§ 3 Mechanical Energy
Mechanical energy is the simplest and the oldest known.
Mechanical Elements: Time, Space, Force, Work, Power.—Mechanical phenomena may be considered under two fundamental conditions—time and space, which are, in a measure, logical elements, to which may be joined a third element, itself experimental, having its foundations in our sensations—namely, force, work, or power.
The ideas of force, work, and power, are drawn from the experience man has of his muscular activity. Nevertheless the greatest mathematical minds from from Descartes to Leibniz have been obliged to define and explain them clearly.
Force.—The prototype of force is weight, universal attraction. Experiment shows us that every body falls as long as no obstacle opposes its fall. This is so universal a property of matter that it serves to define it. The force, weight, is therefore the name given to the cause of the fall of the bodies.
Force in general is the cause of motion. Hence force exists only in so far as there is motion. There would be no force without action. This is Newton’s point of view. It did not prevail, and was not the point of view of his successors. The name of force has been given not only to the cause which produces or modifies motion, but to the cause which resists and prevents it. And then not only have forces in action been considered (dynamics), but forces at rest (statics). Now, to Newton there was no statics. Forces do not continue to exist when they produce no motion; they are not in equilibrium, they are destroyed.
The idea of force therefore is a metaphysical idea which contains the idea of cause. But it becomes experimental immediately it is looked upon as resisting motion, according to the point of view of Newton’s opponents. Its foundations lie in the muscular activity of man.
Man can support a burden without bending or moving. This burden is a weight—that is to say, a mass acted on by the force of weight. Man resists this force so as to prevent its effect. If it were not annihilated by man’s effort, this effect would be the motion or the fall of the heavy body. The effort and the force are thus in equilibrium, and the effort is equal and opposite to the force. It gives to the man who exercises it the conscious idea of force. Thus we know of force through effort. Every clear idea that we can have of force springs from the observation of our muscular effort.
The notion of force is thus an anthropomorphic notion. When an effect is produced in nature outside human intervention, we say that it is by something analogous to what in man is effort, and we give to this something a name which is also analogous, namely force. To give a name to effort and to compare efforts in magnitude, we need not know all about them, nor need we know in what they essentially consist, of what series of physical, chemical, and physiological actions they are the consequence. And so it is with force. It is a resistance to motion or the cause of motion. This cause of motion may be an anterior motion (kinetic force). It may be an anterior physical energy (physical and chemical forces).
Forces are measured in the C.G.S. system by comparing them with the unit called the Dyne.7 In practice they are compared with a much larger unit—the gramme, which is the weight, the force acting on a unit of mass of one centimetre of distilled water at a temperature of 4° C.
Work.—The muscular activity of man may be brought into play in yet another manner. When we employ workmen, as Carnot said in his Essai sur l’équilibre et le mouvement, it is not a question of “knowing the burdens that they can carry without moving from their position,” but rather the burdens that they can carry from one point to another. For instance, a workman may have to lift the water from the bottom of a well to a given height, and the case is the same for the animals we employ. “This is what we understand by force when we say that the force of a horse is equal to the force of seven men. We do not mean that if seven men were pulling in one direction and the horse in another that there would be equilibrium, but that in a piece of work the horse alone would lift, for example, as much water from the bottom of a well to a given height as the seven men together would do in the same time.”8
Here, then, we have to do with the second form of muscular activity, which is called in mechanics, “work”—at least, if in the preceding quotation we attach no particular importance to the words “in the same time,” and retain the employment of muscular activity only “under constant conditions.” Mechanical work is compared with the elevation of a certain weight to a certain height. It is measured by the product of the force (understood in the sense in which it was used just now—that is to say, as causing or resisting motion) and the displacement due to this motion. The unit is the Kilogrammetre—that is to say, the work necessary to lift a weight of one kilogramme to the height of one metre.
It will be remarked that the idea of time does not intervene in our estimation of work. The notion of work is independent of the ideas of velocity and time. “The greater or less time that we take to do a piece of work is of no more assistance in measuring its magnitude than the number of years that a man may have taken to grow rich or to ruin himself can help to estimate the present amount of his fortune.”
