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to know how heavy W must be to balance the one pound on the plane. How
can we fgure 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
circumstanece, (a), the one pound weight is at the bottom and weight W is at
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tạ 1b,
VN ` N
(a) ()
Fig. 4-3. Inclined plane.
the top. When Wƒ 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 ij#ed the one-pound weight only #hree feet and
we lowered W pounds by ƒØioe feet. herefore W = Ỷ of a pound. Note that we
deduced this trom the conseruation oƒ energu, and not from force components.
Cleverness, however, is relative. It can be deduced in a way which is even more
brilliant, discovered by 5tevinus and inscribed on his tombstone. Figure 4-4
explains that it has to be Ỷ of a pound, because the chain does not go around.
lt is evident that the lower part of the chain is balanced by itself, so that the
pull of the ñve 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 š of a pound. (Tf 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 Eig. 4-5. A handle 20 inches long is used to turn the
Fig. 4-4. The epitaph of Stevinus.
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10 =đRĐ—
INCH E=
20” —Ị
Fig. 4-5. A screw Jack.
serew, which has 10 threads to the inch. We would like to know how mụuch force
would be needed at the handle to lift one ton (2000 pounds). IÝ we want to lift
the ton one ¡nch, say, then we must turn the handle around ten times. When it
goes around once i% 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 unknown smaller weight W/ applied to the end of the handle. So we find
out that W is about 1.6 pounds. 'Phis is a result of the conservation of energy.
° /s0\_ joo
_______... —_
Fig. 4-6. Weighted rod supported on one end.
Thake now the somewhat more complicated example shown in Eig. 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 1§
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 distance—tO
make 1% easy for ourselves suppose it goes down 4 inches—how high would the
two load weights rise? 'Phe center rises 2 inches, and the point a quarter of the
way from the fxed 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:
— 4W + (2)(60) + (1)(100) =0, W =5ä lb. (4.5)
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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 oŸ complicated bridge arrangements,
and so on. 'Phis approach is called the principle oƒ uirtual tuork, because in order
to apply this aregument we had to #nag¿ne that the structure moves a little—even
though ït is not reølu moving or even rmooabile. 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). TỶ we
pull the mass aside and release it, it swings back and forth. In its motion, it Ìoses
height in goïng 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 i§ is by virtue of Its mofZon that 16 is able to climb up again,
so we have the conversion of gravitational energy into some other form when it
reaches the bottom.
¬¬+—X
Fig. 4-7. Pendulum.
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 machzner by which it comes up or the pa#h 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. Ít 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 wïll reach a certain height; we do not know what it is yet, but it depends
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on the velocity—there is a formula for that. 'Phen to ñnd the formula for kinetic
energy for an object moving with velocity V, we must calculate the height that
it could reach, and multiply by the weight. We shall soon ñnd that we can write
1t this way:
K.E. =WV3/2g. (4.6)
OŸÝ course, the fact that motion has energy has nothing to do with the fact that
we are in a gravitational fñeld. It makes no diference +øhere the motion came
from. 'This is a general formula for various velocities. Both (4.3) and (4.6) are
approximate formulas, the fñrst because it is incorrect when the heights are great,
1.e., when the heights are so high that gravity is weakening: the second, because
of the relativistic correction at high speeds. However, when we do fñnally get the
exact formula for the energy, then the law of conservation of energy 1s correct.
4-4 Other forms of energy
W© can continue in this way to ïllustrate the existence of energy in other forms.
First, consider elastic energy. If we pull down on a spring, we must do some work,
for when we have it down, we can lift weights with it. Therefore in its stretched