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multiplied by the whole angle through which the shaft turns, measures the whole work done against the friction of the clamp. The same result is much more easily obtained by wrapping a rope or chain several times round the shaft, or round a cylinder or drum carried round by the shaft, and applying measured forces to its two ends in proper directions to keep it nearly steady while the shaft turns round without it. The difference of the moments of these two forces round the axis, multiplied by the angle through which the shaft turns, measures the whole work spent on friction against the rope. If we remove all other resistance to the shaft, and apply the proper amount of force at each end of the rope or chain (which is very easily done in practice), the prime mover is kept running at the proper speed for the test, and having its whole work thus wasted for the time and measured.
391. UNTIL 've know thoroughly the nature of matter and the forces which produce its motions, it will be utterly impossible to submit to mathematical reasoning the exact conditions of any phy. sical question. It has been long understood, however, that an approximate solution of almost any problem in the ordinary branches of Natural Philosophy may be easily obtained by a species of abstraction, or rather limitation of the data, such as enables us easily to solve the modified form of the question, while we are well assured that the circumstances (so modified) affect the result only in a super. ficial manner.
392. Take, for instance, the very simple case of a crowbar em. ployed to move a heavy mass. The accurate mathematical investi, gation of the action would involve the simultaneous treatment of the motions of every part of bar, fulcrum, and mass raised; and from our almost complete ignorance of the nature of matter and molecular forces, it is clear that such a treatment of the problem is impossible.
It is a result of observation that the particles of the bar, fulcrum, and mass, separately, retain throughout the process nearly the same relative positions. Hence the idea of solving, instead of the above impossible question, another, in reality quite different, but, while infinitely simpler, obviously leading to nearly the same results as the former.
393. The new form is given at once by the experimental result of the trial. Imagine the masses involved to be perfectly rigid (i.e. incapable of changing their forms or dimensions), and the infinite multiplicity of the forces, really acting, may be left out of consideration; so that the mathematical investigation deals with a finite (and generally small) number of forces instead of a pràctically infinite number. Our warrant for such a substitution is established thus.
394. The only effects of the intermolecular forces would be ext hibited in molecular alterations of the form or volume of the masses involved. But as these (practically) remain almost unchanged, the forces which produce, or tend to produce, changes in them may be left out of consideration. Thus we are enabled to investigate the action of machinery by supposing it to consist of separate portions whose forms and dimensions are unalterable.
395. If we go a little farther into the question, we find that the lever bends, some parts of it are extended and others compressed.
This would lead us into a very serious and difficult inquiry if we had to take account of the whole circumstances. But (by experience) we find that a sufficiently accurate solution of this more formidable case of the problem may be obtained by supposing (what can never be realized in practice) the mass to be homogeneous, and the forces consequent on a dilatation, compression, or distortion, to be proportional in magnitude, and opposed in direction, to these deformations respectively. By this farther assumption, close approximations may be made to the vibrations of rods, plates, etc., as well as to the statical effects of springs, etc.
We may pursue the process farther. Compression, in general, develops heat, and extension, cold. These alter sensibly the elasticity of a body. By introducing such considerations, we reach, without great difficulty, what may be called a third approximation to the solution of the physical problem considered.
397. We might next introduce the conduction of the heat, so produced, from point to point of the solid, with its accompanying modifications of elasticity, and so on; and we might then consider the production of thermo-electric currents, which (as we shall see) are always developed by unequal heating in a mass if it be not perfectly homogeneous. Énough, however, has been said to show, first, our utter ignorance as to the true and complete solution of any physical question by the only perfect method, that of the consideration of the circumstances which affect the motion of every portion, separately, of each body concerned; and, second, the practically sufficient manner in which practical questions may be attacked by limiting their generality, the limitations introduced being themselves deduced from experience, and being therefore Nature's own solution (to a less or greater degree of accuracy) of the infinite additional number of equations by which we should otherwise have been encumbered.
