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To this point our illustrations of the laws of forceand motion have focused on the effects of externalforces acting on identifiable objects: balls, cars, people,and so forth. We will next turn our attention to illustrat-ing the effects of forces that act inside objects and mate-rials. These internal forces can also be understood interms of the laws we have discussed. Such illustrationsare as important as those involving external forces.

We will begin by discussing forces that occur with-in solid objects such as trees, people, and automobiles.This will establish some useful general principles.Then we will examine forces within fluids, mainlywater and air, and discuss buoyancy and other importantmanifestations of these internal forces. Finally, we willdiscuss some of the forces that occur within the earthitself and that govern some of the changes we see on theearth’s surface.

Forces within Solids

We have seen that objects exert contact forces oneach other whenever they touch. This is also true forindividual samples of matter within any object.Consider any ordinary solid: a tree, for example.Imagine a small piece of wood inside the tree near thecenter of the trunk (Fig. 6.1). What do we know aboutthe forces that act on this sample?

The method for finding forces outlined in Chapter5 applies to this piece of wood as well as to any otherobject. Using this method we first ask about the gravi-tational force on the piece of wood. There must be one,because the sample has mass. Thus we know at leastone force, its weight, is acting on this object.

Because there are no long-range electromagneticforces, we next inquire about possible contact forces.The sample is touching other wood at all of its bound-aries. Electrical contact forces are being exerted at eachpoint of contact. It is difficult to describe these in detailwithout knowing more about the internal structure of thewood. However, we can determine exactly how large thenet force must be due to all of these interactions.

Also notice that the sample is in equilibrium; it isnot accelerating. This means that the total force must bezero. The total force can be zero only if the combinedcontact forces provide a force that exactly balances the

weight of the sample. Thus we know that the combinedcontact forces provide a net upward force on this sampleof wood, and that the strength of the contact forces isexactly 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 thelong-range forces, usually gravity. If the surroundingmaterial is accelerating, these interior contact forces

6. Some Effects Due to Internal Forces

Figure 6.1. What forces act on a piece of wood inside atree trunk?

Figure 6.2. How can we describe the total effect of allthe contact forces?

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change so that a net force is on each piece of the sam-ple, causing it to accelerate in accord with the SecondLaw of Motion.

To better understand this last point, consider theforces that act when you jump. The contact force withthe ground is the external force that accelerates yourbody upward. Gravity, as always, provides a downwardforce. There are no significant long-range electromag-netic interactions. But what force accelerates yourhead? Your head is not in contact with the floor, so thecontact interaction with the floor cannot accelerate yourhead. Your head, in fact, is accelerated by contactforces exerted by that which it touches, namely the topof the spinal column. The spinal column exerts anupward force on your skull that just balances gravitywhen your head is not accelerating and that exceedsgravity (providing a net upward force) when you jump(Fig. 6.3).

Figure 6.3. What force causes your head to acceleratewhen you jump?

The contact force explains the neck pain that a jog-ger sometimes experiences. When a person is running,the head moves up and down. Each time the headcomes down, it must be stopped and then acceleratedupward again by forces exerted by the spine. If thesemotions are very uneven or jerky, the forces the spineexerts on the skull can resemble hammer-blows.Continued over a long time, they can cause some struc-tures of the spinal column to deteriorate. Such damagecan be minimized by running as smoothly as possible,reducing the rate of vertical acceleration by running onsofter surfaces and wearing shoes with soft soles.

The motion of every piece of matter—whether asingle atom, a complete object, or a small part of suchan object—is governed by the laws of motion and force.In the absence of long-range electromagnetic forces, theinterplay between weight and contact forces governsvertical motion, and contact forces alone govern hori-

zontal motion. The contact forces, in turn, depend onwhat a given piece of matter actually touches. It expe-riences no forces from objects that it does not touch.

