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more accurately we measure, the more complicated the truth becomes; so in that |
sense we consider it not to result from a simple, fundamental process, which |
agrees with our original surmise. Eor example, if the velocity 1s extremely low, |
so low that an ordinary airplane is not ñying, as when the airplane is dragged |
slowly through the aïr, then the law changes, and the drag friction depends more |
nearly linearly on the velocity. To take another example, the frictional drag on |
a ball or a bubble or anything that is moving slowly through a viscous liquid |
like honey, is proportional to the velocity, but for motion so fast that the fÑuid |
swirls around (honey does not but water and air do) then the drag becomes more |
nearly proportional to the square of the velocity (F' = cø”), and if the velociEy |
continues to increase, then even this law begins to fail. People who say, “Well |
the coefficient changes slightly,” are dodging the issue. Second, there are other |
great complications: can this force on the airplane be divided or analyzed as a |
force on the wings, a force on the front, and so on? Indeed, this can be done, |
1ƒ we are concerned about the torques here and there, but then we have to get |
special laws for the force on the wings, and so on. It is an amazing fact that |
the force on a wing depends upon the other wing: in other words, if we take |
the airplane apart and put just one wing in the air, then the force is not the |
same as If the rest of the plane were there. The reason, of course, is that some |
of the wind that hits the front goes around to the wings and changes the force |
on the wings. Ít seems a miracle that there is such a simple, rough, empirical |
--- Trang 235 --- |
law that can be used in the design of airplanes, but this law is not in the same |
class as the basic laws of physics, and further study of it will only make it more |
and more complicated. AÁ study of how the coefficient e depends on the shape |
of the front of the airplane is, to put ¡1% mildly, frustrating. 'Phere jusÈ is no |
simple law for determining the coefficient in terms of the shape of the airplane. |
In contrast, the law of gravitation is simple, and further study only indicates its |
greater simplicity. |
We have just discussed bwo cases of friction, resulting from fast movement in |
air and slow movement in honey. There is another kind of friction, called dry |
triction or sliding friction, which occurs when one solid body slides on another. |
In this case a force is needed to maintain motion. 'This is called a frictional force, |
and its origin, also, is a very complicated matter. Both surfaces of contact are |
irregular, on an atomie level. 'Phere are many points of contact where the atoms |
seem to cling together, and then, as the sliding body is pulled along, the atoms |
snap apart and vibration ensues; something like that has to happen. Formerly |
the mechanism of this friction was thought to be very simple, that the surfaces |
were merely full of irregularities and the friction originated in liting the slider |
over the bumps; but this cannot be, for there is no loss of energy in that process, |
whereas power is in facE consumed. “The mechanism of power loss is that as |
the slider snaps over the bumps, the bumps deform and then generate waves |
and atomic motions and, after a while, heat, in the two bodies. NÑow 1È 1s very |
remarkable that again, empirically, this friction can be described approximately |
by a simple law. 'This law is that the force needed to overcome friction and to |
drag one object over another depends upon the normal force (i.e., perpendicular |
to the surface) between the two surfaces that are in contact. Actually, to a fairly |
good approximation, the frictional force is proportional to this normal force, and |
has a more or less constant coefficient; that is, |
†=uN, (12.1) |
where / is called the coeffficient oƒ fricion (Eig. 12-1). Although this coeflicient |
1s not exactly constant, the formula is a good empirical rule for Judging approxi- |
mately the amount of force that will be needed in certain practical or engineering |
circumstances. If the normal force or the speed of motion gets too big, the law |
fails because of the excessive heat generated. lt is important to realize that each |
of these empirical laws has its limitations, beyond which ¡it does not really work. |
'That the formula #' = uN is approximately correct can be demonstrated by |
a simple experiment. We set up a plane, inclined at a small angle Ø, and place a |
--- Trang 236 --- |
—>= DIRECTION OF MOTION |
Fig. 12-1. The relation between frictional force and the normal force |
for sliding contact. |
block of weight W/ on the plane. We then tilt the plane at a steeper angle, until |
the block just begins to slide from its own weight. The component of the weight |
downward along the plane is W sinØ, and this must equal the frictional force #! |
when the block is sliding uniformly. 'Phe component of the weight normail to the |
plane is W cosØ, and this is the normal force /Ú. With these values, the formula |
becomes Wƒ sin Ø = W cosØ, from which we get = sỉn Ø/ cos Ø = tan Ø. TỶ this |
law were exactly true, an object would start to slide at some defñnite inclination. |
T the same block is loaded by putting extra weight on it, then, although W ¡is |
increased, all the forces in the formula are increased in the same proportion, |
and W canecels out. If ð stays constant, the loaded block will slide again at the |
same slope. When the angle Ø is determined by trial with the original weight, it |
is found that with the greater weight the block will slide at about the same angle. |
This will be true even when one weight is many times as great as the other, and |
so we conclude that the coefficient of friction is independent of the weight. |
In performing this experiment it is noticeable that when the plane ïs tilted |
at about the correct angle Ø, the block does not slide steadily but in a halting |
fashion. At one place it may stop, at another it may move with acceleration. This |
behavior indicates that the coefficient of friction is only roughly a constant, and |
varies from place to place along the plane. The same erratic behavior is observed |
whether the block is loaded or not. Such variations are caused by diferent degrees |
of smoothness or hardness of the plane, and perhaps dirt, oxides, or other foreign |
matter. The tables that list purported values of for “steel on sbeel,” “copper |
on copper,” and the like, are all false, because they ignore the factors mentioned |
above, which really determine . “The friction is never due to “copper on copper,” |
etc., but to the impurities clinging to the copper. |
In experiments of the type described above, the friction is nearly independent |
of the velocity. Many people believe that the friction to be overcome to get |
--- Trang 237 --- |
something started (static friction) exceeds the force required to keep it sliding |
(sliding friction), but with dry metals it is very hard to show any diference. The |
opinion probably arises from experiences where small bits of oil or lubricant are |
present, or where blocks, for example, are supported by springs or other fexible |
supports so that they appear to bind. |
Tt ¡is quite dificult to do accurate quantitative experiments In friction, and |
the laws of friction are still not analyzed very well, in spite of the enormous |
engineering value of an accurate analysis. Although the law #' = ðN is fairly |
accurate once the surfaces are standardized, the reason for this form of the law |
is not really understood. To show that the coeficient is nearly independent of |
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