Chapter 2

He had made a careful study of the subject, in his thoroughly practical way, and had become convinced that such a line was feasible, and would be remunerative. At his instance a convention of telegraph men met in the city of New York, to consider the project. The feeling in this convention was extremely unfavorable to it. A committee reported against it unanimously, on three grounds--the country was destitute of timber, the line would be destroyed by the Indians, and if constructed and maintained, it would not pay expenses. Mr. Sibley found himself alone. An earnest appeal which he made from the report of the committee was received with derisive laughter. The idea of running a telegraph line through what was then a wilderness, roamed over for between one and two thousand miles of its breadth by bands of savages, who of course would destroy the line as soon as it was put up, and where repairs would be difficult and useless, even if the other objections to it were out of the way, struck the members of the convention as so exquisitely ludicrous that it seemed as if they would never be done laughing about it. If Mr. Sibley had advocated a line to the moon, they would hardly have seen in it greater evidence of lunacy. When he could be heard, he rose again and said: "Gentlemen, you may laugh, but if I was not so old, I would build the line myself." Upon this, of course, they laughed louder than ever. As they laughed, he grew mad, and shouted: "Gentlemen, I will bar the years, and do it." And he did it. Without help from any one, for every man who claimed a right to express an opinion upon it scouted the project as chimerical, and no capitalist would put a dollar in it, Hiram Sibley built the line of telegraph to San Francisco, risking in it all he had in the world. He set about the work with his customary energy, all obstacles vanished, and the line was completed in an incredibly short time. And from the day it was opened, it has proved probably the most profitable line of telegraph that has ever been constructed. There was the practicability, and there was the demand and the business to be done, and yet no living man could see it, or could be made to see it, except Hiram Sibley. "And to-day," he said, with honest pride, "to-day in New York, men to whom I went almost on my knees for help in building this line, and who would not give me a dollar, have solicited me to be allowed to buy stock in it at the rate of five dollars for one."

"But how about the Indians?" I asked. "Why," he replied, "we never had any trouble from the Indians. I knew we wouldn't have. Men who supposed I was such a fool as to go about this undertaking before that was all settled didn't know me. No Indian ever harmed that line. The Indians are the best friends we have got. You see, we taught the Indians the Great Spirit was in that line; and what was more, we proved it to them. It was, by all odds, the greatest medicine they ever saw. They fairly worshiped it. No Indian ever dared to do it harm."

"But," he added, "there was one thing I didn't count on. The border ruffians in Missouri are as bad as anybody ever feared the Indians might be. They have given us so much trouble that we are now building a line around that State, through Iowa and Nebraska. We are obliged to do it."

This opened another phase of the subject. The telegraph line to the Pacific had a value beyond that which could be expressed in money. It was perhaps the strongest of all the ties which bound California so securely to the Union, in the dark days of its struggle for existence. The secession element in Missouri recognized the importance of the line in this respect, and were persistent in their efforts to destroy it. We have seen by what means their purpose was thwarted.

I have always felt that, among the countless evidences of the ordering of Providence by which the war for the preservation of the Union was signalized, not the least striking was the raising up of this remarkable man, to accomplish alone, and in the very nick of time, a work which at once became of such national importance.

This is the man who has crowned his useful career, and shown again his eminently practical character and wise foresight, by the endowment of this College, which cannot fail to be a perennial source of benefit to the country whose interests he has done so much to promote, and which his remarkable sagacity and energy contributed so much to preserve.

We have an excellent rule, followed by all successful designers of machinery, which is, to make provision for the extreme case, for the most severe test to which, under normal conditions, and so far as practicable under abnormal conditions also, the machinery can be subjected. Then, of course, any demands upon it which are less than the extreme demand are not likely to give trouble. I shall apply this principle in addressing you to-day. In what I have to say, I shall speak directly to the youngest and least advanced minds among my auditors. If I am successful in making an exposition of my subject which shall be plain to them, then it is evident that I need not concern myself about being understood by the higher class men and the professors.

The subject to which your attention is now invited is

THE PRINCIPLES AND METHODS OF BALANCING FORCES DEVELOPED IN MOVING BODIES.

