motion. Harmonic Harmonic Motion, which is of such immense use, not only in ordinary kinetics, but in the theories of sound, light, heat, etc., that we make no apology for entering here into considerable detail regarding it. Simple harmonic motion. A 53. Def. When a point Q moves uniformly in a circle, the perpendicular QP drawn from its position at any instant to a fixed diameter AA' of the circle, intersects the diameter in a point P, whose position changes by a simple harmonic motion. A' 0 Thus, if a planet or satellite, or one of the constituents of a double star, be sup posed to move uniformly in a circular orbit about its primary, and be viewed from a very distant posi tion in the plane of its orbit, it will appear to move backwards and forwards in a straight line with a simple harmonic motion. This is nearly the case with such bodies as the satellites of Jupiter when seen from the earth. Physically, the interest of such motions consists in the fact of their being approximately those of the simplest vibrations of sounding bodies, such as a tuning-fork or pianoforte wire; whence their name; and of the various media in which waves of sound, light, heat, etc., are propagated. 54. The Amplitude of a simple harmonic motion is the range on one side or the other of the middle point of the course, i.e., OA or OA' in the figure. An arc of the circle referred to, measured from any fixed point to the uniformly moving point Q, is the Argument of the harmonic motion. The distance of a point, performing a simple harmonic motion, from the middle of its course or range, is a simple harmonic function of the time. The argument of this function is what we have defined as the argument of the motion. The Epoch in a simple harmonic motion is the interval of time which elapses from the era of reckoning till the moving point first comes to its greatest elongation in the direction reckoned as positive, from its mean position or the middle of its range. Epoch in angular measure is the angle described on the circle of reference in the period of time defined as the epoch harmonic The Period of a simple harmonic motion is the time which Simple elapses from any instant until the moving point again moves motion. in the same direction through the same position. The Phase of a simple harmonic motion at any instant is the fraction of the whole period which has elapsed since the moving point last passed through its middle position in the positive direction. harmonic mechanismi. 55. Those common kinds of mechanism, for producing recti- Simple lineal from circular motion, or vice versa, in which a crank motion in moving in a circle works in a straight slot belonging to a body which can only move in a straight line, fulfil strictly the definition of a simple harmonic motion in the part of which the motion is rectilineal, if the motion of the rotating part is uniform. The motion of the treadle in a spinning-wheel approximates to the same condition when the wheel moves uniformly; the approximation being the closer, the smaller is the angular motion of the treadle and of the connecting string. It is also approximated to more or less closely in the motion of the piston of a steam-engine connected, by any of the several methods in use, with the crank, provided always the rotatory motion of the crank be uniform. in S. H. 56. The velocity of a point executing a simple harmonic Velocity motion is a simple harmonic function of the time, a quarter of a motion. period earlier in phase than the displacement, and having its maximum value equal to the velocity in the circular motion by which the given function is defined. For, in the fig. of § 53, if V be the velocity in the circle, it may be represented by OQ in a direction perpendicular to its own, and therefore by OP and PQ in directions perpendicular to those lines. That is, the velocity of P in the simple harmonic motion is PQ; which, when P is at O, becomes V. 0Q in S. H. 57. The acceleration of a point executing a simple harmonic Acceleration motion is at any time simply proportional to the displacement motion. from the middle point, but in opposite direction, or always towards the middle point. Its maximum value is that with which a velocity equal to that of the circular motion would. be acquired in the time in which an arc equal to the radius is described. Acceleration in S. H. motion. For, in the fig. of § 53, the acceleration of Q (by § 35, a) is along Q0. Supposing, for a moment, QO to represent the magnitude of this acceleration, we may resolve it in QP, PO. The acceleration of P is therefore represented on the same V2 PO Va scale by PO. Its magnitude is therefore = PO, Q09 which is proportional to PO, and has at A its maximum value, , an acceleration under which the velocity V would be Q0 acquired in the time as stated. Let a be the amplitude, e the epoch, and T the period, of a simple harmonic motion. Then if s be the displacement from middle position at time t, we have Composition of S. H. M. in one line. 58. Any two simple harmonic motions in one line, and of one period, give, when compounded, a single simple harmonic motion; of the same period; of amplitude equal to the diagonal of a parallelogram described on lengths equal to their amplitudes measured on lines meeting at an angle equal to their difference of epochs; and of epoch differing from their epochs by angles equal to those which this diagonal makes Composition S R Α' P A P C B with the two sides of the parallelo- B' of S. H. M. in one line. 59. The construction described in the preceding section exhibits the resultant of two simple harmonic motions, whether of the same period or not. Only, if they are not of the same period, the diagonal of the parallelogram will not be constant, but will diminish from a maximum value, the sum of the component amplitudes, which it has at the instant when the phases of the component motions agree; to a minimum, the difference of those amplitudes, which is its value when the phases differ by half a period. Its direction, which always must be nearer to the greater than to the less of the two radii constituting the sides of S. H. M. Composition of the parallelogram, will oscillate on each side of the greater in one line. radius to a maximum deviation amounting on either side to the angle whose sine is the less radius divided by the greater, and reached when the less radius deviates more than this by a quarter circumference from the greater. The full period of this oscillation is the time in which either radius gains a full turn on the other. The resultant motion is therefore not simple harmonic, but is, as it were, simple harmonic with periodically increasing and diminishing amplitude, and with periodical acceleration and retardation of phase. This view is most appropriate for the case in which the periods of the two component motions are nearly equal, but the amplitude of one of them much greater than that of the other. To express the resultant motion, let s be the displacement at time t; and let a be the greater of the two component halfamplitudes. The maximum value of tan in the last of these equations is found by making +=+sin, and is equal to a and hence the maximum value of itself is sina a a' (a2 — a ́2) ! ' The geometrical methods indicated above (§ 58) lead to this conclusion by the following very simple construction. To find the time and the amount of the maximum acceleration or retardation of phase, let CA be the greater half-amplitude. From A as centre, with AB the less half amplitude as radius, describe a circle. CB touching this circle is the generating radius of the most deviated resultant. Hence CBA is a right angle; and sin BCA= AB |