Pressure

When you immerse an object in a fluid, it is sub-jected to forces from the fluid touching it. Because thecontact comes at so many different places, we speak of“pressure” rather than of force. Pressure is defined asthe force per unit area of contact:

pressure 5 force .area

Thus, a 100-pound force distributed over an area of4 square inches exerts a pressure of 25 pounds per squareinch. The same force distributed over an area of 1 squareinch exerts a pressure of 100 pounds per square inch.

For many applications, it is pressure which is thecrucial issue. A pound is a modest force, but if distrib-uted over a very small area it may exert a pressure of100,000 pounds per square inch on that small area.Some materials may not be able to withstand this con-centration of force and may give way, even though thissame force might not cause any problem when spreadover a much larger area (thus producing less pressure).

You can investigate the pressure exerted on objectsin fluids with a device like that shown in Figure 6.4.The device, called a gauge, exposes a small, known areato the contact forces of the fluid. The strength of theforce is measured by how much it compresses thegauge’s spring. Because you are interested in measur-ing pressure at a particular position in the fluid, imaginethe device to be as small as possible. Then you canspeak of the pressure at a certain point.

Figure 6.4. A pressure gauge.

When you investigate pressures with your imag-ined gauge, you find four simple rules:

1. Fluids at rest only exert pressure perpendic-ular to the surface of the object in contact withthe fluid. Fluids at rest do not exert shear(“sideways”) forces, although these are presentwhen the fluid is moving.

2. Pressure does not depend on the orientation ofthe pressure-measuring device. At a given depth,

Head

Contact Force (with spinal column)

Weight

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pressure in the fluid does not depend on direction.

3. Pressure does depend on depth. The deeperthe gauge, the greater the pressure. Indeed, thepressure at any depth in the fluid must equal theweight of the column of fluid above it, if thecolumn is at rest. This insight provides a wayto calculate the pressure at any depth in a fluid:Imagine a fluid column of unit area above thepoint and compute its weight. Also, it may benecessary to add the weight of the air columnabove the fluid if its surface is exposed to theatmosphere. At sea level a column of air witha cross-sectional area of one square inch andreaching to the top of the atmosphere weighs14.7 pounds.

4. The pressure is the same at all points of thesame depth. Pressure does not depend on thetotal surface area of the fluid above the gauge.For example, the pressure on the bottom of adam does not depend on the area of the lake.

It may seem incredible at first to realize that theatmosphere exerts a pressure of almost 15 pounds persquare inch over the surface of your body. Your bodyhas a lot of square inches; the resulting total force onyour body is thousands of pounds, enough to crush youto a pulp. You avoid this fate by keeping air pressureinside your body which balances the air pressure on theoutside. The alternative is not pleasant!

Buoyant Forces

Most of us have noticed that objects immersed in afluid, such as water, seem to weigh less than before.However, from our study of the gravitational interac-tion we know that neither of the factors affectingweight (mass and distance) have changed. Thus,objects really weigh the same when immersed in thefluid. However, they are undeniably easier to lift whenimmersed. Why?

This situation is illustrated in Figure 6.5, where afluid pushes in on an immersed object from all sides.Pressure in the fluid increases with depth, so the forcespushing up on the bottom of the object are larger thanthose pushing down on its top. The total result is a netupward contact force, called a buoyant force. Youshould recognize that the buoyant force is really theresult of all the contact forces between the immersedobject and the surrounding fluid. However, it is easierto think of buoyancy as a single force rather than as alarge number of smaller forces acting in different direc-tions as shown in Figure 6.5.

With a little effort we can discern the strength of thebuoyant force in any situation. Imagine that the space

occupied by the immersed object in Figure 6.5 is filledwith fluid instead. Now recall our earlier discussion ofthe forces on a piece of wood inside a tree. Rememberthat the adjacent wood was exerting a net upward forcethat was just large enough to keep the piece from falling.The same arguments we used there lead us to concludehere that the water adjacent to our sample in Figure 6.5exerts 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 sam-ple of fluid. The surrounding fluid exerts a net upwardforce on the object equaling the force previously exert-ed on the sample of fluid. This force is equal to theweight of the fluid that the object displaced (i.e., tookthe place of). This is the buoyant force. The rule gov-erning the strength of the buoyant force is known asArchimedes’ Principle:

An object immersed in a fluid experiences anupward buoyant force due to contact interac-tions with the surrounding fluid, whosestrength is equal to the weight of the displacedfluid.

