A Treatise on the Cycloid and All Forms of Cycloid Curves; And on the Use of Such Curves in Dealing with the Motions of Planets, Comets, &C. and of Matter Projected from the Sun

A Treatise on the Cycloid and All Forms of Cycloid Curves; And on the Use of Such Curves in Dealing with the Motions of Planets, Comets, &C. and of Matter Projected from the Sun

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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. 1878 edition. Excerpt: ...F = g, a cycloid in which the oscillations under gravity will be the same as the oscillations in the epicycloid must have a generating circle whose radius (F+B)R OC.CB (CK)2, .., =v = QB = v OB; = B 6, obtained by drawing semicircle B k O, taking B k = CK', and drawing k b perp. to BO. Corresponding considerations and constructions apply in the case of hypocycloid. It is manifest (see scholium to lemma) that if the particle in its passage along the epicycloidal, hypocycloidal, or cycloidal arc, be resisted in a degree constantly proportional to the velocity, the periods of oscillation will still be isochronous; the arc of oscillation, however, will no longer be symmetrical on either side of the axis, but will continually be reduced, each complete arc of oscillation being less than the arc last described. A weight may be caused to oscillate in the arc of an inverted cycloid in the manner indicated in fig. 77. Here a A is a string swinging between two cycloidal cheeks apD, ap' T', a being a cusp, and DD', the common tangent at the vertices D, D', being horizontal. The length of the string a A being equal to twice the axis of a p D, or to the arc apD, the weight swings in the cycloidal arc DAD' (Prop. XI. section 1). Such a pendulum would vibrate isochronously, if there were no friction and the string were weightless; but in practice the keloidal pendulum does not vibrate with perfect isochronism. An approach to isochronism is secured in the case of an ordinary pendulum by having the arc of vibration small compared with the length of the pendulum. In this case the small circular arc described by the bob may be regarded as coincident with a small portion of thecycloidal arc DAD' (fig. 75) near to A, and the isochronism thence inferred. But Fio. 77....show more

Product details

  • Paperback | 48 pages
  • 189 x 246 x 3mm | 104g
  • Rarebooksclub.com
  • Miami Fl, United States
  • English
  • black & white illustrations
  • 1236501292
  • 9781236501295