Treatise on dynamics - Wilson

CHAPTER I.

1. Mechanics is the science which treats of the effects produced by force acting on material bodies. When the different forces acting on any body counteract each other so that the body remains at rest the forces are said to be in equilibrium. The consideration of the conditions to which the forces must be subject that this may be the case is the object of the science of Statics.

When these conditions are not fulfilled by the forces the body will not remain at rest. The investigation of the laws which regulate the motion which takes place, and of the nature of the motion itself will form the subject of the present treatise.

Before we proceed, however, there are certain ideas of which it is necessary to form an accurate conception; and, in order to be definite, we will at present confine our attention to bodies so small that they may without sensible error be treated as geometrical points, but we must at the same time consider them to be possessed of all the properties of matter. This is what must be understood when particles, or material points, or molecules are spoken of.

2. It would be useless to attempt to define space and time. No explanation could in any way render the ideas clearer. The measures of them, on the contrary, require the greatest degree of attention. In Dynamics we are only concerned with linear space, or length, or distance, and this is always to be understood when the term space is used. Now every concrete magnitude must be measured by some definite magnitude of the same kind; thus, space must be measured by space, time by time.

With respect to space nothing can be easier. We can fix on some definite lengthy and can take a rule or a piece of string of that lengthy and apply it over and over again to the space which we wish to measure to find how often it is contained in it; and we consider that we know the magnitude of that space when we know the number of times that our measure is contained in it. Thus, if it be the distance between two points A and B that we wish to determine, we consider that we know it when we have found that our measure will go in it a times, a being whole or fractional ; and we say that

AB = a times the measure,

or more commonly, suppressing the measure, we say that AB = a. In this case our measure is called the '' unit of length," and the space AB is said to be numerically represented by a. It is quite obvious that the greater or less our measure or unit is chosen, the less or greater will be the numerical representation of any given space AB, The choice of this unit is perfectly arbitrary ; but we must always bear in mind that it is a space.

3. We cannot apply the same method of proceeding to "time:" for we are not able to take an interval of time and apply it to other intervals to determine how often it is contained in them. An instant of time, corresponding to a point in space, can only be marked by the occurrence of some event. Now we can easily conceive a number of events happening one after the other in such a manner that the interval between any two shall be of the same duration as the interval between any other two. It is possible and easy to conceive, a priori, this equality of intervals, but it is impossible, a priori, to fix on any series of events which we can be sure will so occur at equal intervals. We shall see, when we have advanced somewhat further, that such a series of events can be procured, but till then we must rest satisfied with being able to conceive such a series. The possibility or impossibility of such a series existing will in no wise affect the accuracy of the notion of equal times. If we take one of these intervals as our measure, we shall have any interval which lasts during n of those intervals expressed by n times the interval, where n may be whole or fractional. And after this interval is agreed on, we consider the length of any other interval determined when n is known, and we express it simply by w: in other words, we take the agreed-on interval as our unit. The duration of this "unit of time" is perfectly arbitrary, and as it is greater or less, n the numerical representation of any proposed interval will be less or greater.

4. Having now explained the method of measuring time and space and of representing them numerically, we come to the motion of a material point. The simplest kind of motion that we can conceive is what is called "uniform rectilinear motion." When a point moves so that at every instant during the whole duration of the motion it is found in the same straight line, the motion is said to be rectilinear. If, moreover, it moves so as always to pass over equal spaces in equal intervals of time, at what part soever of the motion those intervals be taken, or however long or short their duration, then the motion is said to be uniform. Hence if it pass over a space s in one interval, it would pass over a space 2 * in two such intervals, and a space ns in such intervals.

5. Like space and time "velocity" is a term which cannot be defined. We may say it is the "rate" of motion or the "pace" or "speed," but we are only giving other words of the same meaning, not explaining the idea. We can however, as in those cases, explain accurately how it is measured.

Suppose two points to be moving uniformly as described above, and suppose that the space which one describes in any interval of time is equal to the space which the other describes in the same interval, then the velocities of these two points are the same.

Now before we can express velocity numerically, it is necessary to know what is meant when one velocity is said to be twice another. We saw in the case of space, that if we took any given length, and at the end of that took another length equal to the former, the whole length so taken was twice the original length.

Similarly, if we supposed an interval of any definite duration, and then at the end of it another interval of the same duration, the whole interval would be twice one of the first intervals.

We cannot however adopt the same method with velocity, we cannot form any distinct conception of what is meant by one velocity being the sum of two other velocities without bringing to our aid ideas which tacitly assume the point in question. The common notion, however, is very simple and sufficiently exact.

Generally one thing is said to move twice as fast as another, or to have twice the velocity of another when it passes over twice the space in the same time. This is the foundation of the method of representing velocity numerically. And this definition of double velocity, or of twice as fast, is at once applicable to any multiple of the velocity. Thus, if a particle moving with a given velocity passes over in a given time a space s (that is, a space whose numerical representation is s), a particle moving with twice that velocity would pass over in the same interval of time a space 2s; and a particle moving with n times the given velocity would in the same interval pass over a space ns. If the first-mentioned velocity be that chosen as a unit, the latter velocity would be represented numerically by n.

The definition just given^ of one velocity being n times another, is also expressed by saying that the velocity varies as the space described in a given interval of time.