The Messenger of Mathematics Volume 8

The Messenger of Mathematics Volume 8

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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1879 edition. Excerpt: ...8. When this wave of contraction reaches B, the contraction of the lowest element will be immediately relaxed, and it will commence to shoot out with a velocity 2 V, which will be communicated in succession to every element of the string, and the stress in the lowest part over which this upgoing wave has passed will be a tension equal to that due to the weight of the string below each oint. To show this, let the unstretched liengths of BP, PQ be ac, dz, and let the contraction and extension of PQ before and after the passage of the wave be df, and a'E, . Assuming the distribution of stress to be that given above the energy surrendered by PQ during the passage of the wave will be Ql (l--: r)"--z) dz: (l'--2lz) dz. 27L 27. The work done by gravity on the string is of two parts, (1) that done on the whole string, considered as rigidly moving with the original velocity V, during the passage of the wave over PQ, (2) that due to the separation of the parts CQ, BP from each other. The velocity of UQ is V, and the time occupied by the wave in passing over PQ= $67-C, and hence the distance through which C' Q falls in this time =-L-55%; the part (1) of the work due to gravity is thus equal If, therefore, V, be the new velocity acquired by PQ, the only change of kinetic energy experienced by the whole string during the passage of the wave over it will be 9. When this wave of extension reaches C' it will be succeeded by a wave of contraction down the string corresponding to velocity 3V, and thereafter by a series of wormlike movements whose velocities, beginning with the first, are V, 2V, 3V, 4V, &c. Suppose, for instance, that a wave of extension has just reached C, the velocity corresponding to it being nV, it will more

Product details

  • Paperback | 32 pages
  • 189 x 246 x 2mm | 77g
  • United States
  • English
  • black & white illustrations
  • 1236891759
  • 9781236891754