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beiïng curious, she counts the blocks very carefully, and discovers a phenomenal |
law—no matter what he does with the blocks, there are always 28 remainingl |
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, |
--- Trang 86 --- |
however, the number appears to change—there are only 26 blocks. Careful |
investigation indicates that the window was open, and upon looking outside, the |
other two blocks are found. Another day, careful count indicates that there are |
30 blocksl "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 |
Bruee 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 sereams. Mother is not allowed to open the toy box. Being extremely |
curious, and somewhat ingenious, she invents a schemel 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: |
( nunber of ) R (weight of box) — 16 ounces _ constant. (41) |
ocks seen 3 ounces |
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 fnd |
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 bloeck raises the water |
a quarter of an inch, this new formula would be: |
number of (weight of box) — 16 ounces |
mm n) 3 ounces |
+ (height TT 6 inches _ constant. (4:2) |
In the gradual increase in the complexity of her world, she ñnds 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 remark- |
able aspect that must be abstracted from this picture is that ứhere are mo blocks. |
Take away the first terms in (4.1) and (4.2) and we find ourselves calculating |
--- Trang 87 --- |
more or less abstract things. he 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. Ín order to verify the conservation oŸ energy, |
we must be careful that we have not put any ¡in or taken any out. Second, the |
energy has a large number of đjƒƒeren‡ ƒorms, 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, InasS ©I©Tgy. |
TỶ we total up the formulas for each of these contributions, it will not change |
except for energy going in and out. |
Tt is important to realize that in physics today, we have no knowledge of what |
energy 2s. We do not have a picture that energy comes in little blobs of a delnite |
amount. lt is not that way. However, there are formulas for calculating some |
numerical quantity, and when we add ït all together it gives “28”——always the |
same number. Ït is an abstract thing in that it does not tell us the mechanism |
or the reøsons 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 ïllustration 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 mmost excellent argument by Mr. Carnot on the efficiency of steam engines.Š |
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 thứng as perpetudl 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 oŸ energy.) WWe must be careful to delne perpetual motion. Eirst, |
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 fnd that the |
net result is to have Ùjfed œ ueight, then we have a perpetual motion machine |
because we can use that lifted weight to run something else. 'That is, prouided the |
* Qur point here is not so mụuch the result, (4.3), which in fact you may already know, as |
the possibility of arriving at it by theoretical reasoning. |
--- Trang 88 --- |
Fig. 4-1. Simple weight-lifting machine. |
machine which lifted the weight is brought back to its exact original condition, |
and furthermore that it is completely self-con#ained—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 liÍt a |
little weight of the left pan. Ôn the other hand, we could lift a one-unit weight |
by lowering the three-unit weight, If we cheat a little by hfting a little weight |
of the other pan. Of course, we realize that with any ac£ual lifting machine, we |
must add a little extra to get it to run. This we disregard, #emporardi. Ideal |
machines, although they do not exist, do not require anything extra. A machine |
that we actually use can be, in a sense, œử”mos‡ reversible: that is, if it will Hft |
the weight of three by lowering a weight of one, then i% will also lift nearly the |
weight of one the same amount by lowering the weight of three. |
W© imagine that there are two classes of machines, those that are oø‡ reversible, |
which ineludes 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, |
Machine , 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 |
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