Practical engineering drawing and third angle projection, for students in scientific, technical and manual training schools and for ..draughtsmen .. . curve is particularlj^ interesting as expressing the law of the vibration of perfectly ? elasticsohds; of the vibratory movement of a particle acted upon by a force which varies directly as thedistance from the origin; apiDroximately, the vibratory movement of a pendulum; and exactly thelaw of vibration of the so-called mathematical pendulum.* 172. From the symmetry of thesinusoid with respect to R R^ and to0 we have area TAO R= E 0 0 R,;adding
Practical engineering drawing and third angle projection, for students in scientific, technical and manual training schools and for ..draughtsmen .. . curve is particularlj^ interesting as expressing the law of the vibration of perfectly ? elasticsohds; of the vibratory movement of a particle acted upon by a force which varies directly as thedistance from the origin; apiDroximately, the vibratory movement of a pendulum; and exactly thelaw of vibration of the so-called mathematical pendulum.* 172. From the symmetry of thesinusoid with respect to R R^ and to0 we have area TAO R= E 0 0 R,;adding area D E L 0 R to both mem-bers we have the area between thesinusoid and T D and D E equal tothe rectangle R E, or one-half the rect-angle D E K T; or to -, ir r x 3 r = IT r ^, the area of the rolling circle. As T A C E is but half of the entire sinusoid it is evident that the total area below the curveis twice that of the generating circle. The area between the cycloid and its companion remains to be determined, but is readilyascertained by noting that as any jjoint of the latter, as A, is on the vertical diameter of the circle ^-igr- * Wood. Elements of Co-ordinate Geometry, p. 209. 58 THEORETICAL AND PRACTICAL GRAPHICS. passing through the then position of the tracing point, as a, the distance, A a, between the twocurves at any level, is merely the semi-chord of the rolhng circle at that level. But this, evidently,equals Ms, the semi-chord at the same level on the equal circle. The equality of Ms and A amakes the elementary • rectangles MsSim^ and A A^ a^ a equal; and considering all the possiblesimilarly constructed rectangles of infinitesimal altitude, the sum of those on semi-chords of therolling circle would equal the area of the semi-circle TDy, which is therefore the extent of the areabetween the two curves under consideration. The figure showing but half of a cycloid, the total area between it and its companion mustbe that of the rolling circle. Adding this t
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