Feynman_Lectures_on_Physics_Volume_1_Chapter_04.pdf

Conservation of Energy

4-1 What is energy?

In this chapter, we begin our more detailed study of the different aspects of

physics, having finished our description of things in general. To illustrate the ideas

and the kind of reasoning that might be used in theoretical physics, we shall now

examine one of the most basic laws of physics, the conservation of energy.

There is a fact, or if you wish, a law, governing all natural phenomena that

are known to date. There is no known exception to this law—it is exact so far as

we know. The law is called the conservation of energy. It states that there is a

certain quantity, which we call energy, that does not change in the manifold

changes which nature undergoes. That is a most abstract idea, because it is a

mathematical principle; it says that there is a numerical quantity which does not

change when something happens. It is not a description of a mechanism, or any-

thing concrete; it is just a strange fact that we can calculate some number and when

we finish watching nature go through her tricks and calculate the number again,

it is the same. (Something like the bishop on a red square, and after a number of

moves—details unknown—it is still on some red square. It is a law of this nature.)

Since it is an abstract idea, we shall illustrate the meaning of it by an analogy.

Imagine a child, perhaps "Dennis the Menace," who has blocks which are

absolutely indestructible, and cannot be divided into pieces. Each is the same as

the other. Let us suppose that he has 28 blocks. His mother puts him with his

28 blocks into a room at the beginning of the day. At the end of the day, being

curious, she counts the blocks very carefully, and discovers a phenomenal law—

no matter what he does with the blocks, there are always 28 remaining! This

continues for a number of days, until one day there are only 27 blocks, but a little

investigating shows that there is one under the rug—she must look everywhere

to be sure that the number of blocks has not changed. One day, however, the

number appears to change—there are only 26 blocks. Careful investigation in-

dicates that the window was open, and upon looking outside, the other two blocks

are found. Another day, careful count indicates that there are 30 blocks! This

causes considerable consternation, until it is realized that Bruce came to visit,

bringing his blocks with him, and he left a few at Dennis' house. After she has

disposed of the extra blocks, she closes the window, does not let Bruce in, and then

everything is going along all right, until one time she counts and finds only 25

blocks. However, there is a box in the room, a toy box, and the mother goes to

open the toy box, but the boy says "No, do not open my toy box," and screams.

Mother is not allowed to open the toy box. Being extremely curious, and somewhat

ingenious, she invents a scheme! She knows that a block weighs three ounces,

so she weighs the box at a time when she sees 28 blocks, and it weighs 16 ounces.

The next time she wishes to check, she weighs the box again, subtracts sixteen

ounces and divides by three. She discovers the following:

4-1 What is energy?

4-2 Gravitational potential energy

4-3 Kinetic energy

4-4 Other forms of energy

There then appear to be some new deviations, but careful study indicates that the

dirty water in the bathtub is changing its level. The child is throwing blocks into

the water, and she cannot see them because it is so dirty, but she can find out how

many blocks are in the water by adding another term to her formula. Since the

original height of the water was 6 inches and each block raises the water a quarter

4-1

of an inch, this new formula would be:

In the gradual increase in the complexity of her world, she finds a whole series of

terms representing ways of calculating how many blocks are in places where she

is not allowed to look. As a result, she finds a complex formula, a quantity which

has to be computed, which always stays the same in her situation.

What is the analogy of this to the conservation of energy? The most re-

markable aspect that must be abstracted from this picture is that there are no blocks.

Take away the first terms in (4.1) and (4.2) and we find ourselves calculating more

or less abstract things. The analogy has the following points. First, when we are

calculating the energy, sometimes some of it leaves the system and goes away,

or sometimes some comes in. In order to verify the conservation of energy, we

must be careful that we have not put any in or taken any out. Second, the energy

has a large number of different forms, and there is a formula for each one. These

are: gravitational energy, kinetic energy, heat energy, elastic energy, electrical

energy, chemical energy, radiant energy, nuclear energy, mass energy. If we total

up the formulas for each of these contributions, it will not change except for energy

going in and out.

