The Conservation of Energy/Chapter 1
THE CONSERVATION OF ENERGY.
CHAPTER I.
WHAT IS ENERGY?
Our Ignorance of Individuals.
1. Very often we know little or nothing of individuals, while we yet possess a definite knowledge of the laws which regulate communities.
The Registrar-General, for example, will tell us that the death-rate in London varies with the temperature in such a manner that a very low temperature is invariably accompanied by a very high death-rate. But if we ask him to select some one individual, and explain to us in what manner his death was caused by the low temperature, he will, most probably, be unable to do so.
Again, we may be quite sure that after a bad harvest there will be a large importation of wheat into the country, while, at the same time, we are quite ignorant of the individual journeys of the various particles of flour that go to make up a loaf of bread.
Or yet again, we know that there is a constant carriage of air from the poles to the equator, as shown by the trade winds, and yet no man is able to individualize a particle of this air, and describe its various motions.
2. Nor is our knowledge of individuals greater in the domains of physical science. We know nothing, or next to nothing, of the ultimate structure and properties of matter, whether organic or inorganic.
No doubt there are certain cases where a large number of particles are linked together, so as to act as one individual, and then we can predict its action—as, for instance, in the solar system, where the physical astronomer is able to foretell with great exactness the positions of the various planets, or of the moon. And so, in human affairs, we find a large number of individuals acting together as one nation, and the sagacious statesman taking very much the place of the sagacious astronomer, with regard to the action and reaction of various nations upon one another.
But if we ask the astronomer or the statesman to select an individual particle and an individual human being, and predict the motions of each, we shall find that both will be completely at fault.
3. Nor have we far to look for the cause of their ignorance. A continuous and restless, nay, a very complicated, activity is the order of nature throughout all her individuals, whether these be living beings or inanimate particles of matter. Existence is, in truth, one continued fight, and a great battle is always and everywhere raging, although the field in which it is fought is often completely shrouded from our view.
4. Nevertheless, although we cannot trace the motions of individuals, we may sometimes tell the result of the fight, and even predict how the day will go, as well as specify the causes that contribute to bring about the issue.
With great freedom of action and much complication of motion in the individual, there are yet comparatively simple laws regulating the joint result attainable by the community.
But, before proceeding to these, it may not be out of place to take a very brief survey of the organic and inorganic worlds, in order that our readers, as well as ourselves, may realize our common ignorance of the ultimate structure and properties of matter.
5. Let us begin by referring to the causes which bring about disease. It is only very recently that we have begun to suspect a large number of our diseases to be caused by organic germs. Now, assuming that we are right in this, it must nevertheless be confessed that our ignorance about these germs is most complete. It is perhaps doubtful whether we ever saw one of these organisms,[1] while it is certain that we are in profound ignorance of their properties and habits.
We are told by some writers[2] that the very air we breathe is absolutely teeming with germs, and that we are surrounded on all sides by an innumerable array of minute organic beings. It has also been conjectured that they are at incessant warfare among themselves, and that we form the spoil of the stronger party. Be this as it may, we are at any rate intimately bound up with, and, so to speak, at the mercy of, a world of creatures, of which we know as little as of the inhabitants of the planet Mars.
6. Yet, even here, with profound ignorance of the individual, we are not altogether unacquainted with some of the habits of these powerful predatory communities. Thus we know that cholera is eminently a low level disease, and that during its ravages we ought to pay particular attention to the water we drink. This is a general law of cholera, which is of the more importance to us because we cannot study the habits of the individual organisms that cause the disease.
Could we but see these, and experiment upon them, we should soon acquire a much more extensive knowledge of their habits, and perhaps find out the means of extirpating the disease, and of preventing its recurrence.
Again, we know (thanks to Jenner) that vaccination will prevent the ravages of small-pox, but in this instance we are no better off than a band of captives who have found out in what manner to mutilate themselves, so as to render them uninteresting to their victorious foe.
