7. Conservation Laws
The laws of motion and force are often difficult to
apply in complicated situations (for example, when a
stick of dynamite explodes). Fortunately, nature seems
to obey some additional rules that are less specific.
These usually do not describe every detail, but they do
predict certain features of the changes in nature and help
us arrive at a correct understanding of them.
Among these general laws are the conservation
laws. A quantity that does not change is said to be conserved. Such a quantity is something like a set of children’s blocks. Although the blocks might be scattered
all over the house, yard, car, neighbors’ property, and
schoolroom, the number of blocks always stays the
same. The “law of conservation of blocks” would
describe the conditions when the number of blocks
would not change, and also the situations when a
change would be expected to occur (for example, when
the parents purchase a new set of blocks).
This chapter discusses several quantities that are conserved: mass, charge, linear momentum, angular momentum, and energy. Several others might have been included, but these are the most important for our purposes.
Figure 7.1. Account for all the mass before and after
these events. Is mass conserved?
The Law of Conservation of Mass is stated here as
it was understood at the beginning of the 20th century.
An amplification of the statement became necessary
when Einstein discovered an unsuspected connection
between mass and energy. We will discuss this modification in Chapter 9. Until that point you should take the
Law of Conservation of Mass to be strictly true. The
modification discussed in Chapter 9 does not affect in a
practical way the applications we will discuss in this
and most subsequent chapters.
Conservation of Mass
Cursory observations tell us that matter comes and
goes in unpredictable and arbitrary ways. A piece of
wood burns and seems to be destroyed, leaving only a
small pile of ashes behind; water seems to disappear
from an open pan on a warm day; a tree grows, apparently from nothing.
However, careful measurements reveal that total
mass does not change in any of these transformations; it
merely changes form. For example, if wood is burned
in a closed box, the total mass of the contents of the box
does not change. The wood and oxygen initially present
have merely changed to carbon dioxide, water vapor,
and ashes (Fig. 7.1). These observations lead to the
Law of Conservation of Mass:
Conservation of Electric Charge
Electric charge is another of the conserved entities
in nature; however, the conservation law is more complicated because there are two kinds of charges. We can
“create” and “destroy” electric charge. For example,
when we rubbed the rods in Chapter 4, we “created”
charge where none was before. Two objects were electrically neutral at first, but charged after rubbing. In
reality the charge was there all the time and the rubbing
merely separated the two kinds from each other. The
process illustrates a general rule about electric charge:
Whenever positive charge appears, an equal amount of
negative charge appears, and vice versa. Whenever
positive charge disappears, an equal amount of negative
charge disappears. The general rule is as follows:
Mass is neither created nor destroyed. The
total amount of mass does not change. Mass
may change from one form to another, but the
total mass after the transformation is always the
same as that before.
The total amount of positive charge minus the
total amount of negative charge does not change.
55
Law of Conservation of (Linear) Momentum:
In the absence of a net force, the linear momentum of an object is conserved, i.e., an object of
fixed mass continues with unchanging speed in
a straight line such that the product of mass and
speed is a constant in time.
The words positive and negative are just labels to identify the two kinds of charge, which gives us an easy way
to keep track of nature’s conservation rule. To understand the rule more completely, try substituting the
labels red and green for positive and negative in the
statement of the conservation law.
Nature obeys the conservation law in some rather
unexpected situations. For example, a neutron isolated
from a nucleus becomes unstable and splits after a short
time into a proton and an electron. One unit of each kind
of charge is produced, so charge is conserved (Fig. 7.2).
All of the examples that we have used to illustrate
Newton’s First Law are, as well, illustrations of the
Law of Conservation of Linear Momentum.
Conservation of Angular Momentum
An object that moves in a circle is not in uniform
motion. For example, the moon’s nearly circular motion
about the earth is an accelerated motion. Although the
speed of the moon in its orbit is constant, the moon is
always changing the direction of its motion. For this reason, linear momentum is not conserved for the motion of
the moon because the requirement for motion of
unchanging direction in a straight line is violated.
However, there is a conserved quantity associated
with revolving motion. You have undoubtedly seen
examples of it as you have watched ice skaters in their
spins, divers or gymnasts in their tucks, or a falling cat
twisting to land on its feet.
Think again of the moon revolving about an axis
through the center of the earth. If we multiply the mass
of the moon by its constant speed and then again by the
constant radius of the circular orbit, we create a quantity that is unchanging during the motion of the moon.
We call this quantity angular momentum,
PROTON
NEUTRON
NEUTRINO
ELECTRON
Figure 7.2. A free neutron becomes a proton and an
electron after a short time. A neutral “neutrino” is also
created in the process. How is charge conserved?
