The laws of motion. We return again and yet again to the subject, for one is not to be deterred therefrom by any concession of inadequacy, no, nor by any degree of respect for the vivid youthful sense of those things which to suit our narrow purpose must be stripped completely bare. It is unfortunate, however, that the most familiar type of motion, the flowing of water or the blowing of the wind, is bewilderingly useless as a basis for the establishment of the simple and precise ideas which are called the "laws of motion," and which are the most important of the fundamental principles of physics. These ideas have in fact grown out of the study of the simple phenomena which are a.s.sociated with the motion of bodies in bulk without perceptible change of form, the motion of rigid bodies, so called.
Before narrowing down the scope of the discussion, however, let us ill.u.s.trate a very general application of the simplest idea of motion, the idea of velocity. Every one has, no doubt, an idea of what is meant by the velocity of the wind; and a sailor, having what he calls a ten-knot wind, knows that he can manage his boat with a certain spread of canvas and that he can accomplish a certain portion of his voyage in a given time; but an experienced sailor, although he speaks glibly of a ten-knot wind, belies his speech by taking wise precaution against every conceivable emergency. He knows that a ten-knot wind is by no means a sure or a simple thing with its incessant blasts and whirls; and a sensitive anemometer, having more regard for minutiae than any sailor, usually registers in every wind a number of almost complete but excessively irregular stops and starts every minute and variations of direction that sweep around half the horizon!
Wer will was Lebendig"s erkennen und beschreiben Sucht erst den Geist heraus zu treiben.
GOETHE.
We must evidently direct our attention to something simpler than the wind. Let us, therefore, consider the drawing of a wagon or the propulsion of a boat. It is a familiar experience that effort is required to start a body moving and that continued effort is required to maintain the motion. Certain very simple facts as to the nature and effects of this effort were discovered by Sir Isaac Newton, and on the basis of these facts Newton formulated the laws of motion.
The effort required to start a body or to keep it moving is called force. Thus, if one starts a box sliding along a table one is said to exert a force on the box. The same effect might be accomplished by interposing a stick between the hand and the box, in which case one would exert a force on the stick and the stick in its turn would exert a force on the box. We thus arrive at the notion of force action between inanimate bodies, between the stick and the box in this case, and Newton pointed out that the force action between the two bodies _A_ and _B_ always consists of two equal and opposite forces, that is to say, if body _A_ exerts a force on _B,_ then _B_ exerts an equal and opposite force on _A,_ or, to use Newton"s words, action is equal to reaction and in a contrary direction.
In leading up to this statement one might consider the force with which a person pushes on the box and the equal and opposite force with which the box pushes back on the person, but if one does not wish to introduce the stick as an intermediary, it is better to speak of the force with which the hand pushes on the box, and the equal and opposite force with which the box pushes back on the hand, because in discussing physical phenomena it is of the utmost importance to pay attention only to impersonal [42] things. Indeed our modern industrial life, in bringing men face to face with an entirely unprecedented array of intricate mechanical and physical problems, demands of every one a great and increasing amount of impersonal thinking, and the precise and rigorous modes of thought of the physical sciences are being forced upon widening circles of men with a relentless insistence--all of which it was intended to imply by referring to the "stress of dryness" which overtakes the little axolotl in his contented existence as a tadpole.
When we examine into the conditions under which a body starts to move and the conditions under which a body once started is kept in motion, we come across a very remarkable fact, if we are careful to consider every force which acts upon the body, and this remarkable fact is that the forces which act upon _a body at rest_ are related to each other in precisely the same way as the forces which act upon _a body moving steadily along a straight path._ Therefore it is convenient to consider, _first_ the relation between the forces which act upon a body at rest, or upon a body in uniform motion, and _second_ the relation between the forces which act upon a body which is starting or stopping or changing the direction of its motion.
Suppose a person _A_ were to hold a box in mid-air. To do so it would of course be necessary for him to push upwards on the box so as to balance the downward pull of the earth, the weight of the box as it is called. If another person _B_ were to take hold of the box and pull upon it in any direction, _A_ would have to exert an equal pull on the box in the opposite direction to keep it stationary. _The forces which act upon a stationary body are always balanced._
Every one, perhaps, realizes that what is here said about the balanced relation of the forces which act upon a stationary box, is equally true of the forces which act on a box similarly held in a steadily moving railway car or boat.
