Bill's School and Mine: A Collection of Essays on Education

by William S. Franklin

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 canal 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 canal 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 canal 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 canal 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 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 assuming 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 illustration 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 tasseled 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 [7].

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 mass 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 mass 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 mass of the body upon which the force acts. In speaking of the mass 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 constitutes a very important discovery: namely, when one body has twice the mass of another, according to the balance method of measuring mass, 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 mass of the body upon which the force acts, and the velocity produced by an unbalanced force is always in the direction of the force.