An object immersed in fluid experiences twoforces: its weight pulling downward and the upwardbuoyant force. The object accelerates in the directionof the net force, which is the stronger of these two. Anobject sinks if its weight is stronger than the buoyantforce, and it rises if the buoyant force is stronger.

Consider a balloon and a solid metal ball that arethe same size and are both submerged in water (Fig.6.6). The buoyant forces on the two have exactly thesame strength, because they both displace the sameamount of fluid. However, the weight of the balloon ismuch less than the buoyant force. Thus the net force isupward and the balloon rises to the surface. Since the

Object

Buoyant Force

Weight

Figure 6.5. Fluid pressure causes a net upward buoyantforce on any object.

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weight of the metal ball is greater than the weight of anequal volume of water, its weight exceeds the strengthof the buoyant force. The net force on the ball is down-ward, and it sinks to the bottom.

Now imagine two objects that have the same weightbut different volumes (Fig. 6.7). One might be a ballmade of wood and the other a much smaller ball made ofiron. Since the wood ball is larger, the buoyant force act-ing on it is also larger. If the wood weighs less than anequal volume of water, the buoyant force will exceed theweight of the ball; the net force will be upward, and theball will rise to the surface. The buoyant force on thesmaller metal ball, however, will be less than that on the

wood ball. The volume of water it displaces weighs lessthe metal. Therefore, the metal ball sinks, even thoughits weight is the same as the weight of the wood ball.

The general rules concerning floating and sinkingare sometimes summarized simply in terms of density.

density 5 mass .volume

The density of an object or sample of fluid is its massper unit volume, or its mass divided by its volume. Anobject sinks if its density is greater than the density ofthe fluid in which it is immersed. It rises if its densityis less than the fluid in which it is immersed. Do yousee why?

Densities of materials are often compared with thedensity of water. This relative density is given the namespecific gravity. Thus, if a rock sample has a densityof 2.3, or a specific gravity of 2.3, its density is 2.3times the density of water.

Floating Objects

We next turn our attention to objects floating on thesurface of fluids, such as objects floating on water. Bynow you should visualize two forces acting on the float-ing boat in Figure 6.8: its weight pulling downward anda buoyant force exerted by the surrounding water press-ing upward. These two forces just balance each other,so that once the boat reaches a certain level, it neitherrises nor sinks.

Figure 6.8. How much of the boat is below water level?

How much of the boat will sink below the waterlevel? We have seen that the strength of the buoyantforce is equal to the weight of displaced fluid. Thus, thevolume of the boat below the water surface must dis-place a weight of water equal to the weight of the boatand passengers. What will happen if the boat is loadedmore heavily? Its weight will increase and it will sinkuntil it displaces more water. How much? Enoughwater must be displaced so that its weight equals theweight of the additional freight. What happens if theship does not have enough volume to displace the extrawater? It sinks.

Icebergs illustrate these same points. They floatbecause ice has a density about 10% less than water. Itonly takes 90% of the iceberg’s volume to displace enoughwater to equal its total weight, so the iceberg floats withabout 10% of its volume above the surface of the water.

Several features of this phenomenon are worth not-

Buoyant Force

Buoyant Force

Weight Weight

���

Buoyant Force

Buoyant Force

Weight Weight

Figure 6.6. A balloon and a metal ball might be thesame size. Why does one float and the other sink?

Figure 6.7. These objects have the same weight. Whydoes one sink and the other float?

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ing. First, compare a large iceberg with a smaller one.Notice that the larger one has more volume below thewater as well as above the water compared to the small-er one. Second, imagine what will happen if more massis added to the part of the iceberg above the water, say bya snowstorm or by visiting walruses (Fig. 6.9). The ice-berg will sink farther into the water, thus increasing thebuoyant force needed to balance the increased mass.Finally, imagine what will happen if some of the materi-al above the water is lost, by melting for example. Theiceberg will rise in the water, since a smaller buoyantforce is needed to balance the iceberg’s lightened weight.