Many such examples have been studied. Every
time a particle with charge appears, no matter what its
source, another particle with the opposite charge also
appears. Whenever a particle with charge seems to disappear, changing its form in any way, another particle
with exactly the opposite charge also disappears. The
Law of Conservation of Electric Charge seems to be
a universal law in nature. No exceptions have ever been
discovered.
angular momentum ! mass " speed " distance
to axis of revolution.
In those cases where the mass is not concentrated in a
small volume, we must think of extended objects subdivided into many small mass “particles” such that the
angular momentum of the whole object is the sum of
angular momenta of its parts.
Just as net forces change linear momentum, some
kinds of forces can change angular momentum. Forces
that impart an additional “twist” (technically called a
torque) to the revolving object can change its orbital
speed and, hence, its angular momentum. Brakes exert
torques on rotating wheels to reduce angular momentum.
However, there is a class of forces that does not
change the angular momentum. These are forces that act
along the line that connects the moving object to the axis
of revolution. In the case of the moon and earth, this is
the line from the center of the moon to the center of the
earth. Forces along this special line, such as the force of
gravity attracting the moon to the earth, impart (nor subtract) no additional “twist” to the system. We say that
Conservation of Linear Momentum
Without realizing it, we have already encountered
another fundamental conservation law: Newton’s First
Law of Motion. In the absence of a net force, an object
continues in unchanging uniform motion in a straight
line at constant speed. The constancy of the speed and
direction represents something that is conserved. To put
it in its more common form, we multiply the speed by
the mass of the object to define (linear) momentum,
momentum ! mass " speed.
Since the mass is also constant, we can state the
56
lost as something else is gained.
The “something” in each of these examples is energy.
Energy appears in the form of motion, position, temperature, and so forth. Our challenge is to discover all the forms
in which energy occurs and to calculate the amount of energy in each possible situation. We will then be in a position
to determine whether energy is a conserved quantity.
Force
Force
Object
Object
X
X
Axis
Axis
(a)
(b)
Kinetic Energy
Figure 7.3. In which case, (a) or (b), does the force
change the angular momentum?
Energy associated with motion is called kinetic
energy. We generally have some experience with the
fact that the motion of moving objects can be converted
into crumpled fenders, broken windshields, and human
injuries, all of which require energy. We know intuitively that the amount of destruction caused by a moving object is somehow connected with the mass of the
object (a truck does more damage than a small car traveling at the same speed) and its speed (a fast baseball
hurts more than a slow one).
The amount of kinetic energy possessed by a moving object can be calculated from the relationship
they exert no torque on the system (see Fig. 7.3).
With this in mind, we may state the Law of
Conservation of Angular Momentum:
In the absence of any net torque, the angular
momentum of an object revolving about an
axis is conserved, i.e., the product of the mass,
the speed, and the distance from the object to
the axis of revolution is a constant in time.
An ice skater makes use of this principle. Using her
feet and with her hands outstretched, she creates enough
torque on her body to begin to spin slowly about a vertical axis through the center of her body. Then she
quickly draws her hands inward to her chest. The force
she uses to draw her hands inward exerts no torque on
her (it is along the line connecting the masses of her
hands to the axis of rotation). Under these conditions her
angular momentum is conserved. But since she has
decreased the distance of her hands (revolving masses)
from the axis of revolution, her speed of revolving
motion must increase accordingly so that the angular
momentum is conserved. If one thing decreases (distance), the other (speed) must increase. Thus, she begins
to spin very rapidly. It is the same conservation law that
operates when divers and cats execute their maneuvers.
1
kinetic energy ! ## " mass " speed 2.
2
If a truck has three times the mass of a small car
going the same speed, it will have three times the kinetic energy; consequently, the truck will cause about three
times as much damage in a collision (Fig. 7.4). The fact
that speed is squared in this formula means that energy
increases rapidly with speed. For example, a car going
80 kilometers/hour has four times as much kinetic energy as one going 40 kilometers/hour.
Forms of Energy
Figure 7.4. Why can a fast truck do more damage than
a slow car?
Another conserved entity in nature is energy. For
example, water stored behind a dam might be allowed
to fall, thereby gaining motion, and then be allowed to
pass through an electric generator. The generator could
then cause the motion of a motor several hundred miles
away. Apparently the water has traded its height for its
motion. The water’s motion was then traded for that of
the generator, and that in turn was traded for the motion
of the motor. In each transformation something is
gained when something else is lost.
Consider a parked car with a full tank of gas. You
can trade some of the gas for the motion of the car. The
motion is lost when you stop, but then the tires and
engine are hotter than they were before. You have traded something in the gasoline for motion which was then
traded for increased temperature. Again, something is
Gravitational Potential Energy
Energy associated with the height of an object is
called gravitational potential energy. The amount of
gravitational potential energy of an object near the surface of the earth can be calculated from
gravitational potential energy ! weight " height.
