will pierce through four, or nearly 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 kilo gramme 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 are 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 velocitylet 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 с velocity. But yet it is very easy to see what is meant by an initial velocity of 98 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 98 metres in one second, will be brought to rest when it has risen 49 metres in height. If, therefore, it be a kilogramme, its upward velocity will have enabled it to raise itself 49 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 441 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 be proportional to the square of the velocity, and we may formularize our conclusion as follows: Letv the initial velocity expressed in metres per second, then the energy in kilogrammetres = 2 v2 19.6 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 m the mass of the body in kilogrammes, we shall have the energy in kilom v 2 grammetres = To test the truth of this formula, 19.6' we have only to apply it to the cases described in Arts. 26 and 27. |