398. To take another case: in the consideration of the propagation of waves on the surface of a fluid, it is impossible, not only on account of mathematical difficulties, but on account of our igno'rance of what matter is, and what forces its particles exert on.each other, to form the equations which would give us the separate motion of each. Our first approximation to a solution, and one sufficient for most practical purposes, is derived from the consideration of the
motion of a homogeneous, incompressible, and perfectly plastic mass; a hypothetical substance which, of course, nowhere exists in nature.
399. Looking a little more closely, we find that the actual motion differs considerably from that given by the analytical solution of the restricted problem, and we introduce farther considerations, such as the compressibility of fluids, their internal friction, the heat generated by the latter, and its effects in dilating the mass, etc. etc. By such successive corrections we attain, at length, to a mathematical result which (at all events in the present state of experimental science) agrees, within the limits of experimental error, with observation.
400. It would be easy to give many more instances substantiating what has just been advanced, but it seems scarcely necessary to do so. We may therefore at once say that there is no question in physical science which can be completely and accurately investigated by mathematical reasoning (in which, be it carefully remembered, it is not necessary that symbols should be introduced), but that there are different degrees of approximation, involving assumptions more and more nearly coincident with observation, which may be arrived at in the solution of any particular question.
401. The object of the present division of this work is to deal with the first and second of these approximations. In it we shall suppose all solids either RIGID, i.e. unchangeable in form and volume, or ELASTIC; but in the latter case, we shall assume the law, connecting pression or a distortion with the force which causes it, to have a particular form deduced from experiment. And we shall also leave out of consideration the thermal or electric effects which compression or distortion generally produce. We shall also suppose fluids, whether liquids or gases, to be either INCOMPRESSIBLE or compressible according to certain known laws; and we shall omit considerations of fluid friction, although we admit the consideration of friction between solids. Fluids will therefore be supposed perfect, i.e. such that any particle may be moved amongst the others by the slightest force.
402. When we come to Properties of Matter and the Physical Forces, we shall give in detail, as far as they are yet known, the modifications which farther approximations have introduced into the previous results.
403. The laws of friction between solids were very ably investi gated by Coulomb; and, as we shall require them in the succeeding, chapters, we give a brief summary of them here; reserving the more careful scrutiny of experimental results to our chapter on Properties of Matter.
404. To produce sliding of 'one solid body on another, the surfaces in contact being plane, requires a tangential force which depends -(1) upon the nature of the bodies; (2) upon their polish, or the species and quantity of lubricant which may have been applied;
(3) upon the notial pressure between them, to which it is in general directly proportional; (4) upon the length of time during which they have been suffered to remain in contact.
It does not (except in extreme cases where scratching or abrasion takes place) depend sensibly upon the area of the surfaces in contact. This, which is called Statical Friction, is thus capable of opposing a tangential resistance to mộtion which may be of any requisite amount up to MR; where R is the whole normal pressure between the bodies; and ji (which depends mainly upon the nature of the surfaces in contact) is the co-efficient of Statical Friction. This co-efficient varies greatly with the circumstances, being in some cases as low as o'03, in others as high as o:80. Later we shall give a table of its values. Where the applied forces are insufficient to produce motion, the whole amount of statical friction is not called into play; its amount then just reaches what is sufficient to equilibrate the other forces, and its direction is the opposite of that in which their resultant tends to produce motion. When the statical friction has been overcome, and sliding is produced, experiment shows that force of friction continues to act, opposing the motion, sensibly proportional to the normal pressure, and independent of the velocity. But for the same two bodies the co-efficient of Kinetic Friction is less than that of Statical Friction, and is approximately the same whatever be the rate of motion.
405. When among the forces acting in any case of equilibrium, there are frictions of solids on solids, the circumstances would not be altered by doing away with all friction, and replacing its forces by forces of mutual action supposed to remain unchanged by any infinitely small relative motions of the parts between which they act. By this artifice all such cases may be brought under the general principle of Lagrange ($ 254).
406. In the following chapters on Abstract Dynamics we will confine ourselves chiefly to such portions of this extensive subject as are likely to be useful to us in the rest of the work.