The formula is more complicated if the object is far
enough away from the earth to decrease its weight
appreciably. However, the general rule is that gravitational potential energy increases with distance—the far-
57
never changed. The energy has changed form from
kinetic to potential and back again, but the sum of the
two was exactly the same at all points in the motion.
Electrical Potential Energy
Kinetic
energy
+
Gravitational
potential
energy
=
Electrical potential energy is associated with the
electrical force. Like gravitational potential energy, the
amount of energy in electrical potential energy depends
on the separation between the interacting objects.
However the electrical case is slightly more complicated because of the possibility of either attraction or
repulsion.
The electrical potential energy associated with
charges that attract each other follows the same rule as
gravitational potential energy of attracting masses. The
potential energy increases when the objects are farther
apart. The only difference is that the amount of electrical potential energy depends on the charges of the
objects, whereas the gravitational potential energy
depends on their masses.
You can visualize the existence of electrical potential energy by imagining two charged particles, one positive and one negative, placed some distance apart. If
the particles are released, they attract each other and
accelerate. Their speed increases as they come closer
together. They lose electrical potential energy as they
gain kinetic energy.
Now imagine two charged particles both positive or
both negative, again placed some distance apart. When
these particles are released, they repel each other. These
too accelerate, but now their speed increases as they
move farther apart. Their kinetic energy increases as
they separate. This must mean that the electrical potential energy is decreasing. In the case of particles with
the same charge, the electrical potential energy is larger
when they are closer together and smaller when they are
farther apart (Fig. 7.6).
Total
energy
Figure 7.5. A ball moves more slowly as it rises, but its
total energy does not change. Describe the energy changes
that occur as the ball falls. Is energy still conserved?
ther the object is from the earth’s center, the greater its
gravitational potential energy.
We can now begin to examine how energy changes
from one form to another. A ball thrown straight up into
the air moves slowly as it rises. The ball has less kinetic energy than before, but it has gained potential energy.
The loss in kinetic energy is exactly equal to the gain in
potential energy (Fig. 7.5). The ball’s kinetic energy is
zero at the highest point, but it has then gained its maximum potential energy which is equal to its original
kinetic energy. As the ball falls, it loses potential energy
and gains kinetic energy until just before it reaches the
ground, where its potential energy is back to its original
value and its kinetic energy is the same as when it was
thrown. The total energy, kinetic plus potential, has
Internal Energy
Recall for a moment the example of the falling ball
in our discussion of gravitational energy. The ball has
its original kinetic energy just before striking the earth.
Figure 7.6. Objects that repel each other go faster as they move apart. In which situation is the kinetic energy largest?
When is the electrical potential energy largest?
58
It then hits the ground, bounces a few times, and eventually comes to rest. What has happened to its energy?
It has no kinetic energy, because it is not moving. It has
no gravitational potential energy, because it has no
height. It appears that the energy has been lost.
This kind of puzzle has confronted people ever
since the concept of energy was suggested. We must
either find the energy somewhere in a different form or
admit that energy is not conserved. The solution to such
puzzles has always been found by carefully examining
what is involved in the situation and finding some other
change that has occurred. The change is associated with
a new form of energy.
In the case of a falling ball which strikes a floor, the
ball (and, to a lesser degree, the floor) is deformed by
the collision. Furthermore, the floor and the ball are
both warmer than they were just before the ball hit.
These temperature changes are so small that we usually
do not notice them. If you doubt their reality, try stopping your car quickly (for example, from 50 kilometers/hour or so) and then stepping out and carefully
touching one of the tires. The tire’s high temperature
into another chemical state (carbon dioxide and water
molecules) at the same temperature.
The discussion to this point might leave the impression that thermal energy is always lost energy, that it is
somehow wasted. Such an impression is only partly
valid. Our society uses thermal energy in many ways.
We use the thermal energy of a natural gas flame or an
electric hot plate to cook our food. Gasoline in our cars
has high chemical potential energy. When gasoline
ignites in the engine, chemical potential energy is converted to the thermal energy which in turn is partly converted into kinetic energy of the car and low-temperature thermal energy of the car and its surroundings.
Conservation of Energy
We have described several forms of energy and
shown how the amount of energy in any particular situation can be calculated. Although all the details of the
calculations are complex, you should be able to identify different kinds of energy and determine if they
increase or decrease.
Figure 7.7. What happens to the kinetic energy of a car when it suddenly stops?
reveals where the original kinetic energy of the car has
gone (Fig. 7.7).
The energy in each of these cases converts to a
form called internal energy. Internal energy gets its
name from the fact that it is energy that seems to be hidden inside materials. We cannot measure it by any
external measurement, such as height or speed; only
careful measurements of the state of the material itself
reveal internal energy.