Therefore, _the forces which act upon a body which moves steadily along a straight path are balanced._
This is evidently true when the moving body is surrounded on all sides by things which are moving along with it, as in a car or a boat; but how about a body which moves steadily along a straight path but which is surrounded by bodies which do not move along with it? Everyone knows that some active agent such as a horse or a steam engine must pull steadily upon such a body to keep it in motion. If left to itself such a moving body quickly comes to rest. Many have, no doubt, reached this further inference from experience, namely, that the tendency of moving bodies to come to rest is due to the dragging forces, or friction, with which surrounding bodies act upon a body in motion. Thus a moving boat is brought to rest by the drag of the water when the propelling force ceases to act; a train of cars is brought to rest because of the drag due to friction when the pull of the locomotive ceases; a box which is moving across a table comes to rest when left to itself, because of the drag due to friction between the box and the table.
We must, therefore, always consider two distinct forces when we are concerned with a body which is kept in motion, namely, the _propelling force_ due to some active agent such as a horse or an engine, and the _dragging force_ due to surrounding bodies. Newton pointed out that when a body is moving steadily along a straight path, the propelling force is always equal and opposite to the dragging force. Therefore, _The forces which act upon a body which is stationary, or which is moving uniformly along a straight path, are balanced forces._
Many hesitate to accept as a fact the complete and exact balance of propelling and dragging forces on a body which is moving steadily along a straight path in the open, but direct experiment shows it to be true, and the most elaborate calculations and inferences based upon this idea of the complete balance of propelling and dragging forces on a body in uniform motion are verified by experiment. One may ask, why a ca.n.a.l boat, for example, should continue to move if the pull of the mule does not exceed the drag of the water; but why should it stop if the drag does not exceed the pull?
Understand that we are not considering the starting of the boat. The fact is that the conscious effort which one must exert to drive a mule, the cost of the mule, and the expense of his keep, are what most people think of, however hard one tries to direct their attention solely to the state of tension in the rope that hitches the mule to the boat after the boat is in full motion; and most people consider that if the function of the mule is simply to balance the drag of the water so as to keep the boat from stopping, then why should there not be some way to avoid the cost of so insignificant an operation? There is, indeed, an extremely important matter involved here, but it has no bearing on the question as to the balance of propulsion and drag on a body which moves steadily along a straight path.
Let us now consider the relation between the forces which act upon a body which is changing its speed, upon a body which is being started or stopped, for example. Everyone has noticed how a mule strains at his rope when starting a ca.n.a.l boat, especially if the boat is heavily loaded, and how the boat continues to move for a long time after the mule ceases to pull. In the first case, the pull of the mule greatly exceeds the drag of the water, and the speed of the boat increases; in the second case, the drag of the water of course exceeds the pull of the mule, for the mule is not pulling at all, and the speed of the boat decreases. When the speed of a body is changing, the forces which act on the body are unbalanced. We may conclude therefore that _the effect of an unbalanced force acting on a body is to change the velocity of the body,_ and it is evident that the longer the unbalanced force continues to act the greater the change of velocity. Thus if the mule ceases to pull on a ca.n.a.l boat for one second the velocity of the boat will be but slightly reduced by the unbalanced drag of the water, whereas if the mule ceases to pull for two seconds the decrease of velocity will be much greater. _In fact the change of velocity due to a given unbalanced force is proportional to the time that the force continues to act._ This is exemplified by a body falling under the action of the unbalanced pull of the earth; after one second it will have gained a certain amount of velocity (about 32 feet per second), after two seconds it will have made a total gain of twice as much velocity (about 64 feet per second), and so on.
Since the velocity produced by an unbalanced force is proportional to the time that the force continues to act, it is evident that the effect of the force should be specified as so-much-velocity-produced-per-second, exactly as in the case of earning money, the amount one earns is proportional to the length of time that one continues to work, and we always specify one"s earning capacity as so-much-money-earned-per-day.
