Dynamics - Lamb

This book is a sequel to a treatise on Statics published a little more than a year ago, and has a similar scope. To avoid repetitions, numerous references to the former volume are made,

A writer who undertakes to explain the elements of Dynamics has the choice, either to follow one or other of the traditional methods which, however effectual from a practical point of view, are open to criticism on logical grounds, or else to adopt a treatment so abstract that it is likely to bewilder rather than to assist the student who looks to learn something about the behaviour of actual bodies which he can see and handle. There is no doubt as to which is the proper course in a work like the present; and I have not hesitated to follow the method adopted by Maxwell, in his Matter and Motion, which forms, I think, the best elementary introduction to the 'absolute' system of Dynamics. Some account of the more abstract, if more logical, way of looking at dynamical questions is, however, given in its proper place, which is at the end, rather than at the beginning of the book.

There is some latitude of judgment as to the order in which the different parts of the subject should be taken. To many students it is more important that they should gain, as soon as possible, some power of dealing with the simpler questions of 'rigid' Dynamics, than that they should master the more intricate problems of 'central forces,' or of motion under various laws of resistance. This consideration has dictated the arrangement here adopted, bat as the later chapters are largely independent of one another, they may be read in a different order without inconvenience.

Some pains have been taken in the matter of examples for practice. The standard collections, and the text-books of several generations, supply at first sight abundant material for appropriation, but they do not always reward the search for problems which are really exercises on dynamical theory, and not merely algebraical or trigonometrical puzzles in disguise. In the present treatise, preference has been given to examples which are simple rather than elaborate from the analytical point of view. Most of those which are in any degree original have been framed with this intention.

I am again greatly indebted to Prof. F. S. Carey, and Mr J. H. C. Searle, for their kindness in reading the proofs, and for various useful suggestions. The latter has moreover verified most of the examples. Miss Mary Taylor has also given kind assistance in the later stages of the passage through the press.

The object of the science of Dynamics is to investigate the motion of bodies as affected by the forces which act upon them. Some system of physical assumptions, to be justified ultimately by comparison with experience, is therefore necessary as a basis. For the present we consider specially cases where the motion and the forces are in one straight line.

The subject may be approached from different points of view, and the fundamental assumptions may consequently be framed in various ways, but the differences must of course be mainly formal, and must lead to the same results when applied to any actual dynamical problem. In the present Article, and the following one, two distinct systems are explained. Both systems start from the idea of force as a primary notion, but they differ as to the principles on which different forces are compared.

The first assumption which we make is in each case that embodied in Newton's First Law, to the effect that a material particle persists in its state of rest, or of motion in a straight line with constant velocity, except in so far as it is compelled to change that state by the action of force upon it. In other words, acceleration is the result of force, and ceases with the force. This is sometimes called the law of inertia.

The first of the two systems to be explained proceeds from a purely terrestrial and local standpoint, and adopts the system of force-measurement, in terms of gravity, with which we are familiar in Statics. The physical assumption now introduced is that the acceleration produced in a given body by the action of any force is proportional to the magnitude of that force. It can therefore be found by comparison with the known acceleration (g) produced in the same body when falling freely under its own gravity. It will be noted that the assumption here made is the simplest that we can frame consistent with the law of inertia, although its validity must rest of course not on its simplicity but on its conformity with experience.

It is convenient here to distinguish between the weight and the gravity of a body. When we say that a body has a weight W, as determined by the balance, we mean that the downward pressure which it exerts on its supports, when at rest, is in the ratio W to the pressure exerted under like circumstances, and therefore in the same locality, by the standard pound or kilogramme, or whatever the unit is. The 'weight' of a body is
therefore, on this definition, a numerical constant attached to it; it is the same at all places, since a variation in the intensity of gravity would affect the body and the standard alike.

By the gravity of a body, on the other hand, is meant the downward pull of the earth upon it. This is, on statical principles, equal to the pressure which the body exerts on its supports when at rest, but is of course to be distinguished from it. It is known to vary somewhat with the latitude, and with altitude above sea-level.

The total variation in the intensity of gravity over the earth's surface is, however, only one-half per cent., and this degree of vagueness is for many practical purposes quite unimportant. The numerical data on which an engineer, for example, has to rely, such as strengths of materials, coefficients of friction, &c., are as a rule affected by much greater uncertainty. For this reason the gravitational system of force-measurement is retained by engineers without inconvenience, even in dynamical questions where gravity is not directly concerned.

But when, as in many scientific measurements, greater precision is desired and is possible, it becomes necessary either to express the results in terms of gravity at some particular station on the earth's surface which is taken as a standard, or to have recourse to some less arbitrary dynamical system, independent of terrestrial or other gravity. The latter procedure is clearly preferable, and in the application to questions of Astronomy almost essential. We proceed accordingly, in the next Article, to give another statement of fundamental dynamical principles, and to explain the absolute system of force-measurement to which it leads.


The Absolute System of Dynamics.

For purposes of explanation it is convenient to appeal to a series of ideal experiments. We imagine that we have some means, independent of gravity, of applying a constant force, and of verifying its constancy, e.g. by a spring-dynamometer which is stretched or deformed to a constant extent; but we do not presuppose any graduated scale by which different forces can be compared numerically.

The first experimental result which we may suppose to be established in this way is that a constant force acting on a body produces a constant acceleration, i.e. the velocity changes by equal amounts in equal times.

It is observed, again, that the same force applied in succession to different bodies produces in general different degrees of acceleration. This is described as due to differences in the inertia, or mass, of the respective bodies. Two bodies which acquire equal velocities in equal times, when acted upon by the same force, are regarded as dynamically equivalent, and their masses are said to be equal. The standard, or unit, of mass must therefore be that of some particular piece of matter, chosen in the first instance arbitrarily, e.g. the standard pound or kilogramme. A body is said to have the mass m (where w is an integer) when it can be divided into m portions each of which is dynamically equivalent to the unit. Similarly, if a unit mass be subdivided into n dynamically equivalent pieces, the mass of each of these is said to be 1/n. It is evident that on these lines a complete scale of mass can be constructed, and that the mass of any body whatever can be indicated by a numerical quantity.