The Elementary Part of a Treatise on the Dynamics of a System of Rigid Bodies; Being Part I. of a Treatise on the Whole Subject Volume 1

The Elementary Part of a Treatise on the Dynamics of a System of Rigid Bodies; Being Part I. of a Treatise on the Whole Subject Volume 1

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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. 1897 edition. Excerpt: ...of gravity of a moving body) referred to the moving axes OA, OB, OC. Let u, v, w be the components of the velocity of 0 parallel to the same axes and X, Y, Z the components of the accelerations. Then since both the resultant velocity and the resultant acceleration are vectors or directed quantities u =-j--yd3 + z6t, X =-r: -v63 + wdi, dz dw n w =-j--xdv + ydj, Z =-j--u02 + v0l. These results will be useful afterwards. The demonstration here given of the fundamental theorem on moving axes is founded on the method used by Prof. Slesser in the Quarterly Journal, 1858, to prove u, =w(. Another very simple proof is given in the chapter on moving axes at the beginning of vol. n. of this treatise. 252. Euler's dynamical equations. To determine the general equations of motion of a body moving about a fixed point 0. Let x, y, z be the coordinates of any particle m referred to axes Ox, Oy, Oz fixed in space. Taking moments about the axis of z we have by D'Alembert's principle "Em (xy--yx) = K. Let tox, toy, ton be the angular velocities of the body about the axes, then x = toyZ--azy, y = (OfX--toxZ, z = oxy--coyX;.'. x = zwy--ytbz + toy (toxy--toyx)--toz (togX--toxZ), y = xaz--Ztox + oz (wyZ--tozy)--ax (axy--ayx). These we shall presently substitute in the equation of moments. Let w, a)2, -, be the angular velocities of the body about three rectangular axes OA, OB, 00 fixed in the body. Let these coincide with the axes fixed in space at the time t; then o, = ax, o2 = toy, to, = oz; a1 = ax, mt = toy, d3 = tot, by Art. 249. The advantage of using axes fixed in the body is that the moments and products of inertia are then constants. If we choose as these axes of coordinates the principal axes at the fixed point, we have the additional...show more

Product details

  • Paperback | 176 pages
  • 189 x 246 x 10mm | 327g
  • Rarebooksclub.com
  • Miami Fl, United States
  • English
  • black & white illustrations
  • 1236534395
  • 9781236534392