7. But if our knowledge of the nature and habits of organized molecules be so small, our knowledge of the ultimate molecules of inorganic matter is, if possible, still smaller. It is only very recently that the leading men of science have come to consider their very existence as a settled point.
In order to realize what is meant by an inorganic molecule, let us take some sand and grind it into smaller and smaller particles, and these again into still smaller. In point of fact we shall never reach the superlative degree of smallness by this operation—yet in our imagination we may suppose the sub-division to be carried on continuously, always making the particles smaller and smaller. In this case we should, at last, come to an ultimate molecule of sand or oxide of silicon, or, in other words, we should arrive at the smallest entity retaining all the properties of sand, so that were it possible to divide the molecule further the only result would be to separate it into its chemical constituents, consisting of silicon on the one side and oxygen on the other.
We have, in truth, much reason to believe that sand, or any other substance, is incapable of infinite subdivision, and that all we can do in grinding down a solid lump of anything is to reduce it into lumps similar to the original, but only less in size, each of these small lumps containing probably a great number of individual molecules.
8. Now, a drop of water no less than a grain of sand is built up of a very great number of molecules, attached to one another by the force of cohesion—a force which is much stronger in the sand than in the water, but which nevertheless exists in both. And, moreover, Sir William Thomson, the distinguished physicist, has recently arrived at the following conclusion with regard to the size of the molecules of water. He imagines a single drop of water to be magnified until it becomes as large as the earth, having a diameter of 8000 miles, and all the molecules to be magnified in the same proportion; and he then concludes that a single molecule will appear, under these circumstances, as somewhat larger than a shot, and somewhat smaller than a cricket ball.
9. Whatever be the value of this conclusion, it enables us to realize the exceedingly small size of the individual molecules of matter, and renders it quite certain that we shall never, by means of the most powerful microscope, succeed in making visible these ultimate molecules. For our knowledge of the sizes, shapes, and properties of such bodies, we must always, therefore, be indebted to indirect evidence of a very complicated nature.
It thus appears that we know little or nothing about the shape or size of molecules, or about the forces which actuate them; and, moreover, the very largest masses of the universe share with the very smallest this property of being beyond the direct scrutiny of the human senses—the one set because they are so far away, and the other because they are so small.
10. Again, these molecules are not at rest, but, on the contrary, they display an intense and ceaseless energy in their motions. There is, indeed, an uninterrupted warfare going on—a constant clashing together of these minute bodies, which are continually maimed, and yet always recover themselves, until, perhaps, some blow is struck sufficiently powerful to dissever the two or more simple atoms that go to form a compound molecule. A new state of things thenceforward is the result.
But a simple elementary atom is truly an immortal being, and enjoys the privilege of remaining unaltered and essentially unaffected amid the most powerful blows that can be dealt against it—it is probably in a state of ceaseless activity and change of form, but it is nevertheless always the same.
11. Now, a little reflection will convince us that we have in this ceaseless activity another barrier to an intimate acquaintance with molecules and atoms, for even if we could see them they would not remain at rest sufficiently long to enable us to scrutinize them.
No doubt there are devices by means of which we can render visible, for instance, the pattern of a quickly revolving coloured disc, for we may illuminate it by a flash of electricity, and the disc may be supposed to be stationary during the extremely short time of the flash But we cannot say the same about molecules and atoms, for, could we see an atom, and could we illuminate it by a flash of electricity, the atom would most probably have vibrated many times during the exceedingly small time of the flash. In fine, the limits placed upon our senses, with respect to space and time, equally preclude the possibility of our ever becoming directly acquainted with these exceedingly minute bodies, which are nevertheless the raw materials of which the whole universe is built.
Action and Reaction, Equal and Opposite.
12. But while an impenetrable veil is drawn over the individual in this warfare of clashing atoms, yet we are not left in profound ignorance of the laws which determine the ultimate result of all these motions, taken together as a whole.
