Analytical mechanics for students of physics and engineering . Fig. ILLUSTRATIVE EXAMPLE Find the center of mass of a parab-oloid of revolution obtained by revolv-ing about the x-axis that part of theparabola y2 = 2 px which lies betweenthe lines x = 0 and x = a. dm = T7T//- dx= T7r 2 px dx; 2tttpC ..•- dx. 2irrp ( xdx = %a. PROBLEMS. Find the center of mass of the homogeneous Bolid of revolution gener-ated by revolving about the .r-axis the area bounded by (1) y = -x, x = h, and y = 0. h (2) x2 = 4ay, x = 0, and y = a. (3) x- + y- = a2, and x = 0. (4) b2x2 + /2 = 0s, x- + //- - //-, and I - 0
Analytical mechanics for students of physics and engineering . Fig. ILLUSTRATIVE EXAMPLE Find the center of mass of a parab-oloid of revolution obtained by revolv-ing about the x-axis that part of theparabola y2 = 2 px which lies betweenthe lines x = 0 and x = a. dm = T7T//- dx= T7r 2 px dx; 2tttpC ..•- dx. 2irrp ( xdx = %a. PROBLEMS. Find the center of mass of the homogeneous Bolid of revolution gener-ated by revolving about the .r-axis the area bounded by (1) y = -x, x = h, and y = 0. h (2) x2 = 4ay, x = 0, and y = a. (3) x- + y- = a2, and x = 0. (4) b2x2 + /2 = 0s, x- + //- - //-, and I - 0. 146 ANALYTICAL MECHANICS 117. Center of Mass of Filaments. — The transverse dimen-sions of a filament are supposed to be negligible; thereforeit can be treated as a geometrical curve. Taking a piece oflength ds as the element of mass and denoting the mass perunit length by X we have dm = X ds. ILLUSTRATIVE EXAMPLE. Find the center of mass of a semicircular filament. (a) Taking x2 + y2 = a2 to be the equation of the circle we get dm = \ils = x« Jo Va2 - .r2 /•a dx Jo Va2 - x2 dx Va2 - x2dx -V^2 sin-1 -a
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