An intermediate course of mechanics

PREFACE

This book has been written as an accompaniment to a First Year Course of College Lectures. It has not been written from the point of view of Examinations. It, however, includes those portions of Mechanics required for the Intermediate Pass Examination of the University of London.

Stress has been laid on fundamental principles rather than on fulness of detail; the latter is apt to overwhelm a Student who is just beginning a subject.

Of all the phenomena of Nature the most direct and most universal are concerned with the motions of bodies. Stones fall, rivers flow, the stars circle in the sky. We ourselves move from place to place, and set other things in motion or keep them at rest.

What are the conditions under which motion takes place? We can give various more or less vague answers to this question. It is the business of the Science of Mechanics to find out what these conditions are, and express them in definite (i.e. scientific) language. The language that is employed is to a large extent the language of Mathematics. Mechanics is therefore often regarded as a branch of Mathematics, and is called Applied Mathematics. But the phenomena which are described are indeed merely the most universal of the phenomena of Nature common to both animate and inanimate things; and the properties of these, when examined in their non vital aspects, form the Science of Physics, of which, therefore, Mechanics must be considered
as a part.

Kinematics. It is convenient to divide this science into two sections. We may answer the question, What is the best way of distinguishing one example of motion from any other, and of describing the difference? The branch in which this question is answered is called Kinematics.

Dynamics. Or we may ask the question, How can a definite motion be produced or prevented from taking place ? The answer to this question is classed under the head of Dynamics.

It is clear that we must know how to describe motion before we can give a satisfactory answer to this second question. We begin, therefore, with Kinematics.

Space and Time. A child soon learns what is meant by here and there. It can reach its toy, or perhaps it is out of reach. It has to wait, and thus acquires a knowledge of time. In scientific language, everything occupies Position in Space and in Time. It is not necessary to inquire here as to what Space and Time really are. This is a very difficult question, to which no certain answer has yet been given. It is sufficient for our purpose to accept the popular conceptions in regard to them. The properties of Space are discussed in Geometry; these will be assumed to be known. The following definitions however must be recalled :

The length of a straight line is the distance or amount of space between its two ends.

Two straight lines are of equal length if when one is placed (or supposed to be placed) in coincidence with the second, their ends also coincide. Any number of straight lines may in this way be tested as to their equality to one another.

The length of a straight line is stated to be an integral multiple of that of another if a number of straight lines equal to the latter and placed end to end along the former are exactly included in it. The smaller is stated to be a fractional part of the larger, the fraction being unity divided by a whole number.

In practice the lines are compared by means of a scale or measure. This is a bar of wood, metal, or other substance, and either two marks are made on it, one near each end, or the ends themselves of the bar are used as points of reference. Such a bar represents a moveable line which can be superposed upon the line under comparison. Similar scales are then placed in succession along the line; as described, or more usually the same scale is placed successively with its first reference point moved to where its second reference point was previously, and the number of such steps counted. Such a scale is called a standard or unit of length.

In general, the length of one straight line will not be an integral multiple of that of the unit. It is found, e.g., that it is greater than m units and less than m + I. The excess over m units is then compared with a smaller unit. If the excess is not an integral multiple of the smaller unit, the fresh excess can be compared with a still smaller unit, and so on. The length of the whole line is the sum of the lengths of its parts. The whole length can, therefore, be specified approximately in terms of this graded series of units; and as the excess at each stage is smaller than that at any preceding stage, the error committed by neglecting to take the final excess into account can theoretically be made as small as we please by a continuation of the process.

The properties, required in order that any such standard as that referred to should be perfect are: (i) Permanence, (ii) Convenience.

By permanence we mean that the distance be- tween the reference points must remain invariable. No substance is known to fulfil this requirement in perfection. All substances change their size with change in temperature. Care has therefore to be taken to use the standard at a definite temperature, or else to know and correct for its changes in length. Invar (an alloy of steel and nickel) undergoes least variation with change in temperature, and is therefore now frequently used for such standards. Every standard requires to be very carefully preserved; copies of it - which can be compared with it from time to time - are made for actual use.

