Download Chapter 6: Some Effects Due to Internal Forces

6. Some Effects Due to Internal Forces
To this point our illustrations of the laws of force
and motion have focused on the effects of external
forces acting on identifiable objects: balls, cars, people,
and so forth. We will next turn our attention to illustrating the effects of forces that act inside objects and materials. These internal forces can also be understood in
terms of the laws we have discussed. Such illustrations
are as important as those involving external forces.
We will begin by discussing forces that occur within solid objects such as trees, people, and automobiles.
This will establish some useful general principles.
Then we will examine forces within fluids, mainly
water and air, and discuss buoyancy and other important
manifestations of these internal forces. Finally, we will
discuss some of the forces that occur within the earth
itself and that govern some of the changes we see on the
earth’s surface.
Forces within Solids
Figure 6.1. What forces act on a piece of wood inside a
tree trunk?
We have seen that objects exert contact forces on
each other whenever they touch. This is also true for
individual samples of matter within any object.
Consider any ordinary solid: a tree, for example.
Imagine a small piece of wood inside the tree near the
center of the trunk (Fig. 6.1). What do we know about
the forces that act on this sample?
The method for finding forces outlined in Chapter
5 applies to this piece of wood as well as to any other
object. Using this method we first ask about the gravitational force on the piece of wood. There must be one,
because the sample has mass. Thus we know at least
one force, its weight, is acting on this object.
Because there are no long-range electromagnetic
forces, we next inquire about possible contact forces.
The sample is touching other wood at all of its boundaries. Electrical contact forces are being exerted at each
point of contact. It is difficult to describe these in detail
without knowing more about the internal structure of the
wood. However, we can determine exactly how large the
net force must be due to all of these interactions.
Also notice that the sample is in equilibrium; it is
not accelerating. This means that the total force must be
zero. The total force can be zero only if the combined
contact forces provide a force that exactly balances the
Figure 6.2. How can we describe the total effect of all
the contact forces?
weight of the sample. Thus we know that the combined
contact forces provide a net upward force on this sample
of wood, and that the strength of the contact forces is
exactly equal to the weight of the sample (Fig. 6.2).
The same is true for the forces within any material.
When the sample is at rest, contact forces balance the
long-range forces, usually gravity. If the surrounding
material is accelerating, these interior contact forces
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change so that a net force is on each piece of the sample, causing it to accelerate in accord with the Second
Law of Motion.
To better understand this last point, consider the
forces that act when you jump. The contact force with
the ground is the external force that accelerates your
body upward. Gravity, as always, provides a downward
force. There are no significant long-range electromagnetic interactions. But what force accelerates your
head? Your head is not in contact with the floor, so the
contact interaction with the floor cannot accelerate your
head. Your head, in fact, is accelerated by contact
forces exerted by that which it touches, namely the top
of the spinal column. The spinal column exerts an
upward force on your skull that just balances gravity
when your head is not accelerating and that exceeds
gravity (providing a net upward force) when you jump
(Fig. 6.3).
zontal motion. The contact forces, in turn, depend on
what a given piece of matter actually touches. It experiences no forces from objects that it does not touch.
Pressure
When you immerse an object in a fluid, it is subjected to forces from the fluid touching it. Because the
contact comes at so many different places, we speak of
“pressure” rather than of force. Pressure is defined as
the force per unit area of contact:
.
pressure 5 force
area
Thus, a 100-pound force distributed over an area of
4 square inches exerts a pressure of 25 pounds per square
inch. The same force distributed over an area of 1 square
inch exerts a pressure of 100 pounds per square inch.
For many applications, it is pressure which is the
crucial issue. A pound is a modest force, but if distributed over a very small area it may exert a pressure of
100,000 pounds per square inch on that small area.
Some materials may not be able to withstand this concentration of force and may give way, even though this
same force might not cause any problem when spread
over a much larger area (thus producing less pressure).
You can investigate the pressure exerted on objects
in fluids with a device like that shown in Figure 6.4.
The device, called a gauge, exposes a small, known area
to the contact forces of the fluid. The strength of the
force is measured by how much it compresses the
gauge’s spring. Because you are interested in measuring pressure at a particular position in the fluid, imagine
the device to be as small as possible. Then you can
speak of the pressure at a certain point.
