Chapter 3

_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.

_Fourth_.--If, on the contrary, I enlarge the angle A O C by moving the radius O C toward O B, then the sine and the versed sine both increase, and the cosine diminishes; and if, finally, I make O C coincide with O B, then the cosine has disappeared, the sine has become the radius O B, and the versed sine has become the radius O A, thus forming the two sides inclosing the right angle A O B. The study of this explanation will make you familiar with these important lines. The sine and the cosine I shall have occasion to employ in the latter part of my lecture. Now you know what the versed sine of an angle is, and are able to observe in Fig. 1 that the versed sine A E, of the angle A O C, represents in a general way the distance that the body A will be deflected from the tangent A D toward the center O while describing the arc A C.

The same law of deflection is shown, in smaller angles, in Fig. 2. In this figure, also, you observe in each of the angles A O B and A O C that the deflection, from the tangential direction toward the center, of a body moving in the arc A C is represented by the versed sine of the angle. The tangent to the arc at A, from which this deflection is measured, is omitted in this figure to avoid confusion. It is shown sufficiently in Fig. 1. The angles in Fig. 2 are still pretty large angles, being 12° and 24° respectively. These large angles are used for convenience of illustration; but it should be explained that this law does not really hold in them, as is evident, because the arc is longer than the tangent to which it would be connected by a line parallel with the versed sine. The law is absolutely true only when the tangent and arc coincide, and approximately so for exceedingly small angles.

In reality, however, we have only to do with the case in which the arc and the tangent do coincide, and in which the law that the deflection is _equal to_ the versed sine of the angle is absolutely true. Here, in observing this most familiar thing, we are, at a single step, taken to that which is utterly beyond our comprehension. The angles we have to consider disappear, not only from our sight, but even from our conception. As in every other case when we push a physical investigation to its limit, so here also, we find our power of thought transcended, and ourselves in the presence of the infinite.

We can discuss very small angles. We talk familiarly about the angle which is subtended by 1" of arc. On Fig. 2, a short line is drawn near to the radius O A'. The distance between O A' and this short line is 1° of the arc A' B'. If we divide this distance by 3,600, we get 1" of arc. The upper line of the Table of versed sines given below is the versed sine of 1" of arc. It takes 1,296,000 of these angles to fill a circular space. These are a great many angles, but they do not make a circle. They make a polygon. If the radius of the circumscribed circle of this polygon is 1,296,000 feet, which is nearly 213 geographical miles, each one of its sides will be a straight line, 6.283 feet long. On the surface of the earth, at the equator, each side of this polygon would be one-sixtieth of a geographical mile, or 101.46 feet. On the orbit of the moon, at its mean distance from the earth, each of these straight sides would be about 6,000 feet long.

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.

SECOND.--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.

_First_.--Directly as the weight of the body.

_Second_.--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.

_Third_.--With a given number of revolutions per minute, or a given angular velocity[1] _directly_ as the radius of the circle; and

_Fourth_.--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.

We come now to the reason for the fourth law, that, with a given actual velocity, the centrifugal force varies _inversely_ as the diameter of the circle. If any of you ever revolved a weight at the end of a cord with some velocity, and let the cord wind up, suppose around your hand, without doing anything to accelerate the motion, then, while the circle of revolution was growing smaller, the actual velocity continuing nearly uniform, you have felt the continually increasing stress, and have observed the increasing angular velocity, the two obviously increasing in the same ratio. That is the operation or action which the fourth law of centrifugal force expresses. An examination of this same figure (Fig. 2) will show you at once the reason for it in the increasing deflection which the body suffers, as its circle of revolution is contracted. If we take the velocity A' B', double the velocity A B, and transfer it to the smaller circle, we have the velocity A C. But the deflection has been increasing as we have reduced the circle, and now with one half the radius it is twice as great. It has increased in the same ratio in which the angular velocity has increased. Thus we see the simple and necessary nature of these laws. They merely express the different rates of deflection of a revolving body in these different cases.

THIRD.--We have a coefficient of centrifugal force, by which we are enabled to compute the amount of this resistance of a revolving body to deflection from a direct line of motion in all cases. This is that coefficient. The centrifugal force of a body making _one_ revolution per minute, in a circle of _one_ foot radius, is 0.000341 of the weight of the body.

