. Differential and integral calculus. dx y. â â¢>X X,= £A$) dy 2 yas Since 2y = chord AC, we see thatthe center of gravity of a circular arcis on its radius of symmetry and at adistance from its center equal to thefourth proportional between the arc, radius and chord. 244. 7b find the center of gravity of a circular dv = d2A = dxdy; hence, rx: 4 n xdxdy V*2 _ ** V««=P xdxdy 381 Mechanical Applications2 j V#2 â x?xdx Jx ~ A -f(^2-^A = %(a*-*y A ira2If the sector is a semicircle then A = â, and # = = o. 2 *, = 40 3^ 245. To find the center of gravity of the area bounded by apa
. Differential and integral calculus. dx y. â â¢>X X,= £A$) dy 2 yas Since 2y = chord AC, we see thatthe center of gravity of a circular arcis on its radius of symmetry and at adistance from its center equal to thefourth proportional between the arc, radius and chord. 244. 7b find the center of gravity of a circular dv = d2A = dxdy; hence, rx: 4 n xdxdy V*2 _ ** V««=P xdxdy 381 Mechanical Applications2 j V#2 â x?xdx Jx ~ A -f(^2-^A = %(a*-*y A ira2If the sector is a semicircle then A = â, and # = = o. 2 *, = 40 3^ 245. To find the center of gravity of the area bounded by aparabola, its axis and one of its Fig. f = 2px be the equation of the parabola; then xdxdy I I xdxdy Jo Jo and fi = A A V2 p I xidx Jo 2 \j2~pX% A l-A \ \ ydxdyJo Jo Jo Jo ydxdy A A p j xdxA Px*2A 382 Integral Calculus But, § 215, A = I I dxdy â I I ^dxdy = V2/ / ofrdxJo Jo Jo Jo Jo 2 ^2~pX% 3Hence, xx = %x and yx = %y. = \xy. 246. To find the center of gravity of a parabolic is, to find the center of gravity of OBC, Fig. , Jo Jo Jo Jo op Jo f Xl~ A A A 4ofA 1 1 ydydx 1 1 ydydx â 1 ysdyJo Jo Jo Jo 2PJ0 y }l A A A 8/^ But A = OBC= OABC-OAB; , A = xy â § xy = \xy. Hence, xx = t3q x, and yx â %y. It will be observed from the limits of integration that the areaOB C is supposed to be generated by a line II to X moving inthe direction of F, the line being limited by the F-axis andthe curve. The student may derive the same result by pro-ceeding as in previous articles. 247. To find the center of gravity of a Pyramid or § 210 (b), we have dV=Adx\ I xdV I Axdx hence, xx = =â
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