This is a subject with which every one who expects to be concerned with machinery, either as designer or constructor, ought to be familiar. The principles which underlie it are very simple, but in order to be of use, these need to be thoroughly understood. If they have once been mastered, made familiar, incorporated into your intellectual being, so as to be readily and naturally applied to every case as it arises, then you occupy a high vantage ground. In this particular, at least, you will not go about your work uncertainly, trying first this method and then that one, or leaving errors to be disclosed when too late to remedy them. On the contrary, you will make, first your calculations and then your plans, with the certainty that the result will be precisely what you intend.

Moreover, when you read discussions on any branch of this subject, you will not receive these into unprepared minds, just as apt to admit error as truth, and possessing no test by which to distinguish the one from the other; but you will be able to form intelligent judgments with respect to them. You will discover at once whether or not the writers are anchored to the sure holding ground of sound principles.

It is to be observed that I do not speak of balancing bodies, but of balancing forces. Forces are the realities with which, as mechanical engineers, you will have directly to deal, all through your lives. The present discussion is limited also to those forces which are developed in moving bodies, or by the motion of bodies. This limitation excludes the force of gravity, which acts on all bodies alike, whether at rest or in motion. It is, indeed, often desirable to neutralize the effect of gravity on machinery. The methods of doing this are, however, obvious, and I shall not further refer to them.

Two very different forces, or manifestations of force, are developed by the motion of bodies. These are

MOMENTUM AND CENTRIFUGAL FORCE.

The first of these forces is exerted by every moving body, whatever the nature of the path in which it is moving, and always in the direction of its motion. The latter force is exerted only by bodies whose path is a circle, or a curve of some form, about a central body or point, to which it is held, and this force is always at right angles with the direction of motion of the body.

Respecting momentum, I wish only to call your attention to a single fact, which will become of importance in the course of our discussion. Experiments on falling bodies, as well as all experience, show that the velocity of every moving body is the product of two factors, which must combine to produce it. Those factors are force and distance. In order to impart motion to the body, force must act through distance. These two factors may be combined in any proportions whatever. The velocity imparted to the body will vary as the square root of their product. Thus, in the case of any given body,

Let force 1, acting through distance 1, impart velocity 1. Then " 1, " " " 4, will " " 2, or " 2, " " " 2, " " " 2, or " 4, " " " 1, " " " 2; And " 1, " " " 9, " " " 3, or " 3, " " " 3, " " " 3, or " 9, " " " 1, " " " 3.

This table might be continued indefinitely. The product of the force into the distance will always vary as the square of the final velocity imparted. To arrest a given velocity, the same force, acting through the same distance, or the same product of force into distance, is required that was required to impart the velocity.

The fundamental truth which I now wish to impress upon your minds is that in order to impart velocity to a body, to develop the energy which is possessed by a body in motion, force must act through distance. Distance is a factor as essential as force. Infinite force could not impart to a body the least velocity, could not develop the least energy, without acting through distance.

This exposition of the nature of momentum is sufficient for my present purpose. I shall have occasion to apply it later on, and to describe the methods of balancing this force, in those cases in which it becomes necessary or desirable to do so. At present I will proceed to consider the second of the forces, or manifestations of force, which are developed in moving bodies--_centrifugal force_.

This force presents its claims to attention in all bodies which revolve about fixed centers, and sometimes these claims are presented with a good deal of urgency. At the same time, there is probably no subject, about which the ideas of men generally are more vague and confused. This confusion is directly due to the vague manner in which the subject of centrifugal force is treated, even by our best writers. As would then naturally be expected, the definitions of it commonly found in our handbooks are generally indefinite, or misleading, or even absolutely untrue.

Before we can intelligently consider the principles and methods of balancing this force, we must get a correct conception of the nature of the force itself. What, then, is centrifugal force? It is an extremely simple thing; a very ordinary amount of mechanical intelligence is sufficient to enable one to form a correct and clear idea of it. This fact renders it all the more surprising that such inaccurate and confused language should be employed in its definition. Respecting writers, also, who use language with precision, and who are profound masters of this subject, it must be said that, if it had been their purpose to shroud centrifugal force in mystery, they could hardly have accomplished this purpose more effectually than they have done, to minds by whom it was not already well understood.

