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Y is not higher than X; that is, it is impossible to build a machine that will lift
a weight an 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 Ö, and that lifts the three-unit weight up
a distance Y.. Thhen we could lower the weight rom Y to X, obfaining [ree pouer,
and use the reversible Machine 4, running backwards, to lower the three-unit
--- Trang 89 ---
weight a distance à 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 ¡s not higher than ÄÃ, so that oŸ alÏ machines that can be designed,
the reversible machine is the best.
We can also see that all reversible machines must liẾt to ezactlU the same
height. Suppose that were really reversible also. The argument that Y is not
higher than Ä 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 ÄX ¡s no‡ húgher than Y.. This, then, is a very remarkable observation
because it permits us to analyze the height to which diferent machines are
going to lift something u#thout looking at the tnterior mmechanism. 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
1s reversible, we also know exactly ho high it will lit. 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. he next question is, of course, what is X?
5uppose 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 Eig. 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 Ñoor and ©wo shelves,
exactly spaced at distance X, and further, that the rack which holds the balls
is spaced at distance X, (a). Eirst we roll the balls horizontally from the rack
to the shelves, (b), and we suppose that this takes no energy because we do no
change the height. “The reversible machine then operates: i% lowers the single
ball to the foor, and it lifts the rack a distance X, (c). Ñow 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 is original condition. NÑow 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 #uo of them at all because, after all, there were
balls on shelves 2 and 3 before. The resulting efect has been to lHiÍt ome bajl a
--- Trang 90 ---
1ft. _ +
(a) START (b) LOAD BALLS
(c) 1 lb. LIFTS 3lb. A (d) UNLOAD BALLS
DISTANCE X
1ft. _
_=—=—— T x
(e) REARRANGE (f) END
Fig. 4-2. A reversible machine.
distance 3X. Now, if 3X exceeds one foot, then we can louer the ball to return
the machine to the initial condition, (), and we can run the apparatus again.
Therefore 3X cannot exceed one foot, for If 3X exceeds one foot we can make
perpetual motion. Likewise, we can prove that ơne ƒoot cannot czcccd 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
li 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. lÝ 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, £here tuiiÏ be no change. (WG have to generalize the example
where we moved only one weight to the case where when we lower one we lift
several đifferent ones——but that is easy.)
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'W© call the sum of the weights times the heights grauitational potential energU——
the energy which an objecE has because of its relationship in space, relative to
the earth. The formula for gravitational energy, then, so long as we are not Eoo
far rom the earth (the force weakens as we go higher) is
gravitational
potential energy | = (weight) x (height). (4.3)
for one object
Tt is a very beautiful line of reasoning. The only problem is that perhaps it is
not true. (After all, nature does not høơue 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. χ turns ouÈ ezperimentaliu, in fact, to be true.
The general name of energy which has to do with location relative to something
else is called po#enlal energy. In this particular case, of course, we call it
grauftatlional potential energu. TỶ ït is a question oŸ 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 elecfrical
potential energu. 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:
Am ") = (force) x bàn van ) (4.4)
cenergy acts through
W©e 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 diferent ways. W©e can now see that these
“laws” are dÌl the same thứng, and that we dịd not have to memorize 7ð rules to
figure i% 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
plane with a pulley, and on the other side of the pulley, a weight W. We want