714 MECHANICS In each of these three cases the complete solution of the problem can be exhibited in terms of elliptic functions. In the last two cases, when the arcs of oscillation are very small, a suiiicient solu tion may easily be obtained by the usual methods of approxima tion. This is a particularly instructive example. 160, As a final example of constrained motion of a particle, let us find the form of a curve such o that a particle will slide down any arc of it, from the origin, in the same time as down the chord of that arc. If OA, OB (fig. 53) be any two chords, it is plain that the difference of the times down these chords must be equal to the time of describing the arc AB. But, if OA make B A Fig. 53. an angle with the vertical, the time of descent alon^ it is
20A V gCOSO And the velocity at A is >J 2gOAcosO, so that the time of describing AB (considered as infinitesimal) is If we put r for OA, our condition gives at once ds_ d_ / 2r de de V g cos e~ g cos e^2g r COS Q where s is the length of the arc OA. This equation is easily integrated, and the resulting relation is which belongs to the well-known lemniscate of Bernoulli. From its form we see that the vertical line from which 6 is measured is a tangent at ; so that the motion in arc commences vertically Disturbed 161. To complete this elementary sketch of the dy- motion. namics of a single particle ws take an instance or two of " disturbed motion." The essence of this question is usually that the disturbing forces are, at any instant, small in com parison with the forces producing the motion ; so that, during any brief period, the motion is practically the same as if no disturbing cause had been at work. But, in time, the effects of the disturbance may become so great as entirely to change the dimensions and form of the orbit described. The mathematical method which has been devised to meet this question depends upon what has just been said. The character of the path is not, at any particular instant, affected by the disturbance ; but its form and dimensions are. Hence, as the first depends upon the form of the equations which represent it, while the latter depend upon the actual and relative magnitudes of the constants involved, we settle, once for all, the form of the equation as if no disturbing cause had acted. But we are thus entitled to assume that the constants which it involves are quantities which vary with the time in consequence of the slight, but persistent, effects of the disturbance. And, as we know that, if at any moment the disturbance were to cease, the motion would forthwith go on for ever in the orbit then being described, we may assume that in the ex pressions for the components of the velocity no terms occur depending on the rate of alteration of the values of the constants. This, as will be seen below, very much simpli fies the mathematical treatment of such questions. Peiulu- Suppose a cycloidal pendulum, or a simple pendulum vibrating lum. through very small arcs, to be subjected to a simple harmonic dis turbance in the direction of its motion. will obviously be of the form The equation of motion where n = l/g, as in 13-1. The integral of this equation is harmonic motion on the natural simple harmonic motion of the undisturbed bob, and that it is altogether independent of the amplitude and phase of the undisturbed motion. So long as the disturbance is very small, this new part of the motion may be neglected, unless m is very nearly equal to n. For in that case the amplitude of the disturbance may become much greater than that of the original motion. AVhen m is equal to n, the integral changes its form, and we have = P cos (nt + Q) + A^- sin nt . . ll> This shows that, in the special case of a disturbance of the same period as the undisturbed motion, the nature of motion the motion is entirely changed. Thus, suppose the pen- J vlth c dulum to be at rest at its lowest point when the disturb- of san ance is applied ; then we have merely period, = A sinnt. 2n a simple harmonic motion whose amplitude increases in proportion to the time elapsed since the disturbance com menced. 162. As another illustration, suppose the point of sus- Point pension of a simple pendulum to have a simple harmonic suspeu motion of small amplitude in a horizontal line. J 101 , Here the equations of motion are (to horizontal and vertical axes) mx = T - -- But if we suppose the oscillations to be small, we may write x - = 10, y = l, where I is the length of the pendulum, and the angle it makes with the vertical. Then we have x = l$+ =ld + Acosmt, suppose, andj/ = 0, Hence m# = T, and W + A cos mt = - gO , which is precisely the equation of the preceding investigation. We see from this how to explain the somewhat puzzling Motion phenomenon which we observe when we produce complete oi a sli rotations of a stone in a sling by a comparatively trifling motion of the hand. All that is necessary is that the hand should have a slight to and fro horizontal motion, in a period nearly equal to that in which, the sling and stone would vibrate as a pendulum. This result of particle kine tics is (like that in 161) of great value in other branches of physics, especially sound, light, and radiant heat. To illustrate the general principle, let us take the case of one Disturl degree of freedom. Then the equation of motion of an unit mass ance must be of the form general where represents the normal force, and Q 1 the abnormal or dis turbing force. Leaving out X for the moment, let the integral of 6 = be 6=f(a, /3, t}, in which a and # are two arbitrary constants. "We may now sup pose a and to be variable in such a way that the equation shall still bo satisfied by this value of 6 when the disturbing forces are included. This imposes only one condition on the two independent quantities a. and /8, so that to determine them completely we must impose a second. This we do, as already explained, by making the expression for the speed independent of the rates of alteration of o and j8, and we gain the advantage th-at our solution will accord at every instant with what would be the actual future motion if I the disturbance were suddenly to cease. The speed is "We therefore assume Taking account of this and differentiating again, we have I j - z cosmt. We see then that the result is the superposition of a new simple
Hence we have, for the determination of a and /3, the equations