Convenience determines the length that shall be selected; it necessitates that there shall be different units for different purposes. The chief units that have been legalised are the following:

1. The Standard Yard. This is the British standard, and is defined as the distance, measured at 62 F., between the centres of the transverse lines in the two gold plugs in the bronze bar deposited in the office of the Exchequer. The fractional parts selected as lesser units are a one-third part, called a foot % and a one-twelfth part of a foot, called an inch. The inch is then variously subdivided into eight, ten, or more equal parts, to which no special names are given.

For long lengths multiples of the yard are taken as units; thus 1,760 yards equal one mile. Other multiples are also sometimes used, but do not require enumeration in this place (see Tables at end).

2. The International Metre. The British units described above do not lend themselves readily to purposes of calculation. Our numbers are based upon the system of ten, and much greater convenience is obtained by making the relation between any two units a multiple of ten. This rule is followed in the International System which will probably supersede all National units. This unit is the outcome of the labours of a committee appointed during the French Revolution (1795)

The unit of length adopted was intended to be the ten-millionth part of a quarter meridian of the Earth, and is called a metre. The present definition of it is the distance at the melting point of ice between the centres of two lines engraved upon the polished surface of a platinum iridium bar of a nearly X-shaped section, called the International Prototype Metre. This is kept at the International Bureau of Weights and Measures at Sevres, near Paris, and copies of it are supplied to the different Governments. One-tenth of this unit is called a decimetre; one-hundredth, a centimetre; one-thousandth, a millimetre. One thousand metres equal one kilometre.

In scientific work this standard has already practically superseded all others; in British commerce it is only an alternative legal standard. The yard or its fractional parts, like any other lengths, can be compared with the metre. One inch is very nearly indeed equal to 2,54 centimetres.

Measurement of Time. Time is another physical magnitude capable of being measured; and exactly as in the case of space, lengths of time can only be compared with one another by finding out how many times some chosen standard length of time is included in each. Thus we may express how many times the sun rose while a certain event proceeded. The length of time between two sunrises is here the unit employed.

This unit, however, is found not to be constant; it can be compared to a scale made out of india-rubber: there would be no consistency amongst measurements made with such a scale. And intervals of time which we have reason to believe are the same appear not to be the same when compared with the time between two sunrises.

The length of time which we have reason to believe is so nearly constant as to serve as a unit is that taken for the earth to turn round once on its axis. This is called a sidereal day because it is the time which a star takes in apparently going once round the earth. The mean value of the time between two passages of the sun across the meridian is compared with this, and is called a mean solar day. This day is divided into twenty-four equal parts, called hours, each hour into sixty equal minutes and each minute into sixty mean solar seconds. Thus, by definition, there are 24 x 60 x 60, that is 86,400, mean solar seconds in a mean solar day, and by observation it is found that there are 86,164 mean solar seconds in a sidereal day. The division of the whole day into equal parts is effected experimentally by means of a clock. Each swing of the pendulum of a clock, which has been compensated so as to be unaffected by temperature, occupies a constant time.

The reader may wonder how we decide that a platinum scale or time of rotation of the earth remains invariable. The reason for asserting their practical invariability is that measurements made on the assumption of their invariability are found to be on the whole more consistent with one another than if the contrary supposition is made; and, moreover, the laws of mechanics deduced from experiments take a simpler form than they otherwise would do. There is no other criterion of invariability.

In reality there are astronomical observations which can be most simply expressed by supposing that the sidereal day is gradually lengthening the increase is, however, so exceedingly slight that it is only during astronomical periods that it can become perceptible. No account need be taken of it in other departments of science or in civil life.

We can assert that when a body moves, any one point in it travels along a continuous path, joining its initial and final positions. This is equivalent to saying that it is never annihilated or re-created. Even if the body becomes invisible, as when it is converted into a gas, evidence of its continuous existence is so usually forthcoming that the above assertion is made with confidence. The path may be either straight or curved. If different characteristic points of a body are examined, their paths in general differ from one another in shape, and may intersect one another. By far the simplest case is that in which all the paths are parallel straight lines; the motion is then called Recti- linear Translation. Throughout the next five chapters the motion will be supposed to be of this simple kind.


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