We have been supposing a body to be firmly held to the center, so as to be compelled to revolve about it in a fixed path. But the bond which holds it to the center may be elastic, and in that case, if the centrifugal force is sufficient, the body will be drawn from the center, stretching the elastic bond. It may be asked if this does not show centrifugal force to be a force tending to produce motion from the center. This question is answered by describing the action which really takes place. The revolving body is now imperfectly deflected. The bond is not strong enough to compel it to leave its direct line of motion, and so it advances a certain distance along this tangential line. This advance brings the body into a larger circle, and by this enlargement of the circle, assuming the rate of revolution to be maintained, its centrifugal force is proportionately increased. The deflecting power exerted by the elastic bond is also increased by its elongation. If this increase of deflecting force is no greater than the increase of centrifugal force, then the body will continue on in its direct path; and when the limit of its elasticity is reached, the deflecting bond will be broken. If, however, the strength of the deflecting bond is increased by its elongation in a more rapid ratio than the centrifugal force is increased by the enlargement of the circle, then a point will be reached in which the centripetal force will be sufficient to compel the body to move again in the circular path.
Sometimes the centripetal force is weak, and opportunity is afforded to observe this action, and see its character exhibited. A common example of weak centripetal force is the adhesion of water to the face of a revolving grindstone. Here we see the deflecting force to become insufficient to compel the drops of water longer to leave their direct paths, and so these do not longer leave their direct paths, but move on in those paths, with the velocity they have at the instant of leaving the stone, flying off on tangential lines.
If, however, a fluid be poured on the side of the revolving wheel near the axis, it will move out to the rim on radial lines, as may be observed on car wheels universally. The radial lines of black oil on these wheels look very much as if centrifugal force actually did produce motion, or had at least a very decided tendency to produce motion, in the radial direction. This interesting action calls for explanation. In this action the oil moves outward gradually, or by inconceivably minute steps. Its adhesion being overcome in the least possible degree, it moves in the same degree tangentially. In so doing it comes in contact with a point of the surface which has a motion more rapid than its own. Its inertia has now to be overcome, in the same degree in which it had overcome the adhesion. Motion in the radial direction is the result of these two actions, namely, leaving the first point of contact tangentially and receiving an acceleration of its motion, so that this shall be equal to that of the second point of contact. When we think about the matter a little closely, we see that at the rim of the wheel the oil has perhaps ten times the velocity of revolution which it had on leaving the journal, and that the mystery to be explained really is, How did it get that velocity, moving out on a radial line? Why was it not left behind at the very first? Solely by reason of its forward tangential motion. That is the answer.
When writers who understand the subject talk about the centripetal and centrifugal forces being different names for the same force, and about equal action and reaction, and employ other confusing expressions, just remember that all they really mean is to express the universal relation between force and resistance. The expression "centrifugal force" is itself so misleading, that it becomes especially important that the real nature of this so-called force, or the sense in which the term "force" is used in this expression, should be fully explained.[1] This force is now seen to be merely the tendency of a revolving body to move in a straight line, and the resistance which it opposes to being drawn aside from that line. Simple enough! But when we come to consider this action carefully, it is wonderful how much we find to be contained in what appears so simple. Let us see.
[Footnote 1: I was led to study this subject in looking to see what had become of my first permanent investment, a small venture, made about thirty-five years ago, in the "Sawyer and Gwynne static pressure engine." This was the high-sounding name of the Keely motor of that day, an imposition made possible by the confused ideas prevalent on this very subject of centrifugal force.]
FIRST.--I have called your attention to the fact that the direction in which the revolving body is deflected from the tangential line of motion is toward the center, on the radial line, which forms a right angle with the tangent on which the body is moving. The first question that presents itself is this: What is the measure or amount of this deflection? The answer is, this measure or amount is the versed sine of the angle through which the body moves.
Now, I suspect that some of you--some of those whom I am directly addressing--may not know what the versed sine of an angle is; so I must tell you. We will refer again to Fig. 1. In this figure, O A is one radius of the circle in which the body A is revolving. O C is another radius of this circle. These two radii include between them the angle A O C. This angle is subtended by the arc A C. If from the point O we let fall the line C E perpendicular to the radius O A, this line will divide the radius O A into two parts, O E and E A. Now we have the three interior lines, or the three lines within the circle, which are fundamental in trigonometry. C E is the sine, O E is the cosine, and E A is the versed sine of the angle A O C. Respecting these three lines there are many things to be observed. I will call your attention to the following only:
First.--Their length is always less than the radius. The radius is expressed by 1, or unity. So, these lines being less than unity, their length is always expressed by decimals, which mean equal to such a proportion of the radius.
Second.--The cosine and the versed sine are together equal to the radius, so that the versed sine is always 1, less the cosine.
Third.--If I diminish the angle A O C, by moving the radius O C toward O A, the sine C E diminishes rapidly, and the versed sine E A also diminishes, but more slowly, while the cosine O E increases. This you will see represented in the smaller angles shown in Fig. 2. If, finally, I make O C to coincide with O A, the angle is obliterated, the sine and the versed sine have both disappeared, and the cosine has become the radius.