Contact Force
(with spinal
column)
Head
Weight
Figure 6.3. What force causes your head to accelerate
when you jump?
The contact force explains the neck pain that a jogger sometimes experiences. When a person is running,
the head moves up and down. Each time the head
comes down, it must be stopped and then accelerated
upward again by forces exerted by the spine. If these
motions are very uneven or jerky, the forces the spine
exerts on the skull can resemble hammer-blows.
Continued over a long time, they can cause some structures of the spinal column to deteriorate. Such damage
can be minimized by running as smoothly as possible,
reducing the rate of vertical acceleration by running on
softer surfaces and wearing shoes with soft soles.
The motion of every piece of matter—whether a
single atom, a complete object, or a small part of such
an object—is governed by the laws of motion and force.
In the absence of long-range electromagnetic forces, the
interplay between weight and contact forces governs
vertical motion, and contact forces alone govern hori-
Figure 6.4. A pressure gauge.
When you investigate pressures with your imagined gauge, you find four simple rules:
1. Fluids at rest only exert pressure perpendicular to the surface of the object in contact with
the fluid. Fluids at rest do not exert shear
(“sideways”) forces, although these are present
when the fluid is moving.
2. Pressure does not depend on the orientation of
the pressure-measuring device. At a given depth,
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pressure in the fluid does not depend on direction.
Buoyant
Force
3. Pressure does depend on depth. The deeper
the gauge, the greater the pressure. Indeed, the
pressure at any depth in the fluid must equal the
weight of the column of fluid above it, if the
column is at rest. This insight provides a way
to calculate the pressure at any depth in a fluid:
Imagine a fluid column of unit area above the
point and compute its weight. Also, it may be
necessary to add the weight of the air column
above the fluid if its surface is exposed to the
atmosphere. At sea level a column of air with
a cross-sectional area of one square inch and
reaching to the top of the atmosphere weighs
14.7 pounds.
Object
Weight
Figure 6.5. Fluid pressure causes a net upward buoyant
force on any object.
4. The pressure is the same at all points of the
same depth. Pressure does not depend on the
total surface area of the fluid above the gauge.
For example, the pressure on the bottom of a
dam does not depend on the area of the lake.
occupied by the immersed object in Figure 6.5 is filled
with fluid instead. Now recall our earlier discussion of
the forces on a piece of wood inside a tree. Remember
that the adjacent wood was exerting a net upward force
that was just large enough to keep the piece from falling.
The same arguments we used there lead us to conclude
here that the water adjacent to our sample in Figure 6.5
exerts a force on it that just balances its weight.
Now, suppose that the immersed object (Fig. 6.5)
exactly fills the space previously occupied by the sample of fluid. The surrounding fluid exerts a net upward
force on the object equaling the force previously exerted on the sample of fluid. This force is equal to the
weight of the fluid that the object displaced (i.e., took
the place of). This is the buoyant force. The rule governing the strength of the buoyant force is known as
Archimedes’ Principle:
It may seem incredible at first to realize that the
atmosphere exerts a pressure of almost 15 pounds per
square inch over the surface of your body. Your body
has a lot of square inches; the resulting total force on
your body is thousands of pounds, enough to crush you
to a pulp. You avoid this fate by keeping air pressure
inside your body which balances the air pressure on the
outside. The alternative is not pleasant!
Buoyant Forces
Most of us have noticed that objects immersed in a
fluid, such as water, seem to weigh less than before.
However, from our study of the gravitational interaction we know that neither of the factors affecting
weight (mass and distance) have changed. Thus,
objects really weigh the same when immersed in the
fluid. However, they are undeniably easier to lift when
immersed. Why?
This situation is illustrated in Figure 6.5, where a
fluid pushes in on an immersed object from all sides.
Pressure in the fluid increases with depth, so the forces
pushing up on the bottom of the object are larger than
those pushing down on its top. The total result is a net
upward contact force, called a buoyant force. You
should recognize that the buoyant force is really the
result of all the contact forces between the immersed
object and the surrounding fluid. However, it is easier
to think of buoyancy as a single force rather than as a
large number of smaller forces acting in different directions as shown in Figure 6.5.
With a little effort we can discern the strength of the
buoyant force in any situation. Imagine that the space
An object immersed in a fluid experiences an
upward buoyant force due to contact interactions with the surrounding fluid, whose
strength is equal to the weight of the displaced
fluid.