In a Vessel of Goldfish.
Let us suppose, for instance, that we have a glass globe containing numerous goldfish standing on the table, and delicately poised on wheels, so that the slightest push, the one way or the other, would make it move. These goldfish are in active and irregular motion, and he would be a very bold man who should venture to predict the movements of an individual fish. But of one thing we may be quite certain: we may rest assured that, notwithstanding all the irregular motions of its living inhabitants the globe containing the goldfish will remain at rest upon its wheels.
Even if the table were a lake of ice, and the wheels were extremely delicate, we should find that the globe would remain at rest. Indeed, we should be exceedingly surprised if we found the globe going away of its own accord from the one side of the table to the other, or from the one side of a sheet of ice to the other, in consequence of the internal motions of its inhabitants. Whatever be the motions of these individual units, yet we feel sure that the globe cannot move itself as a whole. In such a system, therefore, and, indeed, in every system left to itself, there may be strong internal forces acting between the various parts, but these actions and reactions are equal and opposite, so that while the small parts, whether visible or invisible, are in violent commotion among themselves, yet the system as a whole will remain at rest.
In a Rifle.
13. Now it is quite a legitimate step to pass from this instance of the goldfish to that of a rifle that has just been fired. In the former case, we imagined the globe, together with its fishes, to form one system; and in the latter, we must look upon the rifle, with its powder and ball, as forming one system also.
Let us suppose that the explosion takes place through the application of a spark. Although this spark is an external agent, yet if we reflect a little we shall see that its only office in this case is to summon up the internal forces already existing in the loaded rifle, and bring them into vigorous action, and that in virtue of these internal forces the explosion takes place.
The most prominent result of this explosion is the out-rush of the rifle ball with a velocity that may, perhaps, carry it for the best part of a mile before it comes to rest; and here it would seem to us, at first sight, that the law of equal action and reaction is certainly broken, for these internal forces present in the rifle have at least propelled part of the system, namely, the rifle ball, with a most enormous velocity in one direction.
14. But a little further reflection will bring to light another phenomenon besides the out-rush of the ball. It is well known to all sportsmen that when a fowling-piece is discharged, there is a kick or recoil of the piece itself against the shoulder of the sportsman, which he would rather get rid of, but which we most gladly welcome as the solution of our difficulty. In plain terms, while the ball is projected forwards, the rifle stock (if free to move) is at the same moment projected backwards. To fix our ideas, let us suppose that the rifle stock weighs 100 ounces, and the ball one ounce, and that the ball is projected forwards with the velocity of 1000 feet per second; then it is asserted, by the law of action and reaction, that the rifle stock is at the same time projected backwards with the velocity of 10 feet per second, so that the mass of the stock, multiplied by its velocity of recoil, shall precisely equal the mass of the ball, multiplied by its velocity of projection. The one product forms a measure of the action in the one direction, and the other of the reaction in the opposite direction, and thus we see that in the case of a rifle, as well as in that of the globe of fish, action and reaction are equal and opposite.
In a Falling Stone.
15. We may even extend the law to cases in which we do not perceive the recoil or reaction at all. Thus, if I drop a stone from the top of a precipice to the earth, the motion seems all to be in one direction, while at the same time it is in truth the result of a mutual attraction between the earth and the stone. Does not the earth move also? We cannot see it move, but we are entitled to assert that it does in reality move upwards to meet the stone, although quite to an imperceptible extent, and that the law of action and reaction holds here as truly as in a rifle, the only difference being that in the one case the two objects are rushing together, while in the other they are rushing apart. Inasmuch, however, as the mass of the earth is very great compared with that of the stone, it follows that its velocity must be extremely small, in order that the mass of the earth, multiplied into its velocity upwards, shall equal the mass of the stone, multiplied into its velocity downwards.