Buoyancy in the Earth’s Crust: Isostasy

An interesting application of buoyancy occurs inthe earth’s crust. The continents actually float in theearth’s mantle in much the same way that ships or ice-bergs float in water. The outer layer of the mantle is hotenough to have some characteristics of a fluid. In par-ticular, forces within the mantle adjust over long peri-ods, following the general rules for fluids we describedearlier. The crust lies above this semifluid layer. Itsgeneral features are shown in Figure 6.10.

The crust underneath the oceans is quite dense (butless dense than the mantle) and relatively thin. Theoceanic crust sinks just far enough into the mantle that

the mantle’s buoyant force supports its weight, togetherwith that of the water above.

The materials that make up the continents are sig-nificantly less dense than the oceanic crust. Theselighter continental materials sink into the mantle onlyfar enough to displace the weight of mantle materialequal to their own. Each continent, and indeed eachmountain, has “roots” extending far enough below it toprovide the necessary buoyant force.

You can deduce many of the consequences of theseideas by remembering our iceberg example. The tallermountain or continent must have deeper roots, just asthe large iceberg must have more volume below the sur-face. If material is added to a continent—for example,by the formation of a glacier; a flow of lava, or even theconstruction of a large building—the crust will sink,over time, farther into the mantle. If material isremoved—for example, by erosion or the melting of aglacier—the underlying crust will rise. The generalprinciple governing this fluid-like equilibrium in theearth’s crust is called isostasy.

Convection

We now look at one additional illustration of buoy-ant forces. Suppose we have regions of high and lowdensity occurring within the same fluid because of dif-

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.

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ferent temperatures within the fluid. Most fluidsexpand when their temperatures rise. Thus, theybecome less dense because the same mass occupiesgreater volume.

High-density regions of fluid sink, while low-densi-ty regions of fluid rise. Do you see why? High-densityregions displace surrounding fluid, but the displacedfluid does not weigh as much. Thus, the buoyant forceon the high-density region is less than its weight, so itsinks. Low-density regions also displace fluid. In theircase, the displaced fluid weighs more. Thus, the buoy-ant force on the region is greater that its weight, so itrises. The result in both cases is that cooler regions offluid sink as warmer regions rise. These motions causeinteresting and important processes in nature.

Consider the common example shown in Figure6.11 of a large body of water adjacent to land. The tem-perature of the soil is higher than the temperature of thewater during the day. Air above the soil heats andexpands, becoming less dense than the air above; there-fore, it rises. As it rises, it is replaced by the cooler,denser air from the water. A circulation pattern is estab-lished as shown in the figure. The result is a cool breezefrom the body of water during the day.

The situation is reversed at night. The land coolsmore rapidly and, in many cases, becomes cooler thanthe water. Air circulation then proceeds in the oppositedirection, with the surface wind blowing away from theland (Fig. 6.12).

This kind of circulation, caused by differences in tem-

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.

perature and density, is called convection. It is responsi-ble for many of the broad circulation patterns in the earth’satmosphere and oceans. Convection is probably the mech-anism that also drives broad regions of the earth’s crustfrom place to place on the surface of the earth.

Notice that convection depends on the presence ofgravitational forces. Neither weight nor buoyant forceswould be present without gravity, so there would be noconvection.

Summary

This chapter extends applications of the laws ofmotion and force to forces within materials, particular-ly fluids.

STUDY GUIDEChapter 6: Some Effects Due to Internal Forces

A. FUNDAMENTAL PRINCIPLES1. The First Law of Motion: See Chapter 3.2. The Second Law of Motion: See Chapter 3.3. The Third Law of Motion: See Chapter 3.4. The Universal Law of Gravitation:

See Chapter 4.5. The Electric Force Law: See Chapter 4.