According to the above laws, we have only to multiply this coefficient by the square of the number of revolutions made by the body per minute, and this product by the radius of the circle in feet, or in decimals of a foot, and we have the centrifugal force, in terms of the weight of the body. Multiplying this by the weight of the body in pounds, we have the centrifugal force in pounds.

Of course you want to know how this coefficient has been found out, and how you can be sure it is correct. I will tell you a very simple way. There are also mathematical methods of ascertaining this coefficient, which your professors, if you ask them, will let you dig out for yourselves. The way I am going to tell you I found out for myself, and that, I assure you, is the only way to learn anything, so that it will stick; and the more trouble the search gives you, the darker the way seems, and the greater the degree of perseverance that is demanded, the more you will appreciate the truth when you have found it, and the more complete and permanent your possession of it will be.

The explanation of this method may be a little more abstruse than the explanations already given, but it is very simple and elegant when you see it, and I fancy I can make it quite clear. I shall have to preface it by the explanation of two simple laws. The first of these is, that a body acted on by a constant force, so as to have its motion uniformly accelerated, suppose in a straight line, moves through distances which increase as the square of the time that the accelerating force continues to be exerted.

The necessary nature of this law, or rather the action of which this law is the expression, is shown in Fig. 3.

Let the distances A B, B C, C D, and D E in this figure represent four successive seconds of time. They may just as well be conceived to represent any other equal units, however small. Seconds are taken only for convenience. At the commencement of the first second, let a body start from a state of rest at A, under the action of a constant force, sufficient to move it in one second through a distance of one foot. This distance also is taken only for convenience. At the end of this second, the body will have acquired a velocity of two feet per second. This is obvious because, in order to move through one foot in this second, the body must have had during the second an average velocity of one foot per second. But at the commencement of the second it had no velocity. Its motion increased uniformly. Therefore, at the termination of the second its velocity must have reached two feet per second. Let the triangle A B F represent this accelerated motion, and the distance, of one foot, moved through during the first second, and let the line B F represent the velocity of two feet per second, acquired by the body at the end of it. Now let us imagine the action of the accelerating force suddenly to cease, and the body to move on merely with the velocity it has acquired. During the next second it will move through two feet, as represented by the square B F C I. But in fact, the action of the accelerating force does not cease. This force continues to be exerted, and produces on the body during the next second the same effect that it did during the first second, causing it to move through an additional foot of distance, represented by the triangle F I G, and to have its velocity accelerated two additional feet per second, as represented by the line I G. So in two seconds the body has moved through four feet. We may follow the operation of this law as far as we choose. The figure shows it during four seconds, or any other unit, of time, and also for any unit of distance. Thus:

Time 1 Distance 1 " 2 " 4 " 3 " 9 " 4 " 16

So it is obvious that the distance moved through by a body whose motion is uniformly accelerated increases as the square of the time.

But, you are asking, what has all this to do with a revolving body? As soon as your minds can be started from a state of rest, you will perceive that it has everything to do with a revolving body. The centripetal force, which acts upon a revolving body to draw it to the center, is a constant force, and under it the revolving body must move or be deflected through distances which increase as the squares of the times, just as any body must do when acted on by a constant force. To prove that a revolving body obeys this law, I have only to draw your attention to Fig. 2. Let the equal arcs, A B and B C, in this figure represent now equal times, as they will do in case of a body revolving in this circle with a uniform velocity. The versed sines of the angles, A O B and A O C, show that in the time, A C, the revolving body was deflected four times as far from the tangent to the circle at A as it was in the time, A B. So the deflection increased as the square of the time. If on the table already given, we take the seconds of arc to represent equal times, we see the versed sine, or the amount of deflection of a revolving body, to increase, in these minute angles, absolutely so far as appears up to the fifteenth place of decimals, as the square of the time.

The standard from which all computations are made of the distances passed through in given times by bodies whose motion is uniformly accelerated, and from which the velocity acquired is computed when the accelerating force is known, and the force is found when the velocity acquired or the rate of acceleration is known, is the velocity of a body falling to the earth. It has been established by experiment, that in this latitude near the level of the sea, a falling body in one second falls through a distance of 16.083 feet, and acquires a velocity of 32.166 feet per second; or, rather, that it would do so if it did not meet the resistance of the atmosphere. In the case of a falling body, its weight furnishes, first, the inertia, or the resistance to motion, that has to be overcome, and affords the measure of this resistance, and, second, it furnishes the measure of the attraction of the earth, or the force exerted to overcome its resistance. Here, as in all possible cases, the force and the resistance are identical with each other. The above is, therefore, found in this way to be the rate at which the motion of any body will be accelerated when it is acted on by a constant force equal to its weight, and encounters no resistance.