Let us suppose a body to be moving in a circular path, around a center to which it is firmly held; and let us, moreover, suppose the impelling force, by which the body was put in motion, to have ceased; and, also, that the body encounters no resistance to its motion. It is then, by our supposition, moving in its circular path with a uniform velocity, neither accelerated nor retarded. Under these conditions, what is the force which is being exerted on this body? Clearly, there is only one such force, and that is, the force which holds it to the center, and compels it, in its uniform motion, to maintain a fixed distance from this center. This is what is termed centripetal force. It is obvious, that the centripetal force, which holds this revolving body _to_ the center, is the only force which is being exerted upon it.

Where, then, is the centrifugal force? Why, the fact is, there is not any such thing. In the dynamical sense of the term "force," the sense in which this term is always understood in ordinary speech, as something tending to produce motion, and the direction of which determines the direction in which motion of a body must take place, there is, I repeat, no such thing as centrifugal force.

There is, however, another sense in which the term "force" is employed, which, in distinction from the above, is termed a statical sense. This "statical force" is the force by the exertion of which a body keeps still. It is the force of inertia--the resistance which all matter opposes to a dynamical force exerted to put it in motion. This is the sense in which the term "force" is employed in the expression "centrifugal force." Is that all? you ask. Yes; that is all.

I must explain to you how it is that a revolving body exerts this resistance to being put in motion, when all the while it _is_ in motion, with, according to our above supposition, a uniform velocity. The first law of motion, so far as we now have occasion to employ it, is that a body, when put in motion, moves in a straight line. This a moving body always does, unless it is acted on by some force, other than its impelling force, which deflects it, or turns it aside, from its direct line of motion. A familiar example of this deflecting force is afforded by the force of gravity, as it acts on a projectile. The projectile, discharged at any angle of elevation, would move on in a straight line forever, but, first, it is constantly retarded by the resistance of the atmosphere, and, second, it is constantly drawn downward, or made to fall, by the attraction of the earth; and so instead of a straight line it describes a curve, known as the trajectory.

Now a revolving body, also, has the same tendency to move in a straight line. It would do so, if it were not continually deflected from this line. Another force is constantly exerted upon it, compelling it, at every successive point of its path, to leave the direct line of motion, and move on a line which is everywhere equally distant from the center to which it is held. If at any point the revolving body could get free, and sometimes it does get free, it would move straight on, in a line tangent to the circle at the point of its liberation. But if it cannot get free, it is compelled to leave each new tangential direction, as soon as it has taken it.

This is illustrated in the above figure. The body, A, is supposed to be revolving in the direction indicated by the arrow, in the circle, A B F G, around the center, O, to which it is held by the cord, O A. At the point, A, it is moving in the tangential direction, A D. It would continue to move in this direction, did not the cord, O A, compel it to move in the arc, A C. Should this cord break at the point, A, the body would move; straight on toward D, with whatever velocity it had.

You perceive now what centrifugal force is. This body is moving in the direction, A D. The centripetal force, exerted through the cord, O A, pulls it aside from this direction of motion. The body resists this deflection, and this resistance is its centrifugal force.

Centrifugal force is, then, properly defined to be the disposition of a revolving body to move in a straight line, and the resistance which such a body opposes to being drawn aside from a straight line of motion. The force which draws the revolving body continually to the center, or the deflecting force, is called the centripetal force, and, aside from the impelling and retarding forces which act in the direction of its motion, the centripetal force is, dynamically speaking, the only force which is exerted on the body.

It is true, the resistance of the body furnishes the measure of the centripetal force. That is, the centripetal force must be exerted in a degree sufficient to overcome this resistance, if the body is to move in the circular path. In this respect, however, this case does not differ from every other case of the exertion of force. Force is always exerted to overcome resistance: otherwise it could not be exerted. And the resistance always furnishes the exact measure of the force. I wish to make it entirely clear, that in the dynamical sense of the term "force," there is no such thing as centrifugal force. The dynamical force, that which produces motion, is the centripetal force, drawing the body continually from the tangential direction, toward the center; and what is termed centrifugal force is merely the resistance which the body opposes to this deflection, _precisely like any other resistance to a force_.

The centripetal force is exerted on the radial line, as on the line, A O, Fig. 1, at right angles with the direction in which the body is moving; and draws it directly toward the center. It is, therefore, necessary that the resistance to this force shall also be exerted on the same line, in the opposite direction, or directly from the center. But this resistance has not the least power or tendency to produce motion in the direction in which it is exerted, any more than any other resistance has.

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.