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WHITEHEAD, F.Z.S., Mr. A. W. GERRARD, Pharmacist to University College Hospital; Dr. G. MCGOWAN, Messrs. W. K. TOMPKINS, B. Sc., E. P. PERMAN, B.Sc, C. F. BAKER, B.Sc., of University College: Mr. J. T. NORMAN, and other Scientific Contributors. LONDON: 11, NEW BURLINGTON STREET. 1 THURSDAY, APRIL 27, 1893. DYNAMICS IN NUBIBUS. Waterdale Researches; Fresh Light on Dynamics. By theory in order to justify the paying of any serious attention to what can, on general principles, be so easily disproved. It would certainly not be worth while in. vestigating the question in a scientific journal in order to convince the author of the paradox. He could only be convinced by very painstaking and judicious personal interviews of his error and of the unimportance of this question of equal real ponderosities. It would hardly be worth while investigating the question merely because "Waterdale" attributes importance to it, but it is worth while doing so because others may attribute importance to it, and still more so because "Waterdale's" mechanism is interesting and involves a principle that is intimately connected with the second law of thermodynamics, Boltzmann's hypothesis, and a lot of recondite questions which are puzzling the scientific world, so that it is not much wonder that even a clever and ingenious person should get involved in its meshes, especially when that person is involved in a "mission." The general idea involved in "Waterdale's" mechanism is as follows:-Suppose a large body (he objects to the word "mass") M and a small one m, and a spring or other means by which kinetic energy can be given to the bodies. If the spring exert a constant force F through a space s1, it would communicate a velocity V, to (M + m), given by the equation(M + m)V13. = If now it work through a distance s, it will increase this velocity to V2, when = So far all is plain sailing. But we may proceed in another way. We may let the spring work against m alone, and then by suitable mechanism use m's kinetic energy to make the combined system M + m move. In this way we might expect to give m a velocity v1, such that Fs, mv,2, and when this energy was spent on the two bodies M+m, they would acquire a velocity V1 the same as before, given by mv2 }(M + m)V ̧2. Now comes an important assumption, that if the relative velocity of m and M be equal to v1, then by proper mechanism it must always be possible to increase M's velocity by V1, while m's velocity is being reduced to V1. = Suppose now m1, moving with velocity V1, we act upon m by means of the force F, again through the distance So we have for its final velocity v2— Fsm(2 V ̧3). = Hence the relative velocity of M and m is v By choosing s2 = 31, we can arrange that V2 = 2V1, as it simplifies the further argument. In this case "Waterdale" attributes a good deal of importance to this mechanism. He says in his preface: "Let the scientific reader, I would ask, take the trouble first to go through these calculations, and he will then have some idea as to whether the rest of the book is worthy or not of careful perusal." In the body of the work he invents a very complicated hydrodynamic machine to effect his purpose. He there refers to the very much simpler arrangement described in the appendix, and says: Unless the possibility" (of perpetual motion) "is admissible, then I must confess that the theory of equal real ponderosity to all matter can never be accepted." He acknowledges at the same time" that with full knowledge of the liability to error when dealing with the action of forces," all he can reasonably do is to ask and the relative velocity "that . . . pure mathematics be once more applied to the subject." All the same, he asserts that "no disproof can be, or has up to the present been given." "There is no speculation about this, but simple fact, if calculation by figures can be accepted to be true." There are so many things touched on in the work that do not seem in any way necessarily connected with the question of "equal real ponderosity," that it is desirable to show how much interest "Waterdale" feels in this part of his .. V2 = = Vg - Vi which may be much greater than v1, if v1 be much greater than V1, i.e. if m be much smaller than M. This shows that the relative velocity after the second blow may be much greater than after the first, even though the two blows were so chosen as that if applied directly to the combined body they would produce equal increments of velocity in that body. Assuming then that a given relative velocity can always produce a given increase of velocity in the combined system, it appears by our assumption that, as the relative velocity is much greater after the second blow given to m than after the first, the increase of velocity of the system produced by this indirect method of applying the second blow will be much greater than by the first, and consequently much greater than the velocity that could be given to the system by applying the blow directly. By reducing the system to its otherwise produced velocity V2, we could obtain a certain amount of energy, and then repeat the process ad infinitum, thus obtaining a continual supply of energy. An investigator without a mission would be led by this curious result to assume that there must be some mistake in his arguments, and "Waterdale" evidently has some lurking doubts. He sees that it is impossible in the simple case of bodies having only one direction of velocity. Impact can never reduce two bodies of a system to move with the same velocity and conserve energy. We cannot have momentum and energy both