An Elementary Treatise on Mechanics Volume 2

An Elementary Treatise on Mechanics Volume 2

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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. 1833 edition. Excerpt: ... sphere with reference to an axis passing through its centre. Ifthe sphere be cut by a plane EE' perpendicular to the fixed axis AB (Fig. 208), the section will be a. circle whose centre will be found at the point D. Denote by z the absciss AD of this section, and by y the ordinate DE, or the radius of the section. The moment of inertia of the area of this circle taken with reference to the axis AB, will be expressed (Art. 590) by "3/' 3 and if this expression be multiplied by da=DD', the product, wfy'd.z', will express the moment of inertia of the elementary volume EE'F'F bounded by parallel planes drawn through the consecutive points D and D'. The integral of this expression, being taken between the limits.z=0 and a'=AB=2r, will give the moment of inertia of the entire sphere. But by the property of the circle, we have. ' ='2rx-x'; and therefore, fy'dz=%f(2rx-1v2)'da' =centsrf(2r'r'-2111: ' + x")dz;_ or, _/"%centsry'dx=centsrx"( r2- rx-l_ T' .r')+C. The constant C will be equal to zero, since the moment is zero when a=0: and by making a=2r, we obtain for the moment of the whole sphere, rr""5 These examples are sufficient to explain the manner in wh-ich the determination of the moment of inertia is reduced to a simple problem of the integral. calculus. 592. When the moment of inertia of any body with reference to an axis passing through its centre of gravity has been determined, its moment with respect to a parallel axis is readily found. I For let GF and CK (Fig. 209) represent two parallel axes, the firstshow more

Product details

  • Paperback | 60 pages
  • 189 x 246 x 3mm | 127g
  • United States
  • English
  • black & white illustrations
  • 1236817281
  • 9781236817280