16. We have thus, in spite of our ignorance of the ultimate atoms and molecules of matter, arrived at a general law which regulates the action of internal forces. We see that these forces are always mutually exerted, and that if A attracts or repels B, B in its turn attracts or repels A. We have here, in fact, a very good instance of that kind of generalization, which we may arrive at, even in spite of our ignorance of individuals.
But having now arrived at this law of action and reaction, do we know all that it is desirable to know? have we got a complete understanding of what takes place in all such cases—for instance, in that of the rifle which is just discharged? Let us consider this point a little further.
The Rifle further considered.
17. We define quantity of motion to mean the product of the mass by the velocity; and since the velocity of recoil of the rifle stock, multiplied by the mass of the stock, is equal to the velocity of projection of the rifle ball, multiplied by the mass of the ball, we conceive ourselves entitled to say that the quantity of motion, or momentum, generated is equal in both directions, so that the law of action and reaction holds here also. Nevertheless, it cannot but occur to us that, in some sense, the motion of the rifle ball is a very different thing from that of the stock, for it is one thing to allow the stock to recoil against your shoulder and discharge the ball into the air, and a very different thing to discharge the ball against your shoulder and allow the stock to fly into the air. And if any man should assert the absolute equality between the blow of the rifle stock and that of the rifle ball, you might request him to put his assertion to this practical test, with the absolute certainty that he would decline. Equality between the two!—Impossible! Why, if this were the case, a company of soldiers engaged in war would suffer much more than the enemy against whom they fired, for the soldiers would certainly feel each recoil, while the enemy would suffer from only a small proportion of the bullets.
The Rifle Ball possesses Energy.
18. Now, what is the meaning of this great difference between the two? We have a vivid perception of a mighty difference, and it only remains for us to clothe our naked impressions in a properly fitting scientific garb.
The something which the rifle hall possesses in contradistinction to the rifle stock is clearly the power of overcoming resistance. It can penetrate through oak wood or through water, or (alas! that it should be so often tried) through the human body, and this power of penetration is the distinguishing characteristic of a substance moving with very great velocity.
19. Let us define by the term energy this power which the rifle ball possesses of overcoming obstacles or of doing work. Of course we use the word work without reference to the moral character of the thing done, and receive ourselves entitled to sum up, with perfect propriety and innocence, the amount of work done in drilling a hole through a deal board or through a man.
20. A body such as a rifle ball, moving with very great velocity, has, therefore, energy, and it requires very little consideration to perceive that this energy will be proportional to its weight or mass, for a ball of two ounces moving with the velocity of 1000 feet per second will be the same as two balls of one ounce moving with this velocity, but the energy of two similarly moving ounce balls will manifestly be double that of one, so that the energy is proportional to the weight, if we imagine that, meanwhile, the velocity remains the same.
21. But, on the other hand, the energy is not simply proportional to the velocity, for, if it were, the energy of the rifle stock and of the rifle ball would be the same, inasmuch as the rifle stock would gain as much by its superior mass as it would lose by its inferior velocity. Therefore, the energy of a moving body increases with the velocity more quickly than a simple proportion, so that if the velocity be doubled, the energy is more than doubled. Now, in what manner does the energy increase with the velocity? That is the question we have now to answer, and, in doing so, we must appeal to the familiar facts of everyday observation and experience.
22. In the first place, it is well known to artillerymen, that if a ball have a double velocity, its penetrating power or energy is increased nearly fourfold, so that it will pierce through four, or nea,rly four, times as many deal boards as the ball with only a single velocity—in other words, they will tell us, in mathematical language, that the energy varies as the square of the velocity.
Definition of Work.