Everyone knows what it means to give an easy pull or a hard pull on a body. That is to say, we all have the ideas of greater and less as applied to forces. Everybody knows also that if a mule pulls hard on a ca.n.a.l boat, the boat will get under way more quickly than if the pull is easy, that is, the boat will gain more velocity per unit of time under the action of a hard pull than under the action of an easy pull. Therefore, any precise statement of the effect of an unbalanced force on a given body must correlate the precise value of the force and the exact amount of velocity produced per unit of time by the force. This seems a very difficult thing, but its apparent difficulty is very largely due to the fact that we have not as yet agreed as to what we are to understand by the statement that one force is precisely three, or four, or any number of times as great as another. Suppose, therefore, that _we agree to call one force twice as large as another when it will_ as _produce in a given body twice as much velocity in a given time_ (remembering of course that we are now talking about unbalanced forces, or that we are a.s.suming for the sake of simplicity of statement, that no dragging forces exist). As a result of this definition we may state that _the amount of velocity produced per second in a given body by an unbalanced force is proportional to the force._
Of course we know no more about the matter in hand than we did before we adopted the definition, but we do have a good ill.u.s.tration of how important a part is played in the study of physical science, by what we may call making-up one"s mind, in the sense of putting one"s mind in order. This kind of thing is very prominent in the study of elementary physics, and the rather indefinite reference (in the story of the little ta.s.seled tadpole) to an inward growth as needful before one can hope for any measure of success in our modern world of scientific industry was an allusion to this thing, the "making-up" of one"s mind. Nothing is so essential in the acquirement of exact and solid knowledge as the possession of precise ideas, not indeed that a perfect precision is necessary as a means for retaining knowledge, _but that nothing else so effectually opens the mind for the perception even of the simplest evidences of a subject_[G].
We have now settled the question as to the effect of different unbalanced forces on a given body on the basis of very general experience, and by an agreement as to the precise meaning to be attached to the statement that one force is so many times as great as another; but how about the effect of the same force upon different bodies, and how may we identify the force so as to be sure that it is the same? It is required, for example, to exert a given force on body _A_ and then exert the same force on another body _B._ This can be done by causing a third body _C_ (a coiled spring, for example) to exert the force; then the forces exerted on _A_ and _B_ are the same if the reaction in each case produces the same effect on body _C_ (the same degree of stretch, for example). Concerning the effects of the same unbalanced force on different bodies three things have to be settled by experiment as follows:
(a) In the first place let us suppose that a certain force _F_ is twice as large as a certain other force _G,_ according to our agreement, because the force _F_ produces twice as much velocity every second as force _G_ when the one and then the other of these forces is caused to act upon a given body, a piece of lead for example. Then, does the force _F_ produce twice as much velocity every second as the force _G_ whatever the nature and size of the given body, whether it be wood, or ice, or sugar? Experiment shows that it does.
(b) In the second place, suppose that we have such amounts of lead, or iron, or wood, etc., that a certain given force produces the same amount of velocity per second when it is made to act, as an unbalanced force, upon one or another of these various bodies. Then what is the relation between the amounts of these various substances? Experiment shows that they all have the same ma.s.s in grams, or pounds, as determined by a balance. That is, a given force produces the same amount of velocity per second in a given number of grams of any kind of substance. Thus the earth pulls with a certain definite force (in a given locality) upon _M_ grams of any substance and, aside from the dragging forces due to air friction, all kinds of bodies gain the same amount of velocity per second when they fall under action of the unbalanced pull of the earth.
(c) In the third place, what is the relation between the velocity per second produced by a given force and the ma.s.s in grams (or pounds) of the body upon which it acts. Experiment shows that _the velocity per second produced by a given force is inversely proportional to the ma.s.s of the body upon which the force acts._ In speaking of the ma.s.s of the body in grams (or pounds) we here refer to the result which is obtained by weighing the body on a balance scale, and the experimental fact which is here referred to const.i.tutes a very important discovery: namely, when one body has twice the ma.s.s of another, according to the balance method of measuring ma.s.s, it is accelerated half as fast by a given unbalanced force.