It is important to realize that in physics today, we have no knowledge of what

energy is. We do not have a picture that energy comes in little blobs of a definite

amount. It is not that way. However, there are formulas for calculating some

numerical quantity, and when we add it all together it gives "28"'—always the

same number. It is an abstract thing in that it does not tell us the mechanism or

the reasons for the various formulas.

4-2 Gravitational potential energy

Conservation of energy can be understood only if we have the formula for

all of its forms. I wish to discuss the formula for gravitational energy near the

surface of the Earth, and I wish to derive this formula in a way which has nothing

to do with history but is simply a line of reasoning invented for this particular

lecture to give you an illustration of the remarkable fact that a great deal about

nature can be extracted from a few facts and close reasoning. It is an illustration

of the kind of work theoretical physicists become involved in. It is patterned

after a most excellent argument by Mr. Carnot on the efficiency of steam engines.*

Consider weight-lifting machines—machines which have the property that

they lift one weight by lowering another. Let us also make a hypothesis: that

there is no such thing as perpetual motion with these weight-lifting machines.

(In fact, that there is no perpetual motion at all is a general statement of the law

of conservation of energy.) We must be careful to define perpetual motion.

First, let us do it for weight-lifting machines. If, when we have lifted and lowered

a lot of weights and restored the machine to the original condition, we find that

the net result is to have lifted a weight, then we have a perpetual motion machine

because we can use that lifted weight to run something else. That is, provided the

machine which lifted the weight is brought back to its exact original condition,

and furthermore that it is completely self-contained— that it has not received the

energy to lift that weight from some external source—like Bruce's blocks.

A very simple weight-lifting machine is shown in Fig. 4-1. This machine lifts

weights three units "strong." We place three units on one balance pan, and one

unit on the other. However, in order to get it actually to work, we must lift a

little weight off the left pan. On the other hand, we could lift a one-unit weight

* Our point here is not so much the result, (4.3), which in fact you may already know,

as the possibility of arriving at it by theoretical reasoning.

4-2

by lowering the three-unit weight, if we cheat a little by lifting a little weight off

the other pan. Of course, we realize that with any actual lifting machine, we must

add a little extra to get it to run. This we disregard, temporarily. Ideal machines,

although they do not exist, do not require anything extra. A machine that we

actually use can be, in a sense, almost reversible: that is, if it will lift the weight of

three by lowering a weight of one, then it will also lift nearly the weight of one the

same amount by lowering the weight of three.

We imagine that there are two classes of machines, those that are not re-

versible, which includes all real machines, and those that are reversible, which of

course are actually not attainable no matter how careful we may be in our design

of bearings, levers, etc. We suppose, however, that there is such a thing—a

reversible machine—which lowers one unit of weight (a pound or any other unit)

by one unit of distance, and at the same time lifts a three-unit weight. Call this

reversible machine, Machine A. Suppose this particular reversible machine lifts

the three-unit weight a distance X. Then suppose we have another machine, Ma-

chine B, which is not necessarily reversible, which also lowers a unit weight a

unit distance, but which lifts three units a distance Y. We can now prove that Y

is not higher than X; that is, it is impossible to build a machine that will lift a

weight any higher than it will be lifted by a reversible machine. Let us see why.

Let us suppose that Y were higher than X. We take a one-unit weight and lower

it one unit height with Machine B, and that lifts the three-unit weight up a distance

V. Then we could lower the weight from Y to X, obtaining free power, and use

the reversible Machine A, running backwards, to lower the three-unit weight a

distance X and lift the one-unit weight by one unit height. This will put the

one-unit weight back where it was before, and leave both machines ready to be

used again! We would therefore have perpetual motion if Y were higher than X,

which we assumed was impossible. With those assumptions, we thus deduce that

Y is not higher than X, so that of all machines that can be designed, the reversible

machine is the best.

We can also see that all reversible machines must lift to exactly the same height.