It follows that a revolving body, when moving uniformly in any circle at a speed at which its deflection from a straight line of motion is such that in one second this would amount to 16.083 feet, requires the exertion of a centripetal force equal to its weight to produce such deflection. The deflection varying as the square of the time, in 0.01 of a second this deflection will be through a distance of 0.0016083 of a foot.

Now, at what speed must a body revolve, in a circle of one foot radius, in order that in 0.01 of one second of time its deflection from a tangential direction shall be 0.0016083 of a foot? This decimal is the versed sine of the arc of 3°15', or of 3.25°. This angle is so small that the departure from the law that the deflection is equal to the versed sine of the angle is too slight to appear in our computation. Therefore, the arc of 3.25° is the arc of a circle of one foot radius through which a body must revolve in 0.01 of a second of time, in order that the centripetal force, and so the centrifugal force, shall be equal to its weight. At this rate of revolution, in one second the body will revolve through 325°, which is at the rate of 54.166 revolutions per minute.

Now there remains only one question more to be answered. If at 54.166 revolutions per minute the centrifugal force of a body is equal to its weight, what will its centrifugal force be at one revolution per minute in the same circle?

To answer this question we have to employ the other extremely simple law, which I said I must explain to you. It is this: The acceleration and the force vary in a constant ratio with each other. Thus, let force 1 produce acceleration 1, then force 1 applied again will produce acceleration 1 again, or, in other words, force 2 will produce acceleration 2, and so on. This being so, and the amount of the deflection varying as the squares of the speeds in the two cases, the centrifugal force of a body making one revolution per minute in a circle of

1² one foot radius will be ---------- = 0.000341 54.166²

--the coefficient of centrifugal force.

There is another mode of making this computation, which is rather neater and more expeditious than the above. A body making one revolution per minute in a circle of one foot radius will in one second revolve through an arc of 6°. The versed sine of this arc of 6° is 0.0054781046 of a foot. This is, therefore, the distance through which a body revolving at this rate will be deflected in one second. If it were acted on by a force equal to its weight, it would be deflected through the distance of 16.083 feet in the same time. What is the deflecting force actually exerted upon it? Of

0.0054781046 course, it is ------------. 16.083

This division gives 0.000341 of its weight as such deflecting force, the same as before.

In taking the versed sine of 6°, a minute error is involved, though not one large enough to change the last figure in the above quotient. The law of uniform acceleration does not quite hold when we come to an angle so large as 6°. If closer accuracy is demanded, we can attain it, by taking the versed sine for 1°, and multiplying this by 6². This gives as a product 0.0054829728, which is a little larger than the versed sine of 6°.

I hope I have now kept my promise, and made it clear how the coefficient of centrifugal force may be found in this simple way.

We have now learned several things about centrifugal force. Let me recapitulate. We have learned:

1st. The real nature of centrifugal force. That in the dynamical sense of the term force, this is not a force at all: that it is not capable of producing motion, that the force which is really exerted on a revolving body is the centripetal force, and what we are taught to call centrifugal force is nothing but the resistance which a revolving body opposes to this force, precisely like any other resistance.

2d. The direction of the deflection, to which the centrifugal force is the resistance, which is straight to the center.

3d. The measure of this deflection; the versed sine of the angle.

4th. The reason of the laws of centrifugal force; that these laws merely express the relative amount of the deflection, and so the amount of the force required to produce the deflection, and of the resistance of the revolving body to it, in all different cases.

5th. That the deflection of a revolving body presents a case analogous to that of uniformly accelerated motion, under the action of a constant force, similar to that which is presented by falling bodies;[1] and finally,

6th. How to find the coefficient, by which the amount of centrifugal force exerted in any case may be computed.

[Footnote 1: A body revolving with a uniform velocity in a horizontal plane would present the only case of uniformly accelerated motion that is possible to be realized under actual conditions.]

I now pass to some other features.