The effect of an unbalanced force in producing velocity may therefore be summed up as follows: _The velocity per second produced by an unbalanced force is proportional to the force and inversely proportional to the ma.s.s of the body upon which the force acts, and the velocity produced by an unbalanced force is always in the direction of the force._
"We advise all men," says Bacon, "to think of the true ends of knowledge, and that they endeavor not after it for curiosity, contention, or the sake of despising others, nor yet for reputation or power or any other such inferior consideration, but solely for the occasions and uses of life." It is difficult to imagine any other basis upon which the study of physics can be justified than for the occasions and uses of life; in a certain broad sense, indeed, there is no other justification. But the great majority of men must needs be practical in the narrow sense, and physics, as the great majority of men study it, relates chiefly to the conditions which have been elaborated through the devices of industry as exemplified in our mills and factories, in our machinery of transportation, in optical and musical instruments, in the means for the supply of power, heat, light, and water for general and domestic use, and so on.
From this narrow practical point of view it may seem that there can be nothing very exacting in the study of the physical sciences; but what is physics? That is the question. One definition at least is to be repudiated; it is not "The science of ma.s.ses, molecules and the ether." Bodies have ma.s.s and railways have length, and to speak of physics as the _science of ma.s.ses_ is as silly as to define railroading as the _practice of lengths,_ and nothing as reasonable as this can be said in favor of the conception of physics as the science of molecules and the ether; it is the sickliest possible notion of physics, whereas the healthiest notion, even if a student does not wholly grasp it, is that physics is the science of the ways of taking hold of things and pushing them!
Bacon long ago listed in his quaint way the things which seemed to him most needful for the advancement of learning. Among other things he mentioned "A New Engine or a Help to the mind corresponding to Tools for the hand," and the most remarkable aspect of present-day physical science is that aspect in which it const.i.tutes a realization of this New Engine of Bacon{6}. We continually force upon the extremely meager data obtained directly through our senses, an interpretation which, in its complexity and penetration, would seem to be entirely incommensurate with the data themselves, and we exercise over physical things a kind of rational control which greatly transcends the native cunning of the hand. The possibility of this forced interpretation and of this rational control depends upon the use of two complexes: (a) A _logical structure,_ that is to say, a body of mathematical and conceptual theory which is brought to bear upon the immediate materials of sense, and (b) a _mechanical structure,_ that is to say, either (1) a carefully planned _arrangement of apparatus,_ such as is always necessary in making physical measurements, or (2) a carefully planned _order of operations,_ such as the successive operations of solution, reaction, precipitation, filtration, and weighing in chemistry.
These two complexes do indeed const.i.tute a New Engine which helps the mind as tools help the hand; it is through the enrichment of the materials of sense by the operation of this New Engine that the elaborate interpretations of the physical sciences are made possible, and the study of elementary physics is intended to lead to the realization of this New Engine: (a) By the building up in the mind, of the logical structure of the physical sciences; (b) by training in the making of measurements and in the performance of ordered operations, and (c) by exercises in the application of these things to the actual phenomena of physics and chemistry at every step and all of the time with every possible variation.
That, surely, is a sufficiently exacting program; and the only alternative is to place the student under the instruction of Jules Verne where he need not trouble himself about foundations but may follow his teacher pleasantly on a care-free trip to the moon or with easy improvidence embark on a voyage of twenty-thousand leagues under the sea.
What it means to study physical science may be explained further by mentioning the chief difficulties encountered in the teaching of that subject. One difficulty is that the native sense of most men is woefully inadequate without stimulation and direction for supplying the sense material upon which the logical structure of the science is intended to operate. A second difficulty is that the human mind is so in the habit of considering the practical affairs of life that it can hardly be turned to that minute consideration of apparently insignificant details which is so necessary in the scientific a.n.a.lysis even of the most practical things. Everyone knows the capacity of the Indian for long continued and serious effort in his primitive mode of life, and yet it is difficult to persuade an Indian "farmer" to plow.