Thermal energy is a form of internal energy associated with the temperature of materials. The thermal
energy of any object or material increases as its temperature increases. The warmer ball and ground and the
hotter tire in the examples above illustrate increases in
thermal energy.
Chemical potential energy is a form of internal
energy associated with the physical and chemical states
of a material. Chemical potential energy is the name
given to the electrical potential energy of the atoms in a
material. Physical and chemical reactions result in a
rearrangement of the atoms of which a material is composed; this in turn causes changes in the electrical
potential energy of the atoms in the material. Another
example is that the chemical potential energy of gasoline and oxygen molecules is higher than would be the
chemical potential energy of the same atoms rearranged
When precise measurements are taken and exact
calculations performed, the decrease in one form of
energy is always accompanied by an increase in one or
more other forms. If one kind of energy increases by a
certain amount, another kind decreases by exactly the
same amount. These results have led people to conclude that energy is a conserved entity. Nature seems to
obey the Law of Conservation of Energy:
Energy can neither be created nor destroyed.
The total amount of energy in the universe
never changes. Energy can change from one
form to another, or be transferred from one
object to another, but no process, no matter how
violent or sudden or complicated, can change
the total amount of energy.
The law has been tested extensively over an enormous range of circumstances. Situations have been
encountered where it seemed that the law had broken
down. In such cases, new forms of energy were discovered that could account exactly for the changes that
had occurred. The most important of these forms are
the energies associated with waves and nuclear forces.
There are others as well, some of which will be discussed in later chapters.
59
Notice that we have never defined what energy is,
except to say that it is kinetic energy, potential energy,
and so forth. At our present level of understanding,
energy is fairly abstract. It is simply a conserved entity
somehow associated with each condition or situation.
Perhaps someday we will understand energy in terms of
some underlying, more fundamental concept. For now
we will have to be satisfied with identifying its manifestations in terms of motion, position, temperature,
physical state, and so on. Even so, most scientists
regard energy as being more “real” than force or some
other concepts that we use in our description of nature.
Energy is, after all, conserved.
Occasionally an ingenious inventor proposes a perpetual motion machine that apparently violates the Law
of Conservation of Energy by continuously creating
energy without requiring fuel or any other energy
source. Such a machine would be immensely profitable, as you might imagine. Unfortunately, no such
machine has ever worked successfully.
The study of energy has been extremely fruitful
over the years. All processes in nature involve energy
transfer or transformation. These changes often have
provided important clues to a fuller understanding of
both the living and nonliving world. Identify various
forms of energy in situations that you encounter, and
ask yourself where the energy came from and where it
goes. By doing so, you discover more order in the
world than you would have thought possible.
amount of energy that is transferred by work is sometimes referred to as the “work done” or the “amount of
work.” Thus, it is correct to say when lifting a rock that
you “did work on the rock” or when sliding a book that
friction did work on the book. However, it is a conceptual error to refer to the amount of work possessed by the
rock or by the book. They have energy—not work.
The amount of work done by a force, and, hence,
the amount of energy transferred, is given by the product of the strength of the force and the distance that the
Energy Transfer and Transformation Processes
We have noted that energy can be transformed or
transferred, but we have not paid particular attention to
the ways that these changes occur. Nature seems to
have a limited number of energy transfer and transformation mechanisms. Identifying these will help you to
keep track of the energy in the situations you encounter.
The following are most of the important energy-changing mechanisms. We will differentiate between the
basic “simple” processes and complex processes which
may involve a combination of simple processes.
Work is the simple process by which energy is transferred to or from an object by forces acting on the object.
(The use here of the word “work” is a technical usage and
is not meant to be directly related to its more common
usage in everyday conversation.) You “do work” when
you exert a force to lift a large rock and thereby transfer
additional gravitational potential energy to it (Fig. 7.8).
Frictional forces do work on a book sliding across a table,
removing kinetic energy from the book and transforming
the kinetic energy of the book into internal thermal energy in both the book and the table and transferring at least
some of it from the book to the table. Work is both a
transfer and a transformation process.
Work is not a form of energy: it is a process. The
Figure 7.8. You do work when you lift a rock. What
forms of energy are present before and after lifting?
Figure 7.9. Heat conduction, radiation, and convection
all occur when a pan of water is heated on a stove.
Identify the three processes in this figure.
60
object moves in the direction of the force,
that contains the internal energy moves from place to
place. For example, air is heated in a furnace and then
moves through ducts to various parts of a house, thereby transferring the energy and warming the rooms.
Warm ocean currents from the tropics transfer considerable internal energy to northern latitudes. As a result,
many locales near the ocean have much milder climates
than would otherwise be the case. Much of our weather is due to the convective transfer of enormous
amounts of energy through the atmosphere.
