CHAPTER IX Perpetual Motion Devices Attempting Its Attainment by a Misconception of the Relation of Momentum and Energy

The author, within twenty years last past, has had his attention called by two different persons, each ignorant of the efforts of the other, who were seeking to obtain Perpetual Motion by utilizing certain physical facts concerning Momentum and Energy. These facts and the principles out of which they grow are familiar to all who understand thoroughly, even the rudiments of physics; but to persons who are inclined to mechanics, but who have never had the advantages of the presentation of clear principles, they are confusing, and it is surprising that they have not become more fertile fields for Perpetual Motion workers. However, we are unable to find any written or printed account or description of a plan or device of that kind, and our information is confined to instances that have been brought to our personal observation, and concerning which the advice and counsel of the author was sought.

The worker in each case was a man of more than ordinary natural intelligence, and with a bent for mechanical pursuits and reflection. Each had taken a course in what is conventionally called High School Physics.

The idea in each case was so novel and interesting that we deem the presentation worth while. They were so nearly alike that instead of attempting to narrate what they said, we will endeavor in our own way to present the idea, and then to give our explanation, showing wherein lay their error.

The following definitions and laws of physics may be regarded as established:

Momentum

Momentum is the quantity of motion of a moving body, and is the velocity multiplied by the weight.

Thus, a body weighing two pounds, moving at four feet per second, may be represented as having a momentum of eight.

A body weighing two pounds moving at the rate of six feet per second may be said to have a momentum of twelve.

A body weighing ten pounds moving at the rate of ten feet per second will have a momentum of one hundred—and so on.

Now, a step further. A body in motion striking another body free to move will lose part of its motion, and will impart some of its motion to the body moved against. The aggregate momentum after the striking is the same as before—that is to say—if a body weighing ten pounds have a velocity of twenty feet per second, its momentum we will call two hundred. Now, if in moving it strike another body either larger or smaller its motion will be somewhat retarded, and the body struck will possess some motion.

Multiply the weight of each by its motion after the striking, and it will be found that the sum of the products is two hundred. This may be illustrated by swinging balls like pendulums to cords of equal length from a beam, having the arrangement such that balls of different materials and sizes can be substituted at liberty. If a body be drawn back parallel to the beam, and released so as to swing against another swinging body, both will have motion. This motion will, in some cases be a rebounding motion, as in the case of a small elastic body swinging against and striking a larger elastic body, but in all cases the sum total of the momentum after the impingement is the same as before.

The following statement of the law then, is deducible:

The Momentum of one body in motion may be made to impart momentum to another body, the amount of momentum lost by the former being exactly equal to that thus acquired by the latter.

Before leaving these remarks on momentum the reader should observe carefully what momentum is and bear in mind it is the quantity of motion possessed by a moving body, and has to do only with mass and velocity—and takes no account of distance passed through.

Energy

Energy is the capacity to do work, and the energy of a moving body is the amount of work it will do, i.e., the distance it will move against a resistance by virtue of its tendency to move, before being brought to a state of rest.

Now note, and note carefully, that the amount of energy is proportional to the mass, and to the square of the velocity.

Note this carefully: Any body in motion has both momentum and energy. Its momentum is proportional to its velocity; its energy to the square of its velocity. If the velocity be doubled, the momentum will be doubled, but its energy quadrupled. If the velocity be trebled, its momentum will be trebled, but its energy increased nine-fold.

It is important that the student get clearly what is meant by saying that Energy is the capacity to do work, and is proportional to the square of the velocity.

The capacity to do work means the capacity to move against resistance, i.e., to overcome resistance. The word "work" being used in a purely mechanical sense and in that sense it is used whether the result accomplished is destructive or beneficial.

A revolving fly wheel will run machinery for some time after the application of force has ceased. This is doing work, and represents energy.

A bullet fired from a gun will accomplish destruction before having its motion arrested. This is work—energy.

If a boy throw a ball into a snow bank, its motion will sink it into the snow, but not far, the resistance of the snow will soon bring the ball to rest. The ball overcomes resistance in passing through the snow until it is brought to rest, and thus it does the work of forcing itself through the snow, and possesses the energy necessary to do that work.

The overcoming of the resistance of the air by a moving body is work. A steamboat will move for some time in water after the steam has been turned off. The overcoming of the resistance of the water is work, and by virtue of the motion of the boat when the steam was turned off it possessed the energy to do the work of forcing itself for some time through the resistance of the water. The Perpetual Motion worker in each case had reasoned himself into this conclusion: That the same energy will impart the same acceleration of velocity, regardless of the velocity at the beginning of the application of energy. That the same amount of energy or work necessary to impart to a body a velocity of ten feet per second will increase that velocity to twenty feet per second, or from twenty feet per second to thirty feet per second. In other words, that the same amount of energy, and only the same amount of energy is required for a given increase in velocity without regard to the initial velocity. This appears plausible, and almost self-evident. We believe the great majority of people, other than mechanical engineers would, upon presentation of the theory accept it as axiomatic, and as a matter of course. The fallacy becomes manifest only from a critical and technical examination of the Laws of Momentum and Energy.

The Perpetual Motion worker had learned from his text-books that if the velocity be doubled, the energy would be multiplied by four. His idea was to so arrange his mechanism that he would apply the amount of energy to move a fly wheel free to revolve, from a position of rest to a revolving velocity of ten revolutions per second. Then apply again the same amount of energy, and accelerate that velocity from ten revolutions per second to twenty revolutions per second. Thus, the energy at the end of the second second would be four times what it was at the end of the first second. But to make it so, only double the amount of energy had been applied that had been expended at the end of the first second. Thus, he reasoned, his machine was by virtue of its structure, accumulating energy, and this energy could be used one-half to continue the motion of his machine, and the other half to run other machinery, or for any other purpose for which energy might be desired.

