Chapter 9 – Center of Gravity and Centroid Center of gravity- the point at which the weight of the body can be concentrated. CG of 2-D body z y (x1 y1) weight = dW1 (x2 y2) weight = dW2 x y W x Fz W dW1 dW2 ...... dWn M y : x W x1 dW1 x 2 dW2 .... x n dWn M x : yW y1 dW1 y 2 dW2 .... y n dWn If we take the limit as dW 0 W dW xW x dW z W z dW yW y dW Centroids of areas and lines - homogeneous, uniform thickness dw t dA density t thickness dA differential area Substituting into M y , M x M Y : xptA x1 ptdA1 x2 ptdA2 ...... xn ptdAn M x : yptA y1 ptdA1 y 2 ptdA2 ..... y n ptdAn xA x dA yA y dA zA z dA Similarly for wires xL x dL dw pa dL a cross-sectional area dL differential length yL y dL z L z dL First moments of areas and lines The integral x dA is known as the first moment of the area A with respect to the y axis. Denoted by Qy Qy x dA xA Q x y dA yA First moments of areas extremely useful in mechanics of solids. Centroids of common shapes can be found on the inside back cover of the book. These will be useful. Centroids by integration The centroids of areas bounded by analytical curves is usually determined by computing the integrals: XA x dA Y A y dA If the element of area dA is chosen to be a small square of dx by dy, then the determination of the integrals requires a double integration. By choosing a thin rectangular strip, this can be avoided Qy XA xel dA Q x Y A y el dA y y P(x, y) P(x, y) dy xel yel dx xel x y 2 dA y dx y el yel x x xel a ax 2 yel y xel dA (a x) dy Example 1). Given: y y = mx b y = kx3 x a Find: Determine by integration, the centroid. y a A b y2 y1 yel x x y1 kx3 at pt. A y2 mx at pt. A b b y1 3 x 3 3 a a b b b ma m y2 x a a b ka3 k b b dA ( y2 y1 )dx ( x 3 x 3 