An object immersed in fluid experiences two
forces: its weight pulling downward and the upward
buoyant force. The object accelerates in the direction
of the net force, which is the stronger of these two. An
object sinks if its weight is stronger than the buoyant
force, and it rises if the buoyant force is stronger.
Consider a balloon and a solid metal ball that are
the same size and are both submerged in water (Fig.
6.6). The buoyant forces on the two have exactly the
same strength, because they both displace the same
amount of fluid. However, the weight of the balloon is
much less than the buoyant force. Thus the net force is
upward and the balloon rises to the surface. Since the
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weight of the metal ball is greater than the weight of an
equal volume of water, its weight exceeds the strength
of the buoyant force. The net force on the ball is downward, and it sinks to the bottom.
Buoyant
Force
The density of an object or sample of fluid is its mass
per unit volume, or its mass divided by its volume. An
object sinks if its density is greater than the density of
the fluid in which it is immersed. It rises if its density
is less than the fluid in which it is immersed. Do you
see why?
Densities of materials are often compared with the
density of water. This relative density is given the name
specific gravity. Thus, if a rock sample has a density
of 2.3, or a specific gravity of 2.3, its density is 2.3
times the density of water.
Buoyant
Force
Floating Objects
Weight
Weight
We next turn our attention to objects floating on the
surface of fluids, such as objects floating on water. By
now you should visualize two forces acting on the floating boat in Figure 6.8: its weight pulling downward and
a buoyant force exerted by the surrounding water pressing upward. These two forces just balance each other,
so that once the boat reaches a certain level, it neither
rises nor sinks.
Figure 6.6. A balloon and a metal ball might be the
same size. Why does one float and the other sink?
Now imagine two objects that have the same weight
but different volumes (Fig. 6.7). One might be a ball
made of wood and the other a much smaller ball made of
iron. Since the wood ball is larger, the buoyant force acting on it is also larger. If the wood weighs less than an
equal volume of water, the buoyant force will exceed the
weight of the ball; the net force will be upward, and the
ball will rise to the surface. The buoyant force on the
smaller metal ball, however, will be less than that on the
Buoyant
Force
Buoyant
Force
Figure 6.8. How much of the boat is below water level?
Weight
Weight
How much of the boat will sink below the water
level? We have seen that the strength of the buoyant
force is equal to the weight of displaced fluid. Thus, the
volume of the boat below the water surface must displace a weight of water equal to the weight of the boat
and passengers. What will happen if the boat is loaded
more heavily? Its weight will increase and it will sink
until it displaces more water. How much? Enough
water must be displaced so that its weight equals the
weight of the additional freight. What happens if the
ship does not have enough volume to displace the extra
water? It sinks.
Icebergs illustrate these same points. They float
because ice has a density about 10% less than water. It
only takes 90% of the iceberg’s volume to displace enough
water to equal its total weight, so the iceberg floats with
about 10% of its volume above the surface of the water.
Several features of this phenomenon are worth not-
Figure 6.7. These objects have the same weight. Why
does one sink and the other float?
wood ball. The volume of water it displaces weighs less
the metal. Therefore, the metal ball sinks, even though
its weight is the same as the weight of the wood ball.
The general rules concerning floating and sinking
are sometimes summarized simply in terms of density.
density 5
mass .
volume
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Figure 6.9. A floating iceberg sinks when more mass is added and rises when mass leaves or is taken away.
Figure 6.10. The earth’s crust floats in (or on) the mantle much as icebergs float in water.
ing. First, compare a large iceberg with a smaller one.
Notice that the larger one has more volume below the
water as well as above the water compared to the smaller one. Second, imagine what will happen if more mass
is added to the part of the iceberg above the water, say by
a snowstorm or by visiting walruses (Fig. 6.9). The iceberg will sink farther into the water, thus increasing the
buoyant force needed to balance the increased mass.
Finally, imagine what will happen if some of the material above the water is lost, by melting for example. The
iceberg will rise in the water, since a smaller buoyant
force is needed to balance the iceberg’s lightened weight.
the mantle’s buoyant force supports its weight, together
with that of the water above.
The materials that make up the continents are significantly less dense than the oceanic crust. These
lighter continental materials sink into the mantle only
far enough to displace the weight of mantle material
equal to their own. Each continent, and indeed each
mountain, has “roots” extending far enough below it to
provide the necessary buoyant force.