Chemical reactions are processes that transform
energy from one form to another. In a chemical reaction,
atoms are bonded together or released from bonds to one
another in such a way as to store or release “chemical”
potential energy (which is a form of internal energy.)
Combustion is a kind of chemical reaction in which
the stored potential energy is transformed into internal
thermal (i.e., heat) energy. For example, when the molecules in air and natural gas are burned (chemical reaction),
some of the stored potential energy in the molecules is
transformed into the form of internal thermal energy
(heat) which one senses as an increase in temperature.
Heat flow (or heat transfer) is a general term that is
applied to the transfer of energy from one place to another by heat conduction, by convection, and/or by radiation.
Figure 7.10 represents the energy of any object or
collection of objects. On the macroscopic scale (large,
visible objects) we have identified kinetic energy, gravitational potential energy, and electrical potential energy. We have also seen that internal forms of energy are
manifested as thermal energy and chemical potential
energy. If energy neither enters nor leaves the system,
the total energy inside remains constant. Energy of the
system may change from one form to another, but the
total will not change. Energy may enter or leave the
work ! force " distance.
If the object does not move or if the direction of the
force is exactly perpendicular to the direction the object
moves, the force transfers neither kinetic nor potential
energy to or from the object and no work is done.
Heat conduction is a simple transfer process by
which internal energy is transferred because of differences in temperature. A hot object and a cold one in
contact with each other both become warm as internal
energy is transferred from the hot object to the cold one.
Internal energy is transferred from the hot gas flame on
your stove to a pot, from there to water within the pot,
and from there to the eggs in the water, all by heat conduction (Fig. 7.9). Internal energy is transferred from
warm water in a glass to an ice cube floating on the
water. The water becomes cooler, because it has less
internal energy than before, and the increased internal
energy of the ice causes it to melt.
Radiation is a third simple transfer process. Here
we use the word to describe the transfer of energy from
one place to another as light (and its related forms:
infrared light, ultraviolet light, x-rays, radio waves,
etc.). Sometimes the word “radiation” is used to refer to
the energy itself, but here we are stressing its meaning
as the transfer process of the energy, from its emission
at a source to its absorption by another object.
Radiation is the process by which energy is transferred
from the sun to the earth as light.
Convection is a complex transfer process which
often combines the simple processes of work and heat
conduction. Convection is the process by which internal energy is moved from place to place as the matter
ENERGY
MACROSCOPIC
MICROSCOPIC
(Large-scale)
ENERGY INPUT
Heat Conduction
Radiation
Convection
Work
(Internal)
Kinetic energy
Thermal energy
Gravitational
potential energy
Chemical
potential energy
Electrical
potential energy
ENERGY OUTPUT
Heat Conduction
Radiation
Convection
Work
Rest-mass energy
TRANSFORMATION
Work, Chemical Reaction
Figure 7.10. The forms of energy in a system—any object or collection of objects. The total energy of the system can
be changed only by adding or removing energy through one or more of the energy transfer processes. The form of energy in the system can be changed by the transformation processes of work or combustion.
61
Historical Perspectives
system by any of the processes we have described; this
changes the total energy of the system unless identical
amounts of energy leave or enter.
The law we call “Conservation of Energy” had its
origin in the work of a Dutch physicist, Christian
Huygens (1629-1692), who by 1662 had drawn attention to the conservation of kinetic energy (originally
called vis viva, “living force”) in elastic collisions of
very hard objects. But the development of the general
law took nearly 200 years more and the groping work of
many individuals. Indeed, the term energy as we now
use it was first used in 1807.
The real problem was what to make of heat (what
we have called “internal thermal energy”). And this
problem was in turn intertwined with the nature of matter, a problem that we will examine in succeeding chapters. Indeed, there have been two views of heat that have
contested with one another through the centuries. One
view can be traced back to the early Greek philosophers. They maintained that heat was a kind of fluid
which flowed through matter and which had a weight of
its own, although measurements then were not precise
enough to reveal it. The other view, held tentatively by
Bacon and Galileo in the 17th century, was that heat
was really a manifestation of the motion of atoms.
However, at that time atoms were just an idea; there was
no direct evidence for their existence.
The fluid idea gained popularity in the 18th century. The name caloric was coined for it in 1787 by
Lavoisier. The idea seemed to have intuitive evidence.
When people placed a hot piece of metal in water, they
could imagine the caloric flowing out of the metal into
the water, causing the metal to cool and the water to
warm. The flow of the caloric from the metal could
even be thought to cause the slight contraction in volume observed when most objects are cooled. Thus,
there seemed to be a substance, caloric, that was being
conserved as it moved from object to object.
