9 g
mass
19.3 g/cm3
volume
3.00 cm3
348
When 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 material 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 temporarily 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 distorted are said to be inelastic. Clay, putty, and dough are
inelastic materials. Lead is also inelastic, since it is easy to
distort it permanently.
Demonstration
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.
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.
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!
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.
Demonstration
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 compression), 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.
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 be
written as F k x,
where k is expressed
in N/m.
......
CONCEPT What characteristics are described by an
CHECK
object’s elasticity?
think!
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
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.
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.
......
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.
CONCEPT
CHECK
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
Teaching Resources
• Laboratory Manual 49
CHAPTER 18
SOLIDS
349
349
18.4 Compression
and Tension
Physics on the Job
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.
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.
18.4 Compression and Tension
It is interesting to note that
one cubic inch of human bone
can withstand a 2-ton force.
A beam in the position
of the one below in
Figure 18.6 is known as
a cantilever beam.
Steel 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 girders 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 carries 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.
FIGURE 18.6 The top part of the beam
is stretched and the bottom part is compressed.
The middle portion is
neither stretched nor
compressed.
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 (indicated by the red dashed line in the figure).
350
350
FIGURE 18.7
The 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.
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 neutral 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 bottom 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 purpose, comparatively little material is needed. An I-beam is nearly as
strong as a solid bar, and its weight is considerably less.
......
CONCEPT How is a horizontal beam affected by the
CHECK
load it supports?
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.
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.
think!
......
A horizontal beam
CHECK 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).
CONCEPT
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
Teaching Resources
• Reading and Study
Workbook
• PresentationEXPRESS
• Interactive Textbook
CHAPTER 18
SOLIDS
351
351
discover!
MATERIALS
discover!
nails or bolts, seven
sticks
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.
EXPECTED OUTCOME
Demonstration
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.
Hold an egg vertically 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 overlooked 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?
Link
Link to
to ARCHITECTURE
ARCHITECTURE
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
352
352
18.5 Scaling
18.5 Scaling
Key Term
scaling
Did 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 scaling
and described the different bone sizes of
various creatures.
Common Misconception
When 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.
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.
CHAPTER 18
SOLIDS
353
353
Demonstration
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.
Muscular strength
depends on the number of fibers in a particular muscle. Hence the
strength of a muscle
is proportional to its
cross-sectional area.
think!
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.
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 crosssectional surface area? Its
total surface area?
Answer: 18.5
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.
354
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 relationship, let’s consider the simple case of a solid cube of matter, 1 centimeter on a side.
A 1-cubic-centimeter cube has a cross section of 1 square centimeter. 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 dimensions, 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 enlargement, whereas volume and weight grow as the cube of the enlargement. These relationships are illustrated in Figure 18.9.
The volume (and weight) increases much faster than the corresponding 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.
......
CONCEPT If the linear dimensions of an object double, by how
CHECK
354
much will the cross-sectional area grow?
Ask Explain why small cars
are more affected by wind. (They
have more cross-sectional area
compared to weight than larger
cars.)
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
TE bubble
text
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 humanmade devices in the era of
nanotechnology do the same?
FIGURE 18.10 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. (Volume 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 peeling 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.
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.
A sphere has less surface 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.
CHAPTER 18
SOLIDS
355
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.
355
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!
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 fivedegree 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.
......
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.
CONCEPT
CHECK
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.
Teaching Resources
• Reading and Study
Workbook
• Concept-Development
Practice Book 18-1, 18-2
• Problem-Solving Exercises in
Physics 10-2
• Laboratory Manual 50
• Transparency 31
• Next-Time Question 18-1
• Conceptual Physics Alive!
DVDs Scaling
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 nourishment. 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 statement “the bigger they are, the harder they fall” holds true and is a
consequence of the small ratio of surface area to weight. Air resistance 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 leisurely 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.
......
CONCEPT If the linear dimensions of an object double, by how
CHECK
356
much will the volume grow?
REVIEW
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Teaching Resources
• TeacherEXPRESS
• Virtual Physics Lab 19
• Conceptual Physics Alive!
DVDs Scaling
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 enlargement. As the linear size of an object
increases, the volume grows faster than
the total surface area.
Key Terms
crystal (p. 345)
density (p. 346)
weight density
(p. 347)
specific gravity
(p. 347)
think! Answers
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
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:
10 N
15 N
4 cm
x
Then x (15 N) (4 cm)/(10 N) 6 cm.
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 compressed, 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.
••••••
elastic (p. 348)
inelastic (p. 348)
Hooke’s law (p. 349)
elastic limit (p. 349)
scaling (p. 353)
density of gold
19.3 g/cm3
19.3
density of water
1.0 g/cm3
CHAPTER 18
SOLIDS
357
357
ASSESS
Check Concepts
1. Crystalline—regular geometric
shapes; noncrystalline—
nonregular shapes
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2. X-ray patterns (Figure 18.1)
3. Crystals in etched metals
4. no change
Check Concepts
5. The atoms are not the most
closely packed.
Section 18.1
••••••
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.)
1. How does the arrangement of atoms
differ in a crystalline and a noncrystalline
substance?
Section 18.4
8. Mass density—mass/volume;
weight density—weight/
volume
2. What evidence do we have for the
microscopic crystal nature of some solids?
13. Is a steel beam slightly shorter when it
stands vertically? Explain.
9. a. It returns to its original
shape when bent.
b. It doesn’t return to its
shape.
3. What evidence do we have for the visible
crystal nature of some solids?
14. Where is the neutral layer in a horizontal beam that supports a load?
6. same
7. no; yes, decreases; yes,
increases
Section 18.2
10. F ~ Dx (or F 5 kDx), where Dx
is the amount of stretch or
compression.
4. What happens to the density of a uniform piece of wood when we cut it in half?
11. Stretch limit beyond which
the substance is permanently
deformed.
5. Uranium is the heaviest atom found in
nature. Why isn’t uranium metal the most
dense material?
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
15. Why is the cross section of a metal beam
I-shaped and not rectangular?
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?
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?
358
358
20. a. elephant
b. mouse
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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?
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 compression. Which side is which?
Think and Explain
••••••
21. You take 1000 milligrams of a vitamin.
Your friend takes 1 gram of the same vitamin. Who takes more?
22. Your friend says that the primary difference 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?
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.
CHAPTER
CHAPTER18
18
SOLIDS
SOLIDS
359
359
Think and Explain
21. 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.
359
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.
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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 sideways. Why?
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.)
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 supports 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.
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?
40. Large drops fall faster. Large
drops have less surface
area per unit weight, and
therefore less air resistance
per unit weight.
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. 兹2
苶 cm, or 1.26 cm
3
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
360
Think and Solve
••••••
33. Why is cement not needed between the
stone blocks of an arch that has the shape of
an inverted catenary?
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?
34. If you use a batch of cake batter for cupcakes instead of a cake and bake them for
the time suggested for baking a cake, what
will be the result?
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?
360
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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:
4p r2.)
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 previous eight little spheres?
43. a. V 5 4/3pr 3 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 5 L3 5 (0.30 m)3 50.027 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
Teaching Resources
More Problem-Solving Practice
Appendix F
CHAPTER 18
SOLIDS
361
• Computer Test Bank
• Chapter and Unit Tests
361