Going back to Carnot’s comparison, an employer who employed his workmen only on piece-work,—that is to say, who would only care about the amount of work done, and would be indifferent to the time that they took over it,—would be at the same point of view as the advocates of the mechanical theory. M. Bouasse, whom we follow here, has remarked that this idea of mechanical work may be traced back to Descartes. His predecessors, and Galileo in particular, had quite a different idea of the way in which mechanical activity should be measured; and so, among the mathematicians of the eighteenth century, Leibniz and, later, John Bernoulli were almost alone in looking at it from this point of view.
Energy.—Work thus understood is mechanical energy. It represents the lasting and objective effect of the mechanical activity independent of all the circumstances under which it was carried out. The same work may be done under very different conditions of time, velocity, force, and displacement. It is therefore the permanent element in the variety of mechanical aspects. Work, for example, in the collision of bodies when the motion of a body appears to be destroyed on impact with another, reappears as indestructible vis viva. This, then, is exactly what we call energy; and if we agree to give it this name, we may say that the conservation of energy is invariable throughout all mechanical transformations.
Distinction between Work and Force, and between Energy and Work.—The history of mechanics shows us what trouble has been taken and what efforts have been made to distinguish work (now mechanical energy) from force.
It is worth while insisting on this distinction. It could be easily shown that force has no objective existence. It has no duration, no permanence. It does not survive its effect, motion. There is no conservation of force. It passes instantly from infinity to zero. It is a vectorial magnitude—that is to say, it involves the idea of direction. Work, on the other hand, is the real element; it is a scalar magnitude involving the idea of opposite directions, indicated by the signs + and-. Work and force are heterogeneous magnitudes. Energy, and this is the only characteristic by which it is distinguished from work, is an absolute magnitude to which we may not even give opposite signs.
An example may perhaps throw these characteristics into relief—namely, the hydraulic press. We have on the platform exactly the work which has been done on the other side. The machine has only made it change its form. On the contrary, the force has been infinitely multiplied. We may, in fact, consider an infinite number of surfaces equal to that of a small piston, placed and orientated at will within the liquid; each, according to Pascal’s principle, will support a pressure equal to that which is exercised. As soon as we cease to support it, this infinity falls at once to zero. Now what real thing could pass instantly from infinity to zero?
That skilful and very able physiologist, M. Chauveau, has endeavoured to use the same term energy of contraction for the two phenomena of effort (force) and work. It seems, however, from the point of view of the expenditure imposed on the organism, that these two modes of activity, static contraction (effort), and dynamical contraction (work), may be, in fact, perfectly comparable. But although this manner of conceiving the phenomena may certainly be exact, and may be of great value, the idea of force must none the less remain distinct from that of work. The persistence of the author in violating established custom in this connection has prevented him from enabling mechanicians and even some physiologists to understand and accept very useful truths.
Power.—The idea of mechanical power differs from those of force and work. The idea of time must intervene. It is not sufficient, in fact, in order to characterize a mechanical operation, to point to the task accomplished. It may be necessary or useful to know how much time it required. This is true, especially when we are concerned with the circumstances as well as the results of the performance of the work; and this is the case when we wish to compare machines. We say that the machine which does the work in the shortest space of time is the most powerful. The unit of power is the Kilogrammetre-second—that is to say, the power of a machine which does a kilogrammetre in a second. In manufactures we generally employ a unit 75 times greater than this—a horse-power. This is the power of a machine which does 75 kilogrammetres a second. In the electrical industry we measure by kilowatts, which are equivalent to 1.36 horse power, or by a watt, a unit a thousand times smaller.