Wherein lies the fallacy of this supposition?

We will now endeavor to explain. And for the young student to get the explanation fully, it will be necessary for him to pay the closest attention to what we here state.

A force, for instance the pressure of the finger or the hand, equal to one pound against a body free to move, will, we will say, move that body in one second of time through a space of ten feet, and at the end of that second the body will have a velocity of twenty feet. It is manifest that at the end of the second the velocity will be twenty feet per second for its initial velocity is zero, and its average velocity ten feet per second, the acceleration being, of course, presumed uniform.

Now, it is not true as the Perpetual Motion worker had assumed that the same energy—i.e., the same work that is required to increase the velocity from zero to ten feet per second will increase the velocity from ten feet per second to twenty feet per second, and in that assumption lay the fallacy of our friends who were thus seeking Perpetual Motion.

The greater the velocity, the more energy is required to impart a given acceleration. To increase the velocity from ten feet per second to twenty feet per second, the applied force must continue through one second of time, and more energy is required to follow a rapidly moving body, and continue to apply to it a given force for one second than would be required to follow and maintain the application of the same force to a body moving more slowly—the distance traveled is greater in one case than in the other.

It must be plain that if the moving body have a velocity at the end of the first second of twenty feet per second, it will, at the end of the second second, with the same pressure (force) continued against the same resistance, have a velocity of forty feet per second, and at the end of three seconds have a velocity of sixty feet, and at the end of four seconds a velocity of eighty feet, and so on.

Now, at the beginning of the second second it had a velocity of twenty feet, and at the end of that second a velocity of forty feet. It therefore, traveled through that second with an average velocity of thirty feet and, of course, during the second second traveled exactly thirty feet. It traveled ten feet the first second, and if it traveled thirty feet the second, then in the two seconds it traveled forty feet—four times as far as it traveled the first second. At the beginning of the third second it had a velocity of forty feet, and at the end of the third second a velocity of sixty feet. The average velocity then for the third second would be one-half the sum of forty feet and plus sixty feet—that is to say, it would be fifty feet, and that would be the distance traveled during the third second. The first second it traveled ten feet, the second second thirty feet, and the third second fifty feet, making a total in three seconds of ninety feet—that is to say, in three seconds it traveled nine times as far as in one second.

It will be noticed from the above that the velocity is proportional to the number of seconds, but that the distance traveled is proportional to the square of the number of seconds, and also proportional to the square of the velocity.

Momentum is mass multiplied by velocity; energy is measured by the distance through which a body will move against a given resistance.

Should you prop up one wheel of a carriage and revolve the wheel, then with the pressure of the finger or the thumb on the hub as a brake, stop it, it will be found that (omitting the effect of atmospheric resistance), the wheel will make four times as many revolutions before stopping with a doubled velocity; nine times as many with a trebled velocity.

Falling bodies afford the most perfect illustration of the principle of Momentum and Energy, and are so commonly used to illustrate those principles that many students get the idea that the application of those principles is confined to falling bodies, and do not realize that they extend generally through the field of mechanics.

A falling body is, of course, acted upon by gravity with uniform force equal to the weight of the falling body, and that force continues to follow the falling body and to be applied uniformly and equally, however slowly, or rapidly the body may be falling. And, omitting atmospheric resistance, the body is absolutely free to move except for its natural tendency to remain at rest, or at uniform velocity. It is well known that a body falls (almost exactly) sixteen feet in one second, and at the end of one second has a velocity of thirty-two. During the second second it falls through a distance of forty-eight feet, and during the third second a distance of eighty feet. In two seconds it falls sixty-four feet, and in three seconds one hundred twenty-eight feet, and so on. Thus, it will be observed that the velocity is proportional to the time during which it has fallen, but that the distance fallen in any number of seconds is proportional to the square of the time.

This, indeed, is a property of numbers, and results from mathematical law. If the reader will form a series of numbers, setting down any number for the first term of the series, adding to it its double for the second term, and adding to the second term double the first term for the third, and adding double the first term to the third term for the fourth, and so on—in other words, form any increasing arithmetical series with double the first term for the common difference, he will discover that the sum of all the terms is equal to the first term multiplied by the square of the number of terms. Thus:

In the above series the sum of the first two terms is 20, which is 4 times the first term. The sum of the first three terms, i.e., 5 + 15 + 25 = 45-nine times the first term. The sum of the first four terms, i.e., 5 + 15 + 25 + 35 = 80, sixteen times the first term, and so on.

It will thus be seen that Momentum and Energy are entirely different, although co-related; that momentum relates to velocity, which includes the element of time, whereas energy relates to the amount of work done, and may be represented by a force operating against a certain resistance, through a certain distance, entirely irrespective of time. The energy is the same with the same force operating against the same resistance, through the same distance whether the time consumed be great or small. It takes as much energy in the aggregate to wind up a bucket from the bottom of the well if done slowly as if done quickly.

It would seem hardly necessary to do so, and yet it is worth while remarking that the amount of energy necessary to impart a given motion is exactly the amount of Energy that will be required to arrest that motion, and represents the amount of Energy possessed by the moving body by virtue of its motion. Work done, i.e., Energy applied in giving motion is there in that motion, ready to be returned in exactly an equal quantity—no more—no less.

In all the considerations in this chapter no notice is taken of loss by friction or atmospheric resistance. We are considering pure mechanics and the laws governing them only. In actual mechanical devices it is always necessary to make allowance for atmospheric, frictional and other unavoidable resistances.

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