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condition it has a possibility of doing some work. If we were to evaluate the suns
of weights times heights, it would not check out——we must add something else
to account for the fact that the spring is under tension. Elastic energy is the
formula for a spring when it is stretched. How much energy is it? If we let go, the
elastic energy, as the spring passes through the equilibrium poïint, is converted
to kinetic energy and it goes back and forth bebtween compressing or stretching
the spring and kinetic energy of motion. (There is also some gravitational energy
going in and out, but we can do this experiment “sideways” if we like.) It keeps
going until the losses—Ahal We have cheated all the way through by putting
on little weights to move things or saying that the machines are reversible, or
that they go on forever, but we can see that things do stop, eventually. Where is
the energy when the spring has fnished moving up and down? This brings in
another form oŸ energy: heat energ.
Inside a spring or a lever there are crystals which are made up oŸ lots of atoms,
and with great care and delicacy in the arrangement of the parts one can try to
adjust things so that as something rolls on something else, none of the atoms do
any jiggling at all. But one must be very careful. Ordinarily when things roll,
there is bumping and jiggling because of the irregularities of the material, and
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the atoms start to wiggle inside. So we lose track of that energy; we find the
atoms are wigegling inside in a random and confused manner after the motion
slows down. There is still kinetic energy, all right, but iE is not associated with
visible motion. What a dreaml How do we knou there is still kinetic energy? lt
turns out that with thermometers you can find out that, in fact, the spring or
the lever is t0armer, and that there is really an increase of kinetic energy by a
defñnite amount. We call this form oŸ energy heø‡ enerø, but we know that it
is not really a new form, it is just kinetic energy——internal motion. (Ône of the
diñiculties with all these experiments with matter that we do on a large scale
1s that we cannot really demonstrate the conservation of energy and we cannot
really make our reversible machines, because every time we move a large clump of
stuf, the atoms do not remain absolutely undisturbed, and so a certain amount
of random motion goes into the atomic system. We cannot see it, but we can
measure it with thermometers, etc.)
There are many other forms of energy, and oŸ course we cannot describe
them in any more detail just now. There is electrical energy, which has to
do with pushing and pulling by electric charges. 'There is radiant energy, the
energy of light, which we know is a form of electrical energy because light can be
represented as wigglings in the electromagnetic field. 'There is chemical energy,
the energy which is released in chemical reactions. Actually, elastic energy is,
to a certain extent, like chemical energy, because chemical energy is the energy
of the attraction of the atoms, one for the other, and so is elastic energy. Qur
modern understanding is the following: chemical energy has bwo parts, kinetic
energy of the electrons inside the atoms, so part of it is kinetic, and electrical
energy of interaction of the electrons and the protons——the rest of it, therefore,
1s electrical. Next we come to nuclear energy, the energy which is involved with
the arrangement of particles inside the nucleus, and we have formulas for that,
but we do not have the fundamental laws. We know that it is not electrical, not
gravitational, and not purely chemical, but we do not know what it is. It seems
to be an additional form of energy. Pinally, associated with the relativity theory,
there is a modifcation of the laws of kinetic energy, or whatever you wish to call
it, so that kinetic energy is combined with another thing called mass energy. An
object has energy from its sheer ezisfence. Tf Ï have a positron and an electron,
standing still doing nothing—never mind gravity, never mind anything—and
they come together and disappear, radiant energy will be liberated, in a defnite
amount, and the amount can be calculated. All we need know is the mass of the
object. It does not depend on what 1% is—we make two things disappear, and
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we get a certain amount of energy. 'Phe formula was first found by Binstein; it
is E = mcŸ.
Tt is obvious from our discussion that the law of conservation of energy is
enormously useful in making analyses, as we have illustrated in a few examples
without knowing all the formulas. If we had all the formulas for all kinds of
energy, we could analyze how many processes should work without having to go
into the details. 'Pherefore conservation laws are very interesting. The question
naturally arises as to what other conservation laws there are in physics. There
are two other conservation laws which are analogous to the conservation oŸ energy.
One is called the conservation of linear momentum. The other is called the
conservation of angular momentum. We will ñnd out more about these later.
In the last analysis, we do not understand the conservation laws deeply. We do
not understand the conservation of energy. We do not understand energy as a
certain number oŸ little blobs. You may have heard that photons come out in
blobs and that the energy of a photon is Planck's constant times the frequency.
That is true, but since the frequency of light can be anything, there is no law
that says that energy has to be a certain defnite amount. nlike Dennis' blocks,
there can be any amount of energy, at least as presently understood. So we do
not understand this energy as counting something at the moment, but just as a
mmathematical quantity, which is an abstract and rather peculiar circumstance.
In quantum mechanics it turns out that the conservation of energy is very closely
related to another mmportant property of the world, ¿hings do not depend on
the absolute từmec. YWWe can set up an experiment at a given moment and try i§
out, and then do the same experiment at a later moment, and it will behave in
exactly the same way. Whether this is strictly true or not, we do not know. lf
we assume that it 7s true, and add the principles of quantum mechanics, then we
can deduce the principle of the conservation of energy. It is a rather subtle and
interesting thing, and it is not easy to explain. 'Phe other conservation laws are
also linked together. 'The conservation of momentum is associated in quantum
mechanics with the proposition that 1 makes no diference where you do the
experiment, the results will always be the same. As independence in space has
to do with the conservation of momentum, independence of time has to do with
the conservation of energy, and finally, If we #uzn our apparatus, this too makes
no diference, and so the invariance of the world to angular orientation is related
to the conservation of anguÏar rnomentum. Besides these, there are three other
conservation laws, that are exact so far as we can tell today, which are much
simpler to understand because they are in the nature of counting bloecks.
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The first of the three is the conseruation oƒ charge, and that merely means
that you count how many positive, minus how many negative electrical charges
you have, and the number is never changed. You may get rid oŸ a positive with a
negative, but you do not create any net excess of positives over negatives. 'ÏWo
other laws are analogous to this one——one is called the conseruation oj bar0ons.