Let us add that the power of a machine is not an absolute and permanent characteristic of the machine. It depends on the circumstances under which the work is carried out, and that is why, in particular, we cannot appreciate the power of the human machine in comparison with industrial machines. Experience has shown that the mechanical power of living beings depends upon the nature of the work they are doing. In this connection we may mention some very interesting experiments communicated to the Institute, in the year VI., by the celebrated physicist, Coulomb. A man of the average weight of 70 kilogrammes was made to climb the stairs of a house 20 metres high. He ascended at the rate of 14 metres a minute, and he performed this daily task for four effective hours. This work was equivalent to 235,000 kilogrammetres. But if, instead of climbing without a burden, the same man had had to carry a load, the result would have been quite different. Coulomb’s workman took up six loads of wood a day to a height of 12 metres in 66 journeys, corresponding to a maximum work of 109,000 instead of 235,000 kilogrammetres. The mechanical power of the human machine thus varied in the two cases in the ratio of 235 to 109.
The Two Aspects of Mechanical Energy: Kinetic and Potential.—Energy, or mechanical work, may present itself in two forms—kinetic energy, corresponding to the mechanical phenomenon which has really taken place, and potential energy, or the energy of reserve.
A body which has been raised to a certain height will, if it be let fall, perform work which can be exactly measured in kilogrammetres by the product of its weight into the height it falls. Such work may be utilized in many ways. In this way, for instance, public clocks are worked. Now, as long as the clock-weight is raised and not let go, and as long as it is motionless, the physics of early days would say that there is nothing to discuss; the phenomenon is the fall; it is going to take place, but at the present moment there has been no fall.
In energetics we do not reason in this way. We say that the body possesses a capacity for work which will be manifested when the opportunity arises, a storage of energy, a virtual or potential energy, or again, an energy of position, which will be transformed into actual energy or real work as soon as the body falls.
Let us ask whence this energy arises. It proceeds from the previous operation which has raised the weight from the surface of the soil to the position it occupies. For example, if it is a question of the weights of a public clock, which, by its fall, will develop in 15 days the work that is necessary to turn the wheels, to strike the bell, and to turn the hands, this work ought to bring to our minds the exactly equal and opposite work done by the clockmaker, who has to carry the clock-weight and to lift it up from the ground to its point of departure. The work of the fall is the faithful counterpart of the work of elevation. The phenomenon has therefore in reality two phases. We find in the second exactly what was put into the first, the same quantity of energy—i.e., the same work. Between these phases comes the intermediary phase of which we say that it is a period of virtual or potential energy. This is a way of remembering in some measure the preceding phenomenon—i.e., the work of lifting up, and of indicating the phenomena which will follow—i.e., the work of the fall. And thus we connect by our thoughts the present situation with the antecedent and with the consequent position, and it is from this consideration of continuity alone that the conception of energy springs—that is to say, of something which is conserved and is found to be permanent in the succession of phenomena. This energy of which we lose no trace does not appear to us new when it is manifested. Our imagination eventually materializes the idea of it. We follow it as a real thing, having an objective existence, which is asleep during the latent potential period, and is revealed or manifested later.
Among other examples, that of the coiled spring which is unwound is particularly suitable for showing this fundamental character of the idea of mechanical energy, an idea which is the clearest of all. Machines are only transformers and not creators of mechanical energy. They only change one form into another.
In the same way, too, a stream of water or the torrent of a mountainous region may be utilized for setting in motion the wheels and the turbines of the factories situated in the valley. Its descent produces the mechanical work which would be a creation ex nihilo if we do not connect the phenomenon with its antecedents. We look on it as a simple restitution, if we think of the origin of this water which has been transported and lifted in some way to its level by the play of natural forces—evaporation under the action of the sun, the formation of clouds, transport by winds, etc. And we here again see that a complex energy has been transformed, in its first phenomenal condition, into potential energy, and that this potential energy is always expended in the second phase without loss or gain.
The Different Kinds of Mechanical Energy; of Motion, of Position.—There are as many forms of energy as there are distinct categories of phenomena or of varieties in these categories. Physicists distinguish between two kinds of mechanical energy—energy of motion and energy of position. The energy of position presents several variants—energy of distance, which corresponds to force: of this we have just spoken; energy of surface, which corresponds to particular phenomena of surface tension; and energy of volume which corresponds to the phenomena of pressure. Energy of motion, kinetic energy, is measured in two ways: as work (the product of force and displacement, W = fs) or as vis viva (half the product of the mass into the square of the velocity U = mv2∕2.9