Synopsis
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1900 Excerpt: ...of what are termed harmonic equations. As Q moves away from O in either direction, its velocity and acceleration are of opposite sign; it must therefore come to rest. It will next move toward O, the speed becoming numerically greatest at the origin; it vibrates or oscillates about O as a centre, and the motion is evidently symmetrical with reference to that point. Let the velocity at O be ± vl; and the extreme distance from the centre be ± a;,; the latter is called the amplitude of the vibration. Describe a circle with radius a around the centre O, and let a point R (Fig. 19) move in it so that the radius OR turns with constant angular velocity o. The acceleration of R is--u?Xu and being proportional to.RO it may be represented by that line. Consequently the projection of RO upon X represents the acceleration of R parallel to X, which is therefore--a2z, x being the coordinate of R. Provided that we take o2 = fc, it is always possible to make Q the projection of R, and regard simple harmonic motion in a line as a projection of uniform circular motion. If Q is to be continually the projection of R, the points must leave A at the same instant. At that position the velocity parallel to X is zero for each point; and these-velocities remain equal because px has always the same value for both points if u? = k. Then the centre of the circle will be the centre of the vibration, the radius will be the amplitude, and the square of the angular velocity will be the constant of the harmonic equation (cf. § 143). Conveniently for many purposes time is assumed zero for the position A. Then the angle BOA = at, increasing indefinitely with the time as Q goes on repeating its vibration. The position of Q at any time is determined by the equation x = xl cos (af)...
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