Some Effects Due to Internal Forces

8/8/2019 Some Effects Due to Internal Forces

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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 illustrat-

ing the effects of forces that act inside objects and mate-

rials. 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 with-

in 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

earths surface.

Forces within Solids

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 gravi-

tational 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 bound-

aries. 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

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

6. Some Effects Due to Internal Forces

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

tree trunk?

Figure 6.2. How can we describe the total effect of all

the 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 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 electromag-

netic interactions. But what force accelerates yourhead? 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).

Figure 6.3. What force causes your head to accelerate

when 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 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 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 on

softer surfaces and wearing shoes with soft soles.

The motion of every piece of matterwhether a

single atom, a complete object, or a small part of such

an objectis 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-

zontal motion. The contact forces, in turn, depend on

what 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 the

contact comes at so many different places, we speak ofpressure rather than of force. Pressure is defined as

the force per unit area of contact:

pressure 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 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 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

gauges spring. Because you are interested in measur-

ing pressure at a particular position in the fluid, imagine

the 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 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,

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 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 thepoint 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.

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.

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 interac-

tion 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 netupward 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 direc-

tions 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

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 sam-

ple of fluid. The surrounding fluid exerts a net upward

force on the object equaling the force previously exert-

ed 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 gov-erning the strength of the buoyant force is known as

ArchimedesPrinciple:

An object immersed in a fluid experiences an

upward buoyant force due to contact interac-

tions 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. Anobject 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

Object

BuoyantForce

Weight

Figure 6.5. Fluid pressure causes a net upward buoyant

force on any object.


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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 down-

ward, and it sinks to the bottom.

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 act-

ing 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

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 ofdensity.

density mass .volume

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 densityof 2.3, or a specific gravity of 2.3, its density is 2.3

times the density of water.

Floating Objects

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 float-

ing boat in Figure 6.8: its weight pulling downward and

a 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 neither

rises nor sinks.

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

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 dis-

place 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 theweight 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 icebergs 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-

BuoyantForce BuoyantForce

Weight Weight

Buoyant

Force

Buoyant

Force

Weight Weight

Figure 6.6. A balloon and a metal ball might be the

same size. Why does one float and the other sink?

Figure 6.7. These objects have the same weight. Why

does 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 the

water as well as above the water compared to the small-

er 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 ice-berg 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 materi-

al 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 icebergs lightened weight.

Buoyancy in the Earths Crust: Isostasy

An interesting application of buoyancy occurs in

the earths crust. The continents actually float in the

earths mantle in much the same way that ships or ice-

bergs float in water. The outer layer of the mantle is hot

enough 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 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

the mantles buoyant force supports its weight, together

with that of the water above.

The materials that make up the continents are sig-

nificantly less dense than the oceanic crust. These

lighter continental materials sink into the mantle only

far enough to displace the weight of mantle materialequal 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 sur-

face. If material is added to a continentfor example,

by the formation of a glacier; a flow of lava, or even the

construction of a large buildingthe crust will sink,

over time, farther into the mantle. If material is

removedfor example, by erosion or the melting of a

glacierthe underlying crust will rise. The general

principle governing this fluid-like equilibrium in the

earths crust is called isostasy.

Convection

We now look at one additional illustration of buoy-

ant forces. Suppose we have regions of high and low

density 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 earths crust floats in (or on) the mantle much as icebergs float in water.


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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-densi-

ty regions of fluid rise. Do you see why? High-densityregions 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 buoy-

ant 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 tem-

perature 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; 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 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-

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.


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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.

6.5. Do heavy objects always sink? Under what

circumstances would a heavy object not sink?

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 long-

range 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 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 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 deter-

mine how high the helium-filled balloon will rise?

6.12. Explain the meaning ofdensity.

6.13. Explain the meaning of specific gravity.

6.14. Why is a buoyant force always directed

upward?

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.

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 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.

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Some Effects Due to Internal Forces