23. And now, before proceeding further, it will be necessary to tell our readers how to measure work in a strictly scientific manner. We have defined energy to be the power of doing work, and although every one has a general notion of what is meant by work, that notion may not be sufficiently precise for the purpose of this volume. How, then, are we to measure work? Fortunately, we have not far to go for a practical means of doing this. Indeed, there is a force at hand which enables us to accomplish this measurement with the greatest precision, and this force is gravity. Now, the first operation in any kind of numerical estimate is to fix upon our unit or standard. Thus we say a rod is so many inches long, or a road so many miles long. Here an inch and a mile are chosen as our standards. In like manner, we speak of so many seconds, or minutes, or hours, or days, or years, choosing that standard of time or duration which is most convenient for our purpose. So in like manner we must choose our unit of work, but in order to do so we must first of all choose our units of weight and of length, and for these we will take the kilogramme and the metre, these being the units of the metrical system. The kilogramme corresponds to about 15,432.35 English grains, being rather more than two pounds avoirdupois, and the metre to about 39.371 English inches.
Now, if we raise a kilogramme weight one metre in vertical height, we are conscious of putting forth an effort to do so, and of being resisted in the act by the force of gravity. In other words, we spend energy and do work in the process of raising this weight.
Let us agree to consider the energy spent, or the work done, in this operation as one unit of work, and let us call it the kilogrammetre.
24. In the next place, it is very obvious that if we raise the kilogramme two metres in height, we do two units of work—if three metres, three units, and so on. And again, it is equally obvious that if we raise a weight of two kilogrammes one metre high, we likewise do two units of work, while if we raise it two metres high, we do four units, and so on.
From these examples we art entitled to derive the following rule:—Multiply the weight raised (in kilogrammes) by the vertical height (in metres) through which it is raised, and the result will be the work done (in kilogrammetres).
Relation between Velocity and Energy.
25. Having thus laid a numerical foundation for our superstructure, let us next proceed to investigate the relation between velocity and energy. But first let us say a few words about velocity. This is one of the few cases in which everyday experience will aid, rather than hinder, us in our scientific conception. Indeed, we have constantly before us the example of bodies moving with variable velocities.
Thus a railway train is approaching a station and is just beginning to slacken its pace. When we begin to observe, it is moving at the rate of forty miles an hour. A minute afterwards it is moving at the rate of twenty miles only, and a minute after that it is at rest. For no two consecutive moments has this train continued to move at the same rate, and yet we may say, with perfect propriety, that at such a moment the train was moving, say, at the rate of thirty miles an hour. We mean, of course, that had it continued to move for an hour with the speed which it had when we made the observation, it would have gone over thirty miles. We know that, as a matter of fact, it did not move for two seconds at that rate, but this is of no consequence, and hardly at all interferes with our mental grasp of the problem, so accustomed are we all to cases of variable velocity.
26. Let us now imagine a kilogramme weight to be shot vertically upwards, with a certain initial velocity—let us say, with the velocity of 9.8 metres in one second. Gravity will, of course, act against the weight, and continually diminish its upward speed, just as in the railway train the break was constantly reducing the velocity. But yet it is very easy to see what is meant by an initial velocity of 9.8 metres per second; it means that if gravity did not interfere, and if the air did not resist, and, in fine, if no external influence of any kind were allowed to act upon the ascending mass, it would be found to move over 9.8 metres in one second.
Now, it is well known to those who have studied the laws of motion, that a body, shot upwards with the velocity of 9.8 metres in one second, will be brought to rest when it has risen 4.9 metres in height. If, therefore, it be a kilogramme, its upward velocity will have enabled it to raise itself 4.9 metres in height against the force of gravity, or, in other words, it will have done 4.9 units of work; and we may imagine it, when at the top of its ascent, and just about to turn, caught in the hand and lodged on the top of a house, instead of being allowed to fall again to the ground. We are, therefore, entitled to say that a kilogramme, shot upwards with the velocity of 9.8 metres per second, has energy equal to 4.9, inasmuch as it can raise itself 4.9 metres in height.
27. Let us next suppose that the velocity with which the kilogramme is shot upwards is that of 19.6 metres per second. It is known to all who have studied dynamics that the kilogramme will now mount not only twice, but four times as high as it did in the last instance—in other words, it will now mount 19.6 metres in height.
