« PreviousContinue »
DYNAMICAL LAWS AND PRINCIPLES.
205. IN the preceding chapter we considered as a subject of Ideas of pure geometry the motion of points, lines, surfaces, and volumes, force introwhether 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 motions, 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 this experience is to be obtained will form the subject of a subsequent chapter.
206. 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 Motûs, 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.
207. We cannot, of course, give a definition of Matter which Matter. 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
Measurement of mass.
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? And we shall then be in a position to discuss the question of the subjectivity of Force.
208. 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 the volume, of a homogeneous 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 vary from point to point of the body, we have evidently, by the above formula and the elementary notation of the integral calculus,
M = √√√p dxdydz,
where p is supposed to be a known function of x, y, z, and the integration extends to the whole space occupied by the matter of the body whether this be continuous or not.
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.
209. 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 result, 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.
210. The Quantity of Motion, or the Momentum, of a rigid Momentum. 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.
211. Change of Quantity of Motion, or Change of Momen- Change of tum, 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 § 27. 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.
212. Rate of Change of Momentum is proportional to the Rate of mass moving and the acceleration of its velocity conjointly, momentum. Thus (§ 35, b) the rate of change of momentum of a falling body is constant, and in the vertical direction. the rate of change of momentum of a mass circle of radius R, with uniform velocity V,
Again (§ 35, a)
is and is directed to the centre of the circle; that is to say, it is a change of direction, not a change of speed, of the motion. Hence if the mass be compelled to keep in the circle by a cord attached to it and held fixed at the centre of the circle, the MV2 force with which the cord is stretched is equal to : this is R called the centrifugal force of the mass M moving with velocity Vin a circle of radius R.
Generally (29), for a body of mass M moving anyhow in space there is change of momentum, at the rate, Md's
in the direc
Rate of change of momentum.
Particle and point.
tion of motion, and M- towards the centre of curvature of the
dz M or, according to the Newtonian notation, Më, Mij, Mž. " dt
213. The Vis Viva, or Kinetic Energy, of a moving body is proportional to the mass and the square of the velocity, conjointly. If 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.
214. Rate of Change of Kinetic Energy (when defined as above) is the product of the velocity into the component of rate of change of momentum in the direction of motion.
d/Mv2 dt 2
215. It is to be observed that, in what precedes, with the exception of the definition of mass, 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 non-directional, is simply the sum of the kinetic energies of the several parts or particles.
216. Matter has an innate power of resisting external influences, so that every body, as 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 Inertia. to disturb a body's uniformity of motion, or to change its direction from the natural rectilinear path.
217. Force is any cause which tends to alter a body's natural Force. 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 kinds, as pressure, or gravity, or friction, or any of the attractive or repulsive actions of electricity, magnetism, etc.
tion of a
218. The three elements specifying a force, or the three Specificaelements which must be known, before a clear notion of the force. 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 Place of 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 matter. The point of the finest needle, or the edge of the sharpest knife, is still a surface, and acts by pressing over a finite area on 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.