motion is of the most general character for one degree of freedom; that is to say, translation and rotation in any fixed proportions, as of the nut of a screw. If one line of a rigid system be constrained to remain parallel to itself, as for instance, if the body be a three-legged stool standing on a perfectly smooth board fixed to a common window, sliding in its frame with perfect freedom, there remain three displacements and one rotation. But we need not farther pursue this subject, as the number of combinations that might be considered is almost endless; and those already given suffice to show how simple is the determination of the degrees of freedom or constraint in any case that may present itself. 170. One degree of constraint of the most general character, is not producible by constraining one point of the body to a curve surface; but it consists in stopping one line of the body from longitudinal motion, except accompanied by rotation_round this line, in fixed proportion to the longitudinal motion. Every other motion being left unimpeded; there remains free rotation about any axis perpendicular to that line (two degrees freedom); and translation in any direction perpendicular to the same line (two degrees freedom). These last four, with the one degree of freedom to screw, constitute the five degrees of freedom, which, with one degree of constraint, make up the six elements. This condition is realized in the . following mechanical arrangement, which seems the simplest that can be imagined for the purpose : Let a screw be cut on one shaft of a Hooke's joint, and let the other shaft be joined to a fixed shaft by a second Hooke's joint. A nut turning on that screw-shaft has the most general kind of motion admitted when there is one degree of constraint. Or it is subjected to just one degree of constraint of the most general character. It has five degrees of freedom; for it may move, ist, by screwing on its shaft, the two Hooke's joints being at rest; 2nd, it may rotate about either axis of the first Hooke's joint, or any axis in their plane (two more degrees of freedom : being freedom to rotate about two axes through one point); 3rd, it may, by the two Hooke's joints, each bending, have translation without rotation in any direction perpendicular to the link, or shaft between the two Hooke's joints (two more degrees of freedom). But it cannot have a motion of translation parallel to the line of the link without a definite proportion of rotation round this line; nor can it have rotation round this line without a definite proportion of translation parallel to it. CHAPTER II. DYNAMICAL LAWS AND PRINCIPLES. 171. In the preceding chapter we considered as a subject of pure geometry the motion of points, lines, surfaces, and volumes, whether taking place with or without change of dimensions and form; and the results we there arrived at are of course altogether independent of the idea of matter, and of the forces which matter exerts. We have heretofore assumed the existence merely of motion, distortion, etc.; we now come to the consideration, not of how we might consider such motion, etc., to be produced, but of the actual causes which in the material world do produce them. The axioms of the present chapter must therefore be considered to be due to actual experience, in the shape either of observation or experiment. How such experience is to be conducted will form the subject of a subsequent chapter. 172. We cannot do better, at all events in commencing, than follow Newton somewhat closely. Indeed the introduction to the Principia contains in a most lucid form the general foundations of dynamics. The Definitiones and Axiomata, sive Leges Motus, there laid down, require only a few amplifications and additional illustrations, suggested by subsequent developments, to suit them to the present state of science, and to make a much better introduction to dynamics than we find in even some of the best modern treatises. 173. We cannot, of course, give a definition of Matter which will satisfy the metaphysician; but the naturalist may be content to know matter as that which can be perceived by the senses, or as that which can be acted upon by, or can exert, force. The latter, and indeed the former also, of these definitions involves the idea of Force, which, in point of fact, is a direct object of sense; probably of all our senses, and certainly of the muscular sense. To our chapter on Properties of Matter we must refer for further discussion of the question, What is matter? 174. The Quantity of Matter in a body, or, as we now call it, the Mass of a body, is proportional, according to Newton, to the Volume and the Density conjointly. In reality, the definition gives us the meaning of density rather than of mass; for it shows us that if twice the original quantity of matter, air for example, be forced into a vessel of given capacity, the density will be doubled, and so on. But it also shows us that, of matter of uniform density, the mass or quantity is proportional to the volume or space it occupies. Let M be the mass, p the density, and V the volume, of a homo-, géneous body. Then M = Vp; if we so take our units that unit of mass is that of unit volume of a body of unit density. If the density be not uniform, the equation gives the Average ($ 26) density; or, as it is usually called, the Mean density, of the body, It is worthy of particular notice that, in this definition, Newton says, if there be anything which freely pervades the interstices of all bodies, this is not taken account of in estimating their Mass or Density. 175. Newton further states, that a practical measure of the mass of a body is its Weight. His experiments on pendulums, by which he establishes this most important remark, will be described later, in our chapter on Properties of Matter. As will be presently explained, the unit mass most convenient for British measurements is an imperial pound of matter. 