By 1798, a British-American refugee of the
Revolutionary War, Benjamin Thompson (1753-1814),
had become Count Rumford in Bavaria where he supervised the boring of cannons for the Bavarian army.
Throughout a distinguished career in government, he
had maintained a devotion to chemical and mechanical
experiments. He observed that the friction of the boring
process seemed to create caloric almost without limit,
an observation which struck a blow to the idea of conservation of caloric. Within a few years, further observations convinced Rumford that heat was a manifestation of motion.
But Rumford’s idea lost ground when others considered the flow of heat from the sun to the earth through
the vacuum of space. If there were truly no matter in the
vacuum of space, the heat coming to us from the sun
could hardly be thought of as vibrations of matter. In this
instance the light from the sun and the heat that resulted
appeared to be the same thing. Until about 1820, light
Summary
Mass, electric charge, linear momentum, angular
momentum, and energy are quantities that are conserved
within any given system. The laws of conservation are
useful, particularly in phenomena where changes occur,
because they represent a constant in the face of change.
Mass occurs in one form, charge in two forms, and
energy in several forms. Energy exists in various forms:
kinetic energy, gravitational potential energy, electrical
potential energy, thermal energy, and chemical potential
energy. The energy transfer and transformation
processes are work, heat conduction, radiation, convection, and chemical reactions.
Conservation of mass implies that mass does not
change when materials burn, freeze (even though they
change size), evaporate, explode, dissolve in other materials, or undergo any other physical or chemical change.
Electric charge is conserved, for example, when a
rubber rod is rubbed with fur. Before the rubbing, both
rod and fur are neutral; the total charge is zero.
Afterward the rod has a negative charge and the fur an
equal positive charge; the total charge is still zero.
The separation of charge in a thundercloud illustrates the same principle. The water droplets that make
up the forming cloud are originally uncharged. Several
mechanisms cause a separation of charge, with the top
of the cloud positive and the bottom negative.
However, the total charge remains zero.
Conservation of energy is illustrated by virtually
every process that occurs spontaneously in nature or is
caused to happen by man’s technology. The following
are just a few of the many that could be cited:
•
•
•
•
A iceberg melts in the ocean.
An automobile accelerates from rest to 60
miles per hour.
Water stored in a dam is used to generate
electricity.
Water in the ocean evaporates. Later, rainfall
occurs over a mountain range and there is
substantial erosion as the water returns to the
ocean.
In each case you should be able to (1) identify the
various forms of energy that exist and (2) identify the
processes by which energy is transferred or transformed. Energy is conserved in each case, so you
should also be able to identify the forms of energy at the
beginning and end of each process and see that the total
amount does not change.
62
was thought to be a substance. Hence, heat appeared in
this instance to be a flow of substance (i.e., caloric).
Although many others made contributions, the resolution of the two ideas into our modern Law of
Conservation of Energy began to occur in the 1840s primarily because of the work of a German physician,
Julius Robert Mayer (1814-1878), and an English brewer, James Prescott Joule (1818-1889). Mayer established the quantitative connection between macroscopic
energy and heat. In Rumford’s case, it would have been
the connection between the work (in the technical,
physical meaning) of the turning borer and the heat generated in the metal of the cannon. Joule established a
constant ratio between mechanical work done and the
corresponding heat produced by stirring fluids with
paddle wheels in heat-insulated containers, thus demonstrating a conservation principle. The conservation of
energy then began to emerge as a very general and unifying law of physics. One could create heat, it was true,
but at the expense of some other form of energy. Heat
could also be consumed, but it would appear as some
other form of energy.
The German physiologist and physicist Hermann
von Helmholtz (1821-1894) did much to further the
idea by developing its mathematical expression and by
demonstrating its applicability in mechanics, heat, electricity, magnetism, chemistry, and astronomy. In his
hands, the Law of Conservation of Energy provided
explanations for old puzzles, and a wealth of mathematical relationships emerged.
The work of Mayer, Joule, and Helmholtz was also
moving with another strong tide: the growing strength
of the idea that matter is composed of molecules, an
idea to which we will turn in Chapter 11.
(Adapted from Gerald Holton, Introduction to
Concepts and Theories in Physical Science, 2nd
Edition, pp. 263-274.)
3.
4.
5.
The Conservation of Linear Momentum: In the
absence of a net force, the linear momentum of an
object is conserved, i.e., an object of fixed mass
continues with unchanging speed in a straight line
such that the product of mass and speed is a constant in time.
The Conservation of Angular Momentum: In
the absence of any net torque, the angular momentum of an object revolving about an axis is conserved, i.e., the product of the mass, the speed, and
the distance from the object to the axis of revolution is a constant in time.