Everyone knows also that the typical college student is not stupid, and yet it is difficult to persuade the young men of practical and business ideals in our colleges and technical schools to study the abstract elements of science. Indeed it is as difficult to get the average young man to hold abstract things in mind as to get a young Indian to plow, and for almost exactly the same reason. The scientific details of any problem are in themselves devoid of human value, and this quality of detachment is the most serious obstacle to young people in their study of the sciences.
A third difficulty which indeed runs through the entire front-of-progress of the human understanding is that the primitive mind-stuff of a young man must be rehabilitated in entirely new relations in fitting the young man for the conditions of modern life.
Every science teacher knows how much coercion is required for so little of this rehabilitation; but the bare possibility of the process is a remarkable fact, and that it is possible to the extent of bringing a Newton or a Pasteur out of a hunting and fishing ancestry is indeed wonderful. Everyone is familiar with the life history of a b.u.t.terfly, how it lives first as a caterpillar and then undergoes a complete transformation into a winged insect. It is, of course, evident that the bodily organs of a caterpillar are not at all suited to the needs of a b.u.t.terfly, the very food (of those species which take food) being entirely different. As a matter of fact almost every portion of the bodily structure of the caterpillar is dissolved as it were, into a formless pulp at the beginning of the transformation, and the organization of a flying insect then grows out from a central nucleus very much as a chicken grows in the food-stuff of an egg. So it is in the development of a young man. In early childhood the individual, if he has been favored by fortune, exercises and develops more or less extensively the primitive instincts and modes of the race in a free outdoor life, and the result is so much mind-stuff to be dissolved and transformed with more or less coercion and under more or less constraint into an effective mind of the twentieth-century type.
A fourth difficulty is that the possibility of the rehabilitation of mind-stuff has grown up as a human faculty almost solely on the basis of language, and the essence of this rehabilitation lies in the formation of ideas; whereas _a very large part of physical science is a correlation in mechanisms._
The best way of meeting this quadruply difficult situation in the teaching of elementary physics is to relate the teaching as much as possible to the immediately practical and intimate things of life, and to go in for suggestiveness as the only way to avoid a total inhibition of the sense that is born with a young man. Such a method is certainly calculated to limber up our theories and put them all at work, the pragmatic method, our friends the philosophers call it, a method which pretends to a conquering destiny.
THE DISCIPLINE OF WORK.
The first object of all work--not the princ.i.p.al one, but the first and necessary one--is to get food, clothes, lodging, and fuel.
But it is quite possible to have too much of all these things. I know a great many gentlemen, who eat too large dinners; a great many ladies, who have too many clothes. I know there is lodging to spare in London, for I have several houses there myself, which I can"t let.
And I know there is fuel to spare everywhere, since we get up steam to pound the roads with, while our men stand idle; or drink till they can"t stand, idle, or otherwise.
RUSKIN. {5}
Two generations ago school was supplemented by endless opportunity for play, and children had to work about the house and farm more and more as they grew to maturity. Play and work were in those days as plentiful as sunshine and air, and it is no wonder that educational ideals were developed taking no account of them. But we cling to these old ideals at the present time when children have no opportunity to play, when there is an almost complete absence of old fashioned ch.o.r.es about the home, when boys never see their fathers at work, and when the only opportunity for boys and girls to work outside the home is to face the certainty of reckless exploitation! What a piece of stupidity! Our entire educational system, primary and secondary, collegiate and technical, is sick with inconsequential bookishness, and school work has become the most inefficient of all the organized efforts of men.
Yes but we have our Manual Training Schools and our college courses in Shop Work and Shop Inspection. Away with such scholastic shams! The beginnings of manual training must indeed be provided for in school; paper cutting, sewing and whittling. But from the absurdity of an Academic Epitome of Industry may the good Lord deliver us! And he will deliver us, never fear, for the law of economy is His law too.
_The greatest educational problem of our time is how to make use of commercial and industrial establishments as schools to the extent that they are schools._
The first object of all work is indeed to get food and clothes and lodging and fuel, but the essence of work is a human discipline as kindly and beneficent as the sunshine and the rain, and the greatest need of our time is that the discipline of work come again to its own in our entire system of education.