conserved. Unless M = O we cannot have In order to divide the energy mv, between the two bodies and reduce them both to a common velocity, we require a third body, and then what becomes of the principle that seemed so plausible, that the increased velocity that m could impart to M depended on their relative velocity only? "Waterdale" sees the hitch all. right in the simple case, and consequently, in order to cheat nature by inventing a complicated case in which he hopes that she will get as muddled as himself, he interposes bent channels, a third and fourth body to receive the blows, springy arms to absorb energy, and smooth surfaces to divert the motion. He evidently has some doubts about all this, for, notwithstanding his assertion that "Appendix II. is a mechanical demonstration to prove that by the principle of velocity of force, a saving in mechanical work, . . . can be effected," and that "there is no speculation about this, but simple fact," yet he gives only a series of suggestions and vague estimates as unspeculative proofs, that the energy spent in bending his springs, in jumping his bodies about, and so forth, is negligible, while in reality it is an important part of his system. That it is so necessarily is proved conclusively by the impossible result he obtains by neglecting it. This is the really interesting principle in the whole matter, that it is not possible to give energy to a system of bodies by giving a series of impulses to some particles of it, to be transmitted to the rest of the system by actions within the system without some part of the energy being spent on internal motions in the system. It is here that the example touches upon the second law of thermodynamics, Boltzmann's hypothesis, and so forth. In order to minimize the effects of these internal vibrations, &c., "Waterdale" argues thus: "Loss No. 2" (giving rise to internal vibrations of his system) "if it arises" (he himself shows that it would, though he overlooks a more important loss), "would be of the nature of internally asserted work." "This loss of work could not be great, for we see by the diagram that the span of work already done when the ball arrives at ... o is small compared with what it has to do." Notwithstanding his profession of calculating everything he does not calculate here, nor does he calculate with what velocity the ball would rebound after it hit the body B, which ultimately stops it; in fact he omits this important question altogether, and goes to the "third factor, the bending of the arm of the system," which he goes on to say, without calculation, “can be almost neglected if we take the tension of elasticity of the arm to be small.' "I should say that one-eighth internal loss of work would certainly more than cover everything." This blessed "I should say!" Is it thus that "Waterdale gives "a mechanical demonstration to prove. ... a saving in mechanical work"? "There is no speculation about this"! It is "simple fact, if calculation by figures can be accepted as true." Most people would agree that "if calculation by figures can be accepted as true" the velocity that could be given by any mechanism to the system indirectly could not be greater than what would give it kinetic energy corresponding to the work supplied. If "Waterdale" will apply a system of levers, springs, &c., acting on the fixed bodies of his system, so as to reduce all the bodies to relative rest, and thereby gives up as hopeless the task of inventing some method by which he can by internal actions alone transfer kinetic energy from one body of a system to the whole of the system without wasting any of it in internal kinetic or potential energy, then he will see how he has to give up the apparently legitimate assumption that the velocity one body of a system can give to the whole system by being itself reduced to relative rest depends only on the relative velocity of the body and the rest of the system. He will see that it depends also on the velocity of his system relative to those supposed fixed bodies he will require as fulcrums for the mechanism required to transfer the energy of the one body to the rest of the system. He sees that something is required to keep his wedge moving forward. He arranges "that the wedge is supported by a following force . . . during this part." The amount of work required he without calculation assumes to be small, and he is probably right here; but it is only one of several losses that he does not calculate, and there are others, such as the conditions of impact at the end of the flight of m, that he does not even notice, though this is the very first that should strike a person investigating the subject after he had clearly seen, as "Waterdale" appears to do, that it is here, in the laws of impacts, that the simple case of velocity one direction and direct impacts fails. It is interesting how cases of this kind illustrate the warming of a gas by compression, the vibrations produced in a bell when struck, and other such cases where energy is given to one part of a dynamical system for this part to distribute amongst the whole, and also how it illustrates the way in which the amount of this internal energy depends on the mobility of the part originally moved. Of course it is al plain enough when the subject is attacked by means of general principles of conservation of energy and momentum, but when the interactions of the different parts of the system are individually considered and the mind distracted by the complexity of the problem, there is rea danger that what is important may be overlooked as trivial, as has been done by "Waterdale." He is not to |