approxima with no change of temperature. By introducing such considera- Further tions, we reach, without great difficulty, what may be called tions. a third approximation to the solution of the physical problem considered. 444. 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. Enough, 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. 445. To take another case: in the consideration of the propagation of waves at the surface of a fluid, it is impossible, not only on account of mathematical difficulties, but on account of our ignorance 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 may have no existence in nature. 446. 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 further 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 Further approxima tions. Object of the present the work. 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. 447. 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, 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. 448. The object of the present division of this volume is to deal division of 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 a compression or a distortion with the force which causes it, to have a particular form deduced from experiment. And we shall in the latter case neglect the thermal or electric effects which compression or distortion generally cause. 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. Laws of friction. 449. When we come to Properties of Matter and the various forms of Energy, we shall give in detail, as far as they are yet known, the 'modifications which further approximations have introduced into the previous results. 450. The laws of friction between solids were very ably investigated 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. 451. To produce and to maintain sliding of one solid body on another requires a tangential force which depends—(1) upon friction. the nature of the bodies; (2) upon their polish, or the species and Laws of quantity of lubricant which may have been applied; (3) upon the normal pressure between them, to which it is in general directly proportional. It does not (except in some extreme cases where scratching or excessive abrasion takes place) depend sensibly upon the area of the surfaces in contact. When two bodies are pressed together without being caused to slide one on another, the force which prevents sliding is called Statical Friction. It is capable of opposing a tangential resistance to motion which may be of any amount less than or at most equal to μR; where R is the whole normal pressure between the bodies; and μ (which depends mainly upon the nature of the surfaces in contact) is what is commonly called the coefficient of Statical Friction. This coefficient varies greatly with the circumstances, being in some cases as low as 0.03, in others as high as 0.80. Later, we shall give a table of its values. When 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. 452. When the statical friction has been overcome, and sliding is produced, experiment shows that a force of friction. continues to act, opposing the motion; that this force of Kinetic Friction is in most cases considerably less than the extreme force of static friction which had to be overcome before the sliding commenced; that it too is sensibly proportional to the normal pressure; and that it is approximately the same whatever be the velocity of the sliding. of merely 453. In the following Chapters on Abstract Dynamics we con- Rejection fine ourselves mainly to the general principles, and the fundamen- curious speculatal formulas and equations of the mathematics of this extensive tions. subject; and, neither seeking nor avoiding mathematical exercitations, we enter on special problems solely with a view to possible usefulness for physical science, whether in the way of the material of experimental investigation, or for illustrating physical principles, or for aiding in speculations of Natural Philosophy. CHAPTER VI. STATICS OF A PARTICLE.-ATTRACTION. Objects of the chapter. Conditions of equilibrium of a particle. Equili brium of a particle. 454. WE naturally divide Statics into two parts-the equilibrium of a particle, and that of a rigid or elastic body or system of particles whether solid or fluid. In a very few sections we shall dispose of the first of these parts, and the rest of this chapter will be devoted to a digression on the important subject of Attraction. 455. By § 255, forces acting at the same point, or on the same material particle, are to be compounded by the same laws as velocities. Hence, evidently, the sum of their components in any direction must vanish if there is equilibrium; and there is equilibrium if the sums of the components in each of three lines not in one plane are each zero. And thence the necessary and sufficient mathematical equations of equilibrium. Thus, for the equilibrium of a material particle, it is necessary, and sufficient, that the (algebraic) sums of the components of the applied forces, resolved in any three rectangular directions, should vanish. If P be one of the forces, l, m, n its direction-cosines, we have ΣΙΡ = 0, ΣP=0, SnP = 0. If there be not equilibrium, suppose R, with direction-cosines λ, u, v, to be the resultant force. If reversed in direction, it will, with the other forces, produce equilibrium. Hence ΣIP - XR = 0, ΣmP-μR=0, ΣnP-vR = 0. 456. We may take one or two particular cases as examples of the general results above. Thus, (1) If the particle rest on a frictionless curve, the component force along the curve must vanish. If x, y, z be the co-ordinates of the point of the curve at which the particle rests, we have evidently When P, l, m, n are given in terms of x, y, z, this, with the two equations to the curve, determines the position of equilibrum. (2) If the curve be frictional, the resultant force along it must be balanced by the friction. If F be the friction, the condition is This gives the amount of friction which will be called into play; Hence, the limiting positions, between which equilibrium is pos- (3) If the particle rest on a smooth surface, the resultant of the applied forces must evidently be perpendicular to the surface. If p(x, y, z) = 0 be the equation of the surface, we must therefore have and these three equations determine the position of equilibrium. |