Evidently, then, in accordance with our principles of measurement, the kilogramme has now four times as much energy as it had in the last instance, because it can raise itself four times as high, and therefore do four times as much work, and thus we see that the energy is increased four times by doubling the velocity.
Had the initial velocity been three times that of the first instance, or 29.4 metres per second, it might in like manner be shown that the height attained would have been 44.1 metres, so that by tripling the velocity the energy is increased nine times.
28. We thus see that whether we measure the energy of a moving body by the thickness of the planks through which it can pierce its way, or by the height to which it can raise itself against gravity, the result arrived at is the same. We find the energy to he proportional to the square of the velocity, and we may formularize our conclusion as follows:—
Let the initial velocity expressed in metres per second, then the energy in kilogrammetres . Of course, if the body shot upwards weighs two kilogrammes, then everything is doubled, if three kilogrammes, tripled, and so on; so that finally, if we denote by the mass of the body in kilogrammes, we shall have the energy in kilogrammetres . To test the truth of this formula, we have only to apply it to the cases described in Arts 26 and 27.
29. We may further illustrate it by one or two examples. For instance, let it be required to find the energy contained in a mass of five kilogrammes, shot upwards with the velocity of 20 metres per second.
Here we have and , hence—
nearly.
Again, let it be required to find the height to which the mass of the last question will ascend before it stops. We know that its energy is 102.04, and that its mass is 5. Dividing 102.04 by 5, we obtain 20.408 as the height to which this mass of five kilogrammes must ascend in order to do work equal to 102.04 kilogrammetres.
30. In what we have said we have taken no account either of the resistance or of the buoyancy of the atmosphere; in fact, we have supposed the experiments to be made in vacuo, or, if not in vacuo, made by means of a heavy mass, like lead, which will be very little influenced either by the resistance or buoyancy of the air.
We must not, however, forget that if a sheet of paper, or a feather, be shot upwards with the velocities mentioned in our text, they will certainly not rise in the air to nearly the height recorded, but will be much sooner brought to a stop by the very great resistance which they encounter from the air, on account of their great surface, combined with their small mass.
On the other hand, if the substance we make use of be a large light bag filled with hydrogen, it will find its way upwards without any effort on our part, and we shall certainly be doing no work by carrying it one or more metres in height—it will, in reality, help to pull us up, instead of requiring help from us to cause it to ascend. In fine, what we have said is meant to refer to the force of gravity alone, without taking into account a resisting medium such as the atmosphere, the existence of which need not be considered in our present calculations.
31. It should likewise be remembered, that while the energy of a moving body depends upon its velocity, it is independent of the direction in which the body is moving. We have supposed the body to be shot upwards with a given velocity, but it might be shot horizontally with the same velocity, when it would have precisely the same energy as before. A cannon ball, if fired vertically upwards, may either be made to spend its energy in raising itself, or in piercing through a series of deal boards. Now, if the same ball be fired horizontally with the same velocity it will pierce through the same number of deal boards.
In fine, direction of motion is of no consequence, and the only reason why we have chosen vertical motion is that, in this case, there is always the force of gravity steadily and constantly opposing the motion of the body, and enabling us to obtain an accurate measure of the work which it does by piercing its way upwards against this force.
32. But gravity is not the only force, and we might measure the energy of a moving body by the extent to which it would bend a powerful spring or resist the attraction of a powerful magnet, or, in fine, we might make use of the force which best suits our purpose. If this force be a constant one, we must measure the energy of the moving body by the space which it is able to traverse against the action of the force—just as, in the case of gravity, we measured the energy of the body by the space through which it was able to raise itself against its own weight.
33. We must, of course, bear in mind that if this force be more powerful than gravity, a body moved a short distance against it will represent the expenditure of as much energy as if it were moved a greater distance against gravity. In fine, we must take account both of the strength of the force and of the distance moved over by the body against it before we can estimate in an accurate matter the work which has been done.