You can deduce many of the consequences of these
ideas by remembering our iceberg example. The taller
mountain or continent must have deeper roots, just as
the large iceberg must have more volume below the surface. If material is added to a continent—for example,
by the formation of a glacier; a flow of lava, or even the
construction of a large building—the crust will sink,
over time, farther into the mantle. If material is
removed—for example, by erosion or the melting of a
glacier—the underlying crust will rise. The general
principle governing this fluid-like equilibrium in the
earth’s crust is called isostasy.
Buoyancy in the Earth’s Crust: Isostasy
An interesting application of buoyancy occurs in
the earth’s crust. The continents actually float in the
earth’s mantle in much the same way that ships or icebergs float in water. The outer layer of the mantle is hot
enough to have some characteristics of a fluid. In particular, forces within the mantle adjust over long periods, following the general rules for fluids we described
earlier. The crust lies above this semifluid layer. Its
general features are shown in Figure 6.10.
The crust underneath the oceans is quite dense (but
less dense than the mantle) and relatively thin. The
oceanic crust sinks just far enough into the mantle that
Convection
We now look at one additional illustration of buoyant forces. Suppose we have regions of high and low
density occurring within the same fluid because of dif-
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Figure 6.11. Daytime convection pattern near a seashore. Air near the land surface is hotter than air over the water.
Figure 6.12. Nighttime convection pattern near a seashore. Air near the land surface is cooler than air over the water.
ferent temperatures within the fluid. Most fluids
expand when their temperatures rise. Thus, they
become less dense because the same mass occupies
greater volume.
High-density regions of fluid sink, while low-density regions of fluid rise. Do you see why? High-density
regions displace surrounding fluid, but the displaced
fluid does not weigh as much. Thus, the buoyant force
on the high-density region is less than its weight, so it
sinks. Low-density regions also displace fluid. In their
case, the displaced fluid weighs more. Thus, the buoyant force on the region is greater that its weight, so it
rises. The result in both cases is that cooler regions of
fluid sink as warmer regions rise. These motions cause
interesting and important processes in nature.
Consider the common example shown in Figure
6.11 of a large body of water adjacent to land. The temperature of the soil is higher than the temperature of the
water during the day. Air above the soil heats and
expands, becoming less dense than the air above; therefore, it rises. As it rises, it is replaced by the cooler,
denser air from the water. A circulation pattern is established as shown in the figure. The result is a cool breeze
from the body of water during the day.
The situation is reversed at night. The land cools
more rapidly and, in many cases, becomes cooler than
the water. Air circulation then proceeds in the opposite
direction, with the surface wind blowing away from the
land (Fig. 6.12).
This kind of circulation, caused by differences in tem-
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perature and density, is called convection. It is responsible for many of the broad circulation patterns in the earth’s
atmosphere and oceans. Convection is probably the mechanism that also drives broad regions of the earth’s crust
from place to place on the surface of the earth.
Notice that convection depends on the presence of
gravitational forces. Neither weight nor buoyant forces
would be present without gravity, so there would be no
convection.
6.
7.
Summary
earth’s crust seen as “floating” in the underlying
mantle rock.
Pressure: The force applied to the surface of an
object divided by the area over which the force is
applied. Force per unit area.
Specific Gravity: The density of a material divided by the density of a reference substance, usually
water. The specific gravity of water is 1.0, if water
itself is the reference substance.
D. FOCUS QUESTIONS
1. In each of the following situations:
a. Describe the motion.
b. Analyze the motion by applying the procedure of
Chapter 5 (Finding Forces).
c. List the fundamental principles used in coming to
the results of your analysis.
(1) A cube of wood within a tree trunk.
(2) A cube of air above you on a calm day. The
air is not moving.
(3) A part of the surface of a can when the air
was pumped out.
(4) An imaginary, vertical cylinder of water in
a tub of water at rest.
2. Consider three spheres of the same size completely
submerged in a tub of water. One sphere is made of
lead, another is made of wood, and the third is
made of water.
a. How do the weights of the three objects compare
to each other?
b. How do the buoyant forces on the three objects
compare to each other?
c. Analyze the motion of the three objects by
applying the procedure above.
d. List the fundamental principles used in coming
to the results of your analysis.