Suppose that B were really reversible also. The argument that Y is not higher than

X is, of course, just as good as it was before, but we can also make our argument

the other way around, using the machines in the opposite order, and prove that

X is not higher than Y. This, then, is a very remarkable observation because it

permits us to analyze the height to which different machines are going to lift

something without looking at the interior mechanism. We know at once that if

somebody makes an enormously elaborate series of levers that lift three units a

certain distance by lowering one unit by one unit distance, and we compare it

with a simple lever which does the same thing and is fundamentally reversible,

his machine will lift it no higher, but perhaps less high. If his machine is re-

versible, we also know exactly how high it will lift. To summarize: every reversible

machine, no matter how it operates, which drops one pound one foot and lifts

a three-pound weight always lifts it the same distance, X. This is clearly a universal

law of great utility. The next question is, of course, what is XI

Suppose we have a reversible machine which is going to lift this distance X,

three for one. We set up three balls in a rack which does not move, as shown in

Fig. 4-2. One ball is held on a stage at a distance one foot above the ground. The

machine can lift three balls, lowering one by a distance 1. Now, we have arranged

that the platform which holds three balls has a floor and two shelves, exactly spaced

at distance X, and further, that the rack which holds the balls is spaced at distance

X, (a). First we roll the balls horizontally from the rack to the shelves, (b), and

we suppose that this takes no energy because we do not change the height. The

reversible machine then operates: it lowers the single ball to the floor, and it lifts

the rack a distance X, (c). Now we have ingeniously arranged the rack so that

these balls are again even with the platforms. Thus we unload the balls onto the

rack, (d); having unloaded the balls, we can restore the machine to its original

condition. Now we have three balls on the upper three shelves and one at the

bottom. But the strange thing is that, in a certain way of speaking, we have not

lifted two of them at all because, after all, there were balls on shelves 2 and/3

4-3

before. The resulting effect has been to lift one ball a distance 3X. Now, if 3X

exceeds one foot, then we can lower the ball to return the machine to the initial

condition, (f), and we can run the apparatus again. Therefore 3 X cannot exceed

one foot, for if 3 X exceeds one foot we can make perpetual motion. Likewise,

we can prove that one foot cannot exceed 3X, by making the whole machine run

the opposite way, since it is a reversible machine. Therefore 3X is neither greater

nor less than a foot, and we discover then, by argument alone, the law that

X = ^ foot. The generalization is clear: one pound falls a certain distance in

operating a reversible machine; then the machine can lift p pounds this distance

divided by p. Another way of putting the result is that three pounds times the

height lifted, which in our problem was X, is equal to one pound times the distance

lowered, which is one foot in this case. If we take all the weights and multiply

them by the heights at which they are now, above the floor, let the machine operate,

and then multiply all the weights by all the heights again, there will be no change.

(We have to generalize the example where we moved only one weight to the case

where when we lower one we lift several different ones—but that is easy.)

We call the sum of the weights times the heights gravitational potential

energy— the energy which an object has because of its relationship in space, rela-

tive to the earth. The formula for gravitational energy, then, so long as we are

not too far from the earth (the force weakens as we go higher) is

It is a very beautiful line of reasoning. The only problem is that perhaps it is not

true. (After all, nature does not have to go along with our reasoning.) ,For example,

perhaps perpetual motion is, in fact, possible. Some of the assumptions may be

wrong, or we may have made a mistake in reasoning, so it is always necessary to

check. /; turns out experimentally, in fact, to be true.

The general name of energy which has to do with location relative to some-

thing else is called potential energy. In this particular case, of course, we call it

gravitational potential energy. If it is a question of electrical forces against which

we are working, instead of gravitational forces, if we are "lifting" charges away

from other charges with a lot of levers, then the energy content is called electrical

potential energy. The general principle is that the change in the energy is the force

times the distance that the force is pushed, and that this is a change in energy in

general:

We will return to many of these other kinds of energy as we continue the course.

The principle of the conservation of energy is very useful for deducing what

will happen in a number of circumstances. In high school we learned a lot of laws

about pulleys and levers used in different ways. We can now see that these "laws"

are all the same thing, and that we did not have to memorize 75 rules to figure it out.

