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1s weaker. lf we happen to have that kind of friction, then at the end of each
successive cycle the system is in the same condition as it was at the start, except
a little bit smaller. All the forces are smaller in the same proportion: the spring
force 1s reduced, the inertial efects are lower because the accelerations are now
weaker, and the friction is less too, by our careful design. When we actually
have that kind of fiction, we ñnd that each oscillation is exactly the same as the
first one, except reduced in amplitude. If the first cycle dropped the amplitude,
say, to 90 percent of what it was at the start, the next will drop it to 90 percent
of 90 percent, and so on: ứhe sizes öƒ the oscdllatlions are reduced bụ the same
fraclion oj themselues in cuerw cụcÌle. An exponential function is a curve which
does just that. It changes by the same factor in each equal interval of time. That
1s to say, 1ƒ the amplitude of one cycle, relative to the preceding one, is called ø,
then the amplitude of the next is a2, and of the next, ø”. So the amplitude is
some constant raised to a power equal to the number of cycles traversed:
A= Aoad". (25.10)
But of course m œ ứ, so it is perfectly clear that the general solution will be some
kind of an oscillation, sine or cosine œ#, tỉmes an amplitude which goes as ÙÍ
more or less. But ö can be written as e °, 1ƒ b is positive and less than 1. So this
is why the solution looks like e~“ cosœo#. It is very sỉmple.
'What happens ïf the friction is not so artificial; for example, ordinary rubbing
on a table, so that the friction force is a certain constant amount, and is indepen-
dent of the size of the oscillation that reverses its direction each hal£cycle? Then
the equation is no longer linear, i§ becomes hard to solve, and must be solved
by the numerical method given in Chapter 9, or by considering each half-cycle
separately. 'Phe numerical method is the most powerful method of all, and can
solve any equation. lt is only when we have a simple problem that we can use
mathematical analysis.
Mathematical analysis is not the grand thing it is said to be; it solves only
the simplest possible equations. Äs soon as the equations get a little more
complicated, just a shade—they cannot be solved analytically. But the numerical
method, which was advertised at the beginning of the course, can take care of
any equation of physical interest.
Next, what about the resonance curve? Why is there a resonance? First,
Imagine for a moment that there is no friction, and we have something which
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could oscillate by itself. If we tapped the pendulum just right each time it went
by, of course we could make it go like mad. But if we close our eyes and do
not watch ït, and tap at arbitrary equal intervals, what is going to happen?
Sometimes we will fnd ourselves tapping when it is goỉing the wrong way. When
we happen to have the timing just right, of course, each tap is given at just the
right time, and so i% goes higher and higher and higher. So without friction we
get a curve which looks like the solid curve in Eig. 25-5 for diferent frequencies.
Qualitatively, we understand the resonance curve; in order to get the exact shape
of the curve it is probably just as well to do the mathematics. The curve goes
toward infinity as œ —> œọ, where œọ is the natural frequency of the oscillator.
Fig. 25-5. Resonance curves with various amounts of friction present.
Now suppose there is a little bit of friction; then when the displacement of
the oscillator is small, the friction does not affect it much; the resonance curve
1s the same, except when we are near resonance. Instead of becoming infinite
near resonance, the curve is only going to get so hiph that the work done by
our tapping each time is enough to compensate for the loss of energy by friction
during the cycle. 5o the top oŸ the curve is rounded oÑ——it does not go to infnity.
lf there is more friction, the top of the curve is rounded off still more. Now
someone might say, “I thought the widths of the curves depended on the friction.”
'That is because the curve is usually plotted so that the top of the curve ¡s called
one unit. However, the mathematical expression is even simpler to understand 1ƒ
we just plot all the curves on the same scale; then all that happens is that the
friction cuts down the topl T there is less friction, we can go farther up into that
little pinnacle before the friction cuts it of, so it looks relatively narrow. That is,
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the higher the peak of the curve, the narrower the width at half the maximum
height.
Jinally, we take the case where there is an enormous amount of friction. lt
turns out that iŸ there is too much friction, the system does not oscillate at all.
The energy in the spring is barely able to move it against the frictional force,
and so it slowly oozes down to the equilibrium poiïnt.
25-4 Analogs in physics
'The next aspect of this review is to note that masses and springs are not the
only linear systems; there are others. In particular, there are electrical systems
called linear circuits, in which we fnd a complete analog to mechanical systems.
W© dịd not learn exactly hy each of the objects in an electrical cireuit works in
the way it does—that is not to be understood at the present moment; we may
assert 1% as an experimentally verifable fact that they behave as stated.
For example, let us take the sinplest possible cireumstance. We have a piece
of wire, which is just a resistance, and we have applied to it a difference In
potential, V. Now the V means this: if we carry a charge g through the wire
from one terminal to another terminal, the work done is gV. 'Phe higher the
voltage diference, the more work was done when the charge, as we say, “falls”
from the hiph potential end of the terminal to the low potential end. So charges
release energy in goiïng om one end to the other. Now the charges do not simply
fñy om one end straight to the other end; the atoms in the wire ofer some
resistance to the current, and this resistance obeys the following law for almost
all ordinary substances: If there is a current 7, that is, so and so many charges
per second tumbling down, the number per second that comes tumbling through
the wire is proportional to how hard we push them——in other words, proportional
to how much voltage there is:
V =1TR= R(dq/d). (25.11)
'The coefficient ?‡ ¡is called the resisfance, and the equation is called Ohm's Law.
'The unit of resistanee is the ohm; it is equal to one volt per ampere. In mechanical
situations, to get such a frictional force in proportion to the velocity is dificult;
in an electrical system it is very easy, and this law is extremely accurate for most
mnetals.
W© are often interested in how much work is done per second, the power Ìoss,
or the energy liberated by the charges as they tumble down the wire. When
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we carry a charge g through a voltage V, the work is gV, so the work done per
second would be V(dqg/đ£), which is the same as Vĩ, or also IR- I = I2R. This
is called the heøat#ng loss—this is how much heat is generated in the resistance
per second, by the conservation of energy. It is this heat that makes an ordinary
incandescent light bulb work.