176. The Quantity of Motion, or the Momentum, of a rigid body moving without rotation is proportional to its mass and velocity conjointly. The whole motion is the sum of the motions of its several parts. Thus a doubled mass, or a doubled velocity, would correspond to a double quantity of motion; and so on. Hence, if we take as unit of momentum the momentum of a unit of matter moving with unit velocity, the momentum of a mass M moving with velocity v is Mv. 177. Change of Quantity of Motion, or Change of Momentum, is proportional to the mass moving and the change of its velocity conjointly. Change of velocity is to be understood in the general sense of $ 31. Thus, in the figure of that section, if a velocity represented by OA be changed to another represented by OC, the change of velocity is represented in magnitude and direction by AC. 178. Rate of Change of Momentum, or Acceleration of Momentum, is proportional to the mass moving and the acceleration of its velocity conjointly. Thus ($ 44) the rate of change of momentum of a falling body is constant, and in the vertical direction. Again (9 36) the rate of change of momentum of a mass M, describing a circle of MV2 radius R, with uniform velocity V, is and is directed to the R centre of the circle; that is to say, it depends upon a change of direction, not a change of speed, of the motion. 179. The Vis Viva, or Kinetic Energy, of a moving body is proportional to the mass and the square of the velocity, conjointly. If т T - 09) == we adopt the same units of mass and velocity as before, there is particular advantage in defining kinetic energy as half the product of the mass and the square of its velocity. 180. Rate of Change of Kinetic Energy (when defined as above) is the product of the velocity into the component of acceleration of momentum in the direction of motion. Suppose the velocity of a mass M to be changed from v to v, in any time t; the rate at which the kinetic energy has changed is IM (0,2 M (v, – v). Į (v, + v). Now M (0. M(0, – v) is the rate of change of momentum in the direction of motion, and į (v, + v) is equal to v, if be infinitely small. Hence the above statement. It is often convenient to use Newton's Fluxional notation for the rate of change of any quantity per unit of 1 time. In this notation (§ 28) • stands for = (0,—v); so that the rate of change of {Mvʻ, the kinetic energy, is Mö.v. (See also $$ 229, 241.) 181. It is to be observed that, in what precedes, with the exception of the definition of density, we have taken no account of the dimensions of the moving body. This is of no consequence so long as it does not rotate, and so long as its parts preserve the same relative positions amongst one another. In this case we may suppose the whole of the matter in it to be condensed in one point or particle. We thus speak of a material particle, as distinguished from a geometrical point. If the body rotate, or if its parts change their relative positions, then we cannot choose any one point by whose motions alone we may determine those of the other points. In such cases the momentum and change of momentum of the whole body in any direction are, the sums of the momenta, and of the changes of momentum, of its parts, in these directions; while the kinetic energy of the whole, being nondirectional, is simply the sum of the kinetic energies of the several parts or particles. 182. Matter has an innate power of resisting external influences, so that every body, so far as it can, remains at rest, or moves uniformly in a straight line. This, the Inertia of matter, is proportional to the quantity of matter in the body. And it follows that some cause is requisite to disturb a body's uniformity of motion, or to change its direction from the natural rectilinear path. 183. Impressed Force, or Force simply, is any cause which tends to alter a body's natural state of rest, or of uniform motion in a straight line. Force is wholly expended in the Action it produces; and the body, after the force ceases to act, retains by its inertia the direction of motion and the velocity which were given to it. Force may be of divers kind, as pressure, or gravity, or friction, or any of the attractive or repulsive actions of electricity, magnetism, etc. matter. 184. The three elements specifying a force, or the three elements which must be known, before a clear notion of the force under consideration can be formed, are, its place of application, its direction, and its magnitude. (a) The place of application of a force. The first case to be considered is that in which the place of application is a point. It has been shown already in what sense the term 'point' is to be taken, and, therefore, in what way a force may be imagined as acting at a point. In reality, however, the place of application of a force is always either a surface or a space of three dimensions occupied by The point of the finest needle, or the edge of the sharpest knife, is still a surface, and acts as such on the bodies to which it may be applied. Even the most rigid substances, when brought together, do not touch at a point merely, but mould each other so as to produce a surface of application. On the other hand, gravity is a force of which the place of application is the whole matter of the body whose weight is considered; and the smallest particle of matter that has weight occupies some finite portion of space. Thus it is to be remarked, that there are two kinds of force, distinguishable by their place of application-force whose place of application is a surface, and force whose place of application is a solid. When a heavy body rests on the ground, or on a table, force of the second character, acting downwards, is balanced by force of the first character acting upwards. (6) The second element in the specification of a force is its direction. The direction of a force is the line in which it acts. If the place of application of a force be regarded as a point, a line through that point, in the direction in which the force tends to move the body, is the direction of the force. In the case of a force distributed over a surface, it is frequently possible and convenient to assume a single point and a single line, such that a certain force acting at that point in that line would produce the same effect as is really produced. (c) The third element in the specification of a force is its magnitude. This involves a consideration of the method followed in dynamics for measuring forces. Before measuring anything it is necessary to have a unit of measurement, or a standard to which to refer, and a principle of numerical specification, or a mode of referring to the standard. These will be supplied presently. See also $ 224, below. 185. The Measure of a Force is the quantity of motion which it produces in unit of time. The reader, who has been accustomed to speak of a force of so many pounds, or so many tons, may be reasonably startled when he finds that Newton gives no countenance to such expressions. The method is not correct unless it be specified at what part of the earth's surface the pound, or other definite quantity of matter named, is to be weighed; for the weight of a given quantity of matter differs in different latitudes. |