A simple example is a smooth inclined plane which is, happily, a three-four-five

triangle (Fig. 4-3). We hang a one-pound weight on the inclined plan.e with a

pulley, and on the other side of the pulley, a weight W. We want to know how

heavy W must be to balance the one pound on the plane. How can we figure that

out? If we say it is just balanced, it is reversible and so can move up and down,

and we can consider the following situation. In the initial circumstance, (a),

the one pound weight is at the bottom and weight W is at the top. When W has

slipped down in a reversible way, we have a one-pound weight at the top and the

weight W the slant distance, (b), or five feet, from the plane in which it was before.

We lifted the one-pound weight only three feet and we lowered W pounds by

five feet. Therefore W = f of a pound. Note that we deduced this from the

conservation of energy, and not from force components. Cleverness, however, is

relative. It can be deduced in a way which is even more brilliant, discovered by

4-4

Stevinus and inscribed on his tombstone. Figure 4-4 explains that it has to be

^ of a pound, because the chain does not go around. It is evident that the lower

part of the chain is balanced by itself, so that the pull of the five weights on one

side must balance the pull of three weights on the other, or whatever the ratio of

the legs. You see, by looking at this diagram, that W must be ^ of a pound.

(If you get an epitaph like that on your gravestone, you are doing fine.)

Let us now illustrate the energy principle with a more complicated problem,

the screw jack shown in Fig. 4-5. A handle 20 inches long is used to turn the screw,

which has 10 threads to the inch. We would like to know how much force would

be needed at the handle to lift one ton (2000 pounds). If we want to lift the ton

one inch, say, then we must turn the handle around ten times. When it goes around

once it goes approximately 126 inches. The handle must thus travel 1260 inches,

and if we used various pulleys, etc., we would be lifting our one ton with an un-

known smaller weight W applied to the end of the handle. So we find out that W

is about 1.6 pounds. This is a result of the conservation of energy.

Take now the somewhat more complicated example shown in Fig. 4-6. A rod

or bar, 8 feet long, is supported at one end. In the middle of the bar is a weight

of 60 pounds, and at a distance of two feet from the support there is a weight of

100 pounds. How hard do we have to lift the end of the bar in order to keep

it balanced, disregarding the weight of the bar? Suppose we put a pulley at one

end and hang a weight on the pulley. How big would the weight W have to be

in order for it to balance? We imagine that the weight falls any arbitrary dis-

tance—to make it easy for ourselves suppose it goes down 4 inches—how high

would-the two load weights rise? The center rises 2 inches, and the point a quarter

of the way from the fixed end lifts 1 inch. Therefore, the principle that the sum of

the heights times the weights does not change tells us that the weight W times

4 inches down, plus 60 pounds times 2 inches up, plus 100 pounds times 1 inch

has to add up to nothing:

Thus we must have a 55-pound weight to balance the bar. In this way we can work

out the laws of "balance"—the statics of complicated bridge arrangements, and so

on. This approach is called the principle of virtual work, because in order to apply

this argument we had to imagine that the structure moves a little—even though

it is not really moving or even movable. We use the very small imagined motion

to apply the principle of conservation of energy.

4-3 Kinetic energy

To illustrate another type of energy we consider a pendulum (Fig. 4-7).

If we pull the mass aside and release it, it swings back and forth. In its motion,

it loses height in going from either end to the center. Where does the potential

energy go? Gravitational energy disappears when it is down at the bottom;

nevertheless, it will climb up again. The gravitational energy must have gone into

another form. Evidently it is by virtue of its motion that it is able to climb up again,

so we have the conversion of gravitational energy into some other form when it

reaches the bottom.

We must get a formula for the energy of motion. Now, recalling our arguments

about reversible machines, we can easily see that in the motion at the bottom

must be a quantity of energy which permits it to rise a certain height, and which

has nothing to do with the machinery by which it comes up or the path by which

it comes up. So we have an equivalence formula something like the one we wrote

for the child's blocks. We have another form to represent the energy. It is easy to

say what it is. The kinetic energy at the bottom equals the weight times the height

that it could go, corresponding to its velocity: K.E. = WH. What we need is

the formula which tells us the height by some rule that has to do with the motion

of objects. If we start something out with a certain velocity, say straight up, it

will reach a certain height; we do not know what it is yet, but it depends on the

velocity—there is a formula for that. Then to find the formula for kinetic energy

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