3. Consider a floating iceberg:
a. Describe why it floats by using the procedure
given to analyze motion.
b. What happens when some of the iceberg melts?
Why does it happen?
c. What happens when it snows on the top of the
iceberg? Why?
This chapter extends applications of the laws of
motion and force to forces within materials, particularly fluids.
STUDY GUIDE
Chapter 6: Some Effects Due to Internal Forces
A.
1.
2.
3.
4.
FUNDAMENTAL PRINCIPLES
The First Law of Motion: See Chapter 3.
The Second Law of Motion: See Chapter 3.
The Third Law of Motion: See Chapter 3.
The Universal Law of Gravitation:
See Chapter 4.
5. The Electric Force Law: See Chapter 4.
B. MODELS, IDEAS, QUESTIONS, OR APPLICATIONS
1. How can you use the laws of force and motion to
analyze the motion of a part of a stationary object?
2. How can you use the laws of force and account for
the motion of a part of a fluid? What is a buoyant
force? What is Archimedes’ Principle?
3. How can you apply this understanding to explain:
a. floating objects?
b. buoyancy in the crust of the earth?
c. convection in a liquid or a gas?
C. GLOSSARY
1. Archimedes’ Principle: An object immersed in a
fluid experiences an upward buoyant force due to
contact interactions with the surrounding fluid,
whose strength is equal to the weight of the displaced fluid.
2. Buoyant Force: For an object at rest which is
wholly or partially immersed in a fluid, the buoyant
force is the net or resultant force of the contact
forces exerted on the object by the fluid.
3. Convection: The circulation (movement) of the
matter of a fluid because of differences in temperature and pressure throughout the fluid.
4. Density: The mass per unit volume of a substance.
5. Isostasy: The word means “balance.” In geology,
the term is applied to the balance of the buoyant
force and the force of gravity on a piece of the
E. EXERCISES
6.1. Consider a piece of air 20 meters above the
ground. What keeps it from falling?
6.2. Imagine yourself running. What forces govern the motion of each foot from the time it leaves the
ground until it returns to the ground?
6.3. Imagine yourself in an elevator. What force
accelerates your stomach when the elevator begins to
move up?
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6.4. Explain why it is easier to lift a rock when it
is under water than when it is above water.
6.20. Explain why buoyant forces cause convection when a fluid such as air is unevenly heated.
6.5. Do heavy objects always sink? Under what
circumstances would a heavy object not sink?
6.21. An object sinks in oil but floats in water.
Which is true?
(a) The above situation is not possible.
(b) Oil decreases the gravitational force on an object.
(c) Water increases the gravitational force on an
object.
(d) The buoyant force on the object immersed in
oil is greater than the gravitational force.
(e) The buoyant force on the object floating in
water is equal to the gravitational force.
6.6. Explain why an object sinks if it is more dense
than the fluid it is immersed in.
6.7. Explain why the strength of a buoyant force is
equal to the weight of displaced fluid.
6.8. Describe the interaction which causes the
buoyant force on an immersed object. Is this a longrange or a contact interaction? What other interaction is
important when describing the behavior of objects
immersed in a fluid?
6.9. Does the strength of the buoyant force depend
on the weight of an immersed object, its size, its density, the density of the fluid, all of these, or none of these?
Explain your answer.
6.10. Explain why a kilogram of wood can float in
water when a kilogram of iron sinks.
6.11. Explain why a helium-filled balloon rises but
why an air-filled balloon does not. What factors determine how high the helium-filled balloon will rise?
6.12. Explain the meaning of “density.’’
6.13. Explain the meaning of “specific gravity.’’
6.14.
upward?
Why is a buoyant force always directed
6.15. Outdoor swimming pools in certain areas of
California sometimes rise out of the ground when they
are drained for the winter. How could this occur?
6.16. Explain why an aircraft carrier can float
while a small ball made of the same steel will sink.
6.17. Explain why an oil tanker sits lower in the
water when loaded than when it is empty.
6.18. Why are the most dense materials of the earth
mostly found near its center?
6.19. Some parts of the U.S. require the excavation of
soft rocks and sediment before laying foundations for large
buildings. What would you expect to happen to such a
building if its weight were greater than that of the removed
rock? What would happen if the building weighed less?
Explain why you would expect such behavior.
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