The Conservation of Energy: Energy can neither
be created nor destroyed. The total amount of energy in the universe never changes. Energy can
change from one form to another, or be transferred
from one object to another, but no process, no matter how violent or sudden or complicated, can
change the total amount of energy. (This principle of
the 19th century had to be modified when the
Special Theory of Relativity established an unsuspected connection between mass and energy.
Strictly speaking, it is mass-energy that is conserved.)
B. MODELS, IDEAS, QUESTIONS, OR APPLICATIONS
1. What does it mean for a quantity to be conserved?
2. Is it possible for mass to be either created or
destroyed?
3. Is it possible for electric charge to be either created
or destroyed?
4. Is it possible for energy to be either created or
destroyed?
5. What are some important ways that energy can be
transferred or transformed?
C. GLOSSARY
1. Angular Momentum: The quantity of motion of a
revolving object formed from the product of the
mass of an object, the speed of the object, and the
distance of the object from the axis of revolution.
2. Chemical Potential Energy: The form of internal
energy associated with the physical and chemical
states of matter.
3. Chemical Reaction: The process in which bonds
between atoms are made or broken with an accompanying storing or release of chemical potential
energy. Combustion (“burning”) is a kind of
chemical reaction.
4. Conservation: Unchanging in time. A quantity is
“conserved” if the amount of that quantity does not
change in time, even though processes may be
changing its form.
5. Convection: The process by which energy is
moved from one place to another by being stored in
STUDY GUIDE
Chapter 7: Conservation Laws
A. FUNDAMENTAL PRINCIPLES
1. The Conservation of Mass: Mass is neither created nor destroyed. The total amount of mass of an
isolated system does not change. Mass may change
from one form to another, but the total mass after
the transformation is always the same as that
before. (This principle of the 19th century had to be
modified when the Special Theory of Relativity
established an unsuspected connection between
mass and energy. Strictly speaking, it is mass-energy that is conserved.)
2. The Conservation of Electric Charge: The total
amount of positive charge minus the total amount
of negative charge of an isolated system does not
change.
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6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
matter as internal energy, then moving the matter
from one place to another.
Electric Charge: See Chapter 4.
Electrical Potential Energy: The form of energy
associated with the relative positions of charged
objects. Objects with opposite charges have maximum electrical potential energy when they are separated by greatest distance, but objects with the same
charge have maximum electrical potential energy
when they are separated by the least distance.
Energy: The name given to a quantity observed to
be conserved in nature. Energy takes many forms
associated with motion, position, temperature, etc.
Gravitational Potential Energy (GPE): The
form of energy associated with the relative positions of gravitating objects. Near the surface of the
earth, the increase of gravitational potential energy
of an object that is lifted is given by GPE =
(weight)(height lifted).
Heat Conduction: The process by which energy
moves from a hot object to a cold object placed in
contact with it.
Heat Flow (or heat transfer): A general term used
to describe the transfer of energy from one place to
another by heat conduction, by convection, and/or
by radiation.
Internal Energy: A name given to energy hidden
within matter but manifest by the temperature of
the matter, the shape of the matter, the physical
state of the matter (solid, liquid, or gas), the chemical composition of the matter (i.e., the kind of
energy that might be released by burning or explosion of a substance), etc.
Kinetic Energy (KE): The form of energy associated with motion. The kinetic energy of an object
in motion is given by KE = 1/2(mass)(speed)2.
Linear Momentum: The quantity of motion
formed from the product of the mass of an object
and its speed.
Mass: See Chapter 3.
Net Electric Charge: The amount of positive
charge minus the amount of negative charge of a
system or object. An atom has a zero net charge
because it has as many protons carrying positive
units of charge as it has electrons carrying negative
units of charge.
Radiation: The process by which energy is moved
from one place to another in the form of light or
related forms such as x-ray, gamma rays,
microwaves, etc.
Thermal Energy: The form of internal energy
associated with the temperature of matter.
Torque: A twist (technically, a force multiplied by
the distance from an axis of rotation) that is
applied in such a way as to increase or decrease
rotational motion (angular momentum) of an
object about that axis.
20. Work: The technical name given to the process by
which energy is transferred to or from an object by
an agent that exerts a force on the object and the
object moves along the direction of the force.
D. FOCUS QUESTIONS
1. In each of the following situations:
a. Describe what would be observed.
b. Name and state in your own words the fundamental conservation principle that would explain
what would happen.
c. Explain what would happen in terms of the fundamental conservation principle. Be explicit and
accurate in describing what changes occur and
what is conserved.
(1) Suppose a piece of wood is burned. All the
products produced are carefully caught and
their masses are measured. (Ignore any effects
suggested by the Special Theory of Relativity.)
(2) Parts of a thundercloud are electrically
charged because of internal motions and friction within a cloud. Lightning is observed.