B. MODELS, IDEAS, QUESTIONS, OR APPLICA-TIONS

1. How can you use the laws of force and motion toanalyze the motion of a part of a stationary object?

2. How can you use the laws of force and account forthe motion of a part of a fluid? What is a buoyantforce? 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. GLOSSARY1. Archimedes’ Principle: An object immersed in a

fluid experiences an upward buoyant force due tocontact interactions with the surrounding fluid,whose strength is equal to the weight of the dis-placed fluid.

2. Buoyant Force: For an object at rest which iswholly or partially immersed in a fluid, the buoyantforce is the net or resultant force of the contactforces exerted on the object by the fluid.

3. Convection: The circulation (movement) of thematter of a fluid because of differences in tempera-ture 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 buoyantforce and the force of gravity on a piece of the

earth’s crust seen as “floating” in the underlyingmantle rock.

6. Pressure: The force applied to the surface of anobject divided by the area over which the force isapplied. Force per unit area.

7. Specific Gravity: The density of a material divid-ed by the density of a reference substance, usuallywater. The specific gravity of water is 1.0, if wateritself is the reference substance.

D. FOCUS QUESTIONS1. In each of the following situations:

a. Describe the motion.b. Analyze the motion by applying the procedure ofChapter 5 (Finding Forces).c. List the fundamental principles used in coming tothe results of your analysis.

(1) A cube of wood within a tree trunk.(2) A cube of air above you on a calm day. Theair is not moving.(3) A part of the surface of a can when the airwas pumped out.(4) An imaginary, vertical cylinder of water ina tub of water at rest.

2. Consider three spheres of the same size completelysubmerged in a tub of water. One sphere is made oflead, another is made of wood, and the third ismade of water.a. How do the weights of the three objects compareto each other?b. How do the buoyant forces on the three objectscompare to each other?c. Analyze the motion of the three objects byapplying the procedure above.d. List the fundamental principles used in comingto the results of your analysis.

3. Consider a floating iceberg:a. Describe why it floats by using the proceduregiven 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 theiceberg? Why?

E. EXERCISES6.1. Consider a piece of air 20 meters above the

ground. What keeps it from falling?

6.2. Imagine yourself running. What forces gov-ern the motion of each foot from the time it leaves theground until it returns to the ground?

6.3. Imagine yourself in an elevator. What forceaccelerates your stomach when the elevator begins tomove up?

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6.4. Explain why it is easier to lift a rock when itis under water than when it is above water.

6.5. Do heavy objects always sink? Under whatcircumstances would a heavy object not sink?

6.6. Explain why an object sinks if it is more densethan the fluid it is immersed in.

6.7. Explain why the strength of a buoyant force isequal to the weight of displaced fluid.

6.8. Describe the interaction which causes thebuoyant force on an immersed object. Is this a long-range or a contact interaction? What other interaction isimportant when describing the behavior of objectsimmersed in a fluid?

6.9. Does the strength of the buoyant force dependon the weight of an immersed object, its size, its densi-ty, 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 inwater when a kilogram of iron sinks.

6.11. Explain why a helium-filled balloon rises butwhy an air-filled balloon does not. What factors deter-mine 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. Why is a buoyant force always directedupward?

6.15. Outdoor swimming pools in certain areas ofCalifornia sometimes rise out of the ground when theyare drained for the winter. How could this occur?

6.16. Explain why an aircraft carrier can floatwhile a small ball made of the same steel will sink.

6.17. Explain why an oil tanker sits lower in thewater when loaded than when it is empty.

6.18. Why are the most dense materials of the earthmostly found near its center?

6.19. Some parts of the U.S. require the excavation ofsoft rocks and sediment before laying foundations for largebuildings. What would you expect to happen to such abuilding if its weight were greater than that of the removedrock? What would happen if the building weighed less?Explain why you would expect such behavior.

6.20. Explain why buoyant forces cause convec-tion when a fluid such as air is unevenly heated.

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 anobject.(d) The buoyant force on the object immersed inoil is greater than the gravitational force.(e) The buoyant force on the object floating inwater is equal to the gravitational force.

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