The best we are able to do is to conceive of a polygon having an infinite number of sides, and so an infinite number of angles, the versed sines of which are infinitely small, and having, also, an infinite number of tangential directions, in which the body can successively move. Still, we have not reached the circle. We never can reach the circle. When you swing a sling around your head, and feel the uniform stress exerted on your hand through the cord, you are made aware of an action which is entirely beyond the grasp of our minds and the reach of our analysis.
So always in practical operation that law is absolutely true which we observe to be approximated to more and more nearly as we consider smaller and smaller angles, that the versed sine of the angle is the measure of its deflection from the straight line of motion, or the measure of its fall toward the center, which takes place at every point in the motion of a revolving body.
Then, assuming the absolute truth of this law of deflection, we find ourselves able to explain all the phenomena of centrifugal force, and to compute its amount correctly in all cases.
We have now advanced two steps. We have learned the direction and the measure of the deflection, which a revolving body continually suffers, and its resistance to which is termed centrifugal force. The direction is toward the center, and the measure is the versed sine of the angle.
We next come to consider what are known as the laws of centrifugal force. These laws are four in number. They are, that the amount of centrifugal force exerted by a revolving body varies in four ways.
Directly as the weight of the body.
In a given circle of revolution, as the square of the speed or of the number of revolutions per minute; which two expressions in this case mean the same thing.
With a given number of revolutions per minute, or a given angular velocity directly as the radius of the circle; and
With a given actual velocity, or speed in feet per minute, inversely as the radius of the circle.
[Footnote 1: A revolving body is said to have the same angular velocity, when it sweeps through equal angles in equal times. Its actual velocity varies directly as the radius of the circle in which it is revolving.]
Of course there is a reason for these laws. You are not to learn them by rote, or to accept them on any authority. You are taught not to accept any rule or formula on authority, but to demand the reason for it--to give yourselves no rest until you know the why and wherefore, and comprehend these fully. This is education, not cramming the mind with mere facts and rules to be memorized, but drawing out the mental powers into activity, strengthening them by use and exercise, and forming the habit, and at the same time developing the power, of penetrating to the reason of things.
In this way only, you will be able to meet the requirement of a great educator, who said: "I do not care to be told what a young man knows, but what he can do." I wish here to add my grain to the weight of instruction which you receive, line upon line, precept on precept, on this subject.
The reason for these laws of centrifugal force is an extremely simple one. The first law, that this force varies directly as the weight of the body, is of course obvious. We need not refer to this law any further. The second, third, and fourth laws merely express the relative rates at which a revolving body is deflected from the tangential direction of motion, in each of the three cases described, and which cases embrace all possible conditions.
These three rates of deflection are exhibited in Fig. 2. An examination of this figure will give you a clear understanding of them. Let us first suppose a body to be revolving about the point, O, as a center, in a circle of which A B C is an arc, and with a velocity which will carry it from A to B in one second of time. Then in this time the body is deflected from the tangential direction a distance equal to A D, the versed sine of the angle A O B. Now let us suppose the velocity of this body to be doubled in the same circle. In one second of time it moves from A to C, and is deflected from the tangential direction of motion a distance equal to A E, the versed sine of the angle, A O C. But A E is four times A D. Here we see in a given circle of revolution the deflection varying as the square of the speed. The slight error already pointed out in these large angles is disregarded.
The following table will show, by comparison of the versed sines of very small angles, the deflection in a given circle varying as the square of the speed, when we penetrate to them, so nearly that the error is not disclosed at the fifteenth place of decimals.
The versed sine of 1" is 0.000,000,000,011,752 " " " " 2" is 0.000,000,000,047,008 " " " " 3" is 0.000,000,000,105,768 " " " " 4" is 0.000,000,000,188,032 " " " " 5" is 0.000,000,000,293,805 " " " " 6" is 0.000,000,000,423,072 " " " " 7" is 0.000,000,000,575,848 " " " " 8" is 0.000,000,000,752,128 " " " " 9" is 0.000,000,000,951,912 " " " " 10" is 0.000,000,001,175,222 " " " " 100" is 0.000,000,117,522,250
You observe the deflection for 10" of arc is 100 times as great, and for 100" of arc is 10,000 times as great as it is for 1" of arc. So far as is shown by the 15th place of decimals, the versed sine varies as the square of the angle; or, in a given circle, the deflection, and so the centrifugal force, of a revolving body varies as the square of the speed.
The reason for the third law is equally apparent on inspection of Fig. 2. It is obvious, that in the case of bodies making the same number of revolutions in different circles, the deflection must vary directly as the diameter of the circle, because for any given angle the versed sine varies directly as the radius. Thus radius O A' is twice radius O A, and so the versed sine of the arc A' B' is twice the versed sine of the arc A B. Here, while the angular velocity is the same, the actual velocity is doubled by increase in the diameter of the circle, and so the deflection is doubled. This exhibits the general law, that with a given angular velocity the centrifugal force varies directly as the radius or diameter of the circle.