086 MECHANICS Ampli tude. Let a point P (fig. 14) move uniformly in the circle APA . Then, drawing any diameter AOA , and PM per pendicular to it, the motion of M is simple harmonic. The speed and acceleration of M are obviously the resolved parts, along AA , ac- of the speed and celeration of P. Hence if V be the speed of P we have speed of M= T, PO Fig. 14. MO acceleration of M = -p^ x acceleration of P MO V 2 _V2 "PO PO~P0 2MO< From these expressions we see that, if we call w the angular velocity of OP, so that w = V/PO, we have speed of M = PM. co , acceleration of M = MO. 2 . Thus the speed of M increases from A to O, being zero at A, and V at O ; then it falls off to zero at A , and goes through the same numerical values in the opposite order, when the direction of motion is reversed at A . The acceleration of M is always directed towards 0. It has its greatest value at A and again at A , and is always proportional to the distance from 0. If T be the period of the simple harmonic motion, i.e., the period of rotation of P in the circle, we have where a is the radius of the circle, or, as it is also called, the "amplitude" of the simple harmonic motion. We may now write as the characteristic of this species of motion acceleration = x displacement ; T _ /displacement / acceleration Phase 52. In our further remarks about simple harmonic ld motion the following terms will be found convenient. P is the position at time t of the point moving in the circle. Let E be its position at the zero of reckoning, when t = 0. Then the angle AOP may be called the "phase" of the simple harmonic motion, and AOE the " epoch." In time units the values of the phase and epoch are found from their circular measure by dividing by co. If the position of the point moving with simple harmonic motion be denoted by x, we obviously have aj-OM-OPcosPOA, = OPcos(POE + EOA), This expression is to be found, perhaps more frequently than any other, in all branches of mathematical physics. It is in terms, or series of terms, of this form that every periodic phenomenon can be described mathematically, as will be seen later. From the expressions for the longitude and radius-vector of a planet or a satellite to those of the most complex undulations whether in water, in air, or in the luminiferous medium, all are alike dependent upon it. The results obtained geometrically above are easily reproduced from this form : thus x = - aw sin (ut + e) ; Graphic 53. The simplest graphical method of exhibiting the nature of any kind of rectilineal motion is to compound it tiou. with a uniform velocity in a direction perpendicular to the line in which it is executed. This is, in fact, what is done in the majority of self-registering instruments, where a slip of paper is drawn by clock-work uniformly past the moving point, in a direction perpendicular to its line of motion, and a record is made by mechanical means, by a pencil, by an electric spark, or (best of all) by photo graphic processes. When this process is applied to a simple harmonic motion the record is of the general form of the curve in fig. 15. This curve has long been known as Fig. 15. the " curve of sines," or the " harmonic " curve. All its forms can be deduced from any one of them by mere extension or foreshortening in the vertical or horizontal directions in the figure. It represents the simplest forms into which a vibrating string can be thrown, as well as the instantaneous form of a section of the surface of water along which a simple series of oscillatory waves or ripples is pass ing. In this case the form of the section remains the same as time goes on, but the whole figure moves steadily onwards in the direction in which the waves are travelling. This is expressed analytically by the form Analj y = acos(nt - mx), cal ex pressi where x and y are horizontal and vertical coordinates of a point at f or a the surface of the water, y being measured from the level of the W ave. undisturbed surface. When x is constant we study, for all time, the simple harmonic rise and fall at a particular place. When t is constant we have the above-figured instantaneous glance of a sec tion of the whole water-surface. The rate at which the wave travels is obviously n/m ; for, if we increase t by any quantity r, and x by the corresponding quantity nr/m, the value of y is unaltered. 54. We have next to consider the result of superposing Comp or compounding two simple harmonic motions which take sition place in the same line. The geometrical method amply simpl< suffices for this purpose provided the periods of the two are m c m equal, however different may be their amplitudes and their tions i phases. For, if we suppose PQ (fig. 16) to turn about P in one li: the same plane and with the same angular velocity as OP about 0, the angle OPQ will remain unaltered, and therefore the triangle OPQ will remain of constant size and form while turn ing about 0. Thus Q describes a circle about in the given period. The resolved parts of OP, PQ, along any diameter OA, together make up the resolved part of OQ along the Fig. it>. same line. Hence two simple harmonic motions, of the same period and in the same line, are equivalent to a single simple harmonic motion of the common period. The amplitude of the resultant simple harmonic motion is OQ, and depends only upon OP, PQ, and the angle OPQ, the amplitudes of the two component simple harmonic motions and the supplement of the difference of their phases. 55, When the difference of phase is nil, or any whole Ampli numberof circumferences, the resultant amplitude is thesum tu( ^ e of the amplitudes of the components, which is its greatest -J^ i? value. When the difference of phase is an odd number of ail t semi-circumferences, the amplitude of the resultant is the difference of those of the components. If we produce QP to meet OA in K, we see that QOA, the phase of the resultant simple harmonic motion, is inter mediate in value to the phases of the components, which are POA and QUA respectively. Its excess over the one, and its defect from the other, are the angles at Oand Q in the triangle OPQ ; and their sines are to one another as the separate amplitudes QP, PO. Hence, when these am- pliLides differ, the phase of the resultant coincides more nearly with that of the component whose amplitude is the
greater.