(Use the conservation principle associated
with electric charge to answer this question.)
(3) A real pendulum is set in motion. It gradually comes to rest.
(4) Suppose a pendulum were set in motion and
there were no friction anywhere in the system.
2. Describe and illustrate the important energy transfer and transformation mechanisms (work, heat
conduction, radiation, and convection).
E. EXERCISES
7.1. Account for all the mass before and after
burning a piece of wood. Show how mass is conserved
in this case.
7.2. Account for all the mass before and after
water evaporates from a lake. Show how mass is conserved in this case.
7.3. Identify the processes by which mass enters
and leaves your body during a complete day. How do
the “inputs” compare with the “outputs” when your
weight remains constant? When your weight is increasing? When it is reducing?
7.4. What do we mean when we say that mass is
“conserved”?
7.5. Account for all the mass before and after a
stick of dynamite explodes. Show how mass is conserved in this case.
7.6. An ice cube is larger (has more volume) than
the water from which it was created. Does it also have
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more mass? Explain your answer.
to crash into a building.
7.7. Account for all the electric charge before and
after a person walks across a carpet and then touches a
metal doorknob, thereby receiving an electric shock.
Show how electric charge is conserved in this case.
7.20. An object falls from the roof of a building.
(a) What kind(s) of energy does it have before it
falls?
(b) What kind(s) of energy does it lose as it falls?
(c) What kind(s) of energy does it gain as it falls?
(d) What kind(s) of energy does it have just before
it hits the ground?
(e) What kind(s) of energy does it have just after it
hits the ground and stops?
(f) How does the total amount of energy in (e) compare with that in (a)? Explain your answer.
(g) What does it mean to say that energy is conserved in this situation?
7.8. Show how electric charge is conserved when
two charged objects, one positive and one negative, touch
each other and then become electrically neutral.
7.9. What do we mean when we say that electric
charge is “conserved”?
7.10. A thundercloud becomes electrically charged
because of internal motions and friction within the cloud.
Show how electric charge is conserved in this case.
7.21. Explain the meaning of the term “energy.”
7.11. How is electric charge conserved when an
electric current flows in a copper wire?
7.22. Describe the process (or processes) by which
a warm house becomes cooler on a cold day. Does energy leave, or does the cold enter?
7.12. Describe the energy changes that occur in
and around an automobile when
(a) the car travels on a level road at constant speed,
(b) the car climbs a hill at constant speed, and
(c) the car speeds up on a level road. How is energy conserved in these situations?
7.23. Describe the energy changes that occur in
your body during normal activities of a day.
7.24. What would you call the process by which
the potential energy of a falling rock is converted to
kinetic energy?
7.13. Describe the energy changes that occur as
water evaporates from a lake or ocean, later falls as rain,
and eventually returns to the ocean, perhaps causing
erosion in the process. How is energy conserved in this
situation?
7.25. Describe and illustrate the important energy
transfer and transformation mechanisms (work, heat
conduction, radiation, convection, and combustion).
7.14. What was the function of the “heat shield”
used on the Apollo spacecraft during its return to the
earth?
7.26. Energy transfers and transformations occur
from step to step in Exercise 7.20. Which are due to
work and which to heat conduction? Are there any
other energy transfer or transformation processes in this
situation?
7.15. Electrical energy plays an important role in
our technology. What forms of energy can be “created”? What does it mean to say that energy is conserved
in such processes?
7.27. Name and describe all of the processes by
which energy is transferred or transformed in the energy changes implied in Exercise 7.12.
7.16. List the various forms in which energy may
occur. Briefly describe each form of energy.
7.28. Describe the energy transfer and transformation processes that occur in Exercise 7.13.
7.17. State the Law of Conservation of Energy in
your own words. Explain its meaning.
7.18. Explain why a truck would do more damage
than a small car if both were to crash into a building at
the same speed.
7.29. It seems that kinetic energy and potential
energy are always converted into internal energy in
most of the processes we have studied. Describe a
process in which the opposite transformation takes
place—where internal energy is converted to kinetic or
potential energy. (Hint: Exercise 7.12.)
7.19. Explain why a car traveling 60 miles per
hour would be expected to do more damage than would
an identical car traveling 30 miles per hour if both were
7.30. Which of the following is false?
(a) Mass is conserved in chemical reactions.
(b) Mass is conserved in “mechanical” reactions.
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(c) Internal energy is conserved in all reactions.
(d) Charge is conserved in all reactions.
(e) Charge is conserved in chemical reactions.
7.31. Which of the following involves a radiative
energy transfer process?
(a) Airflow in a warm air furnace.
(b) Falling object.
(c) Bonfire heat reaching a camper.
(d) Light passing through a window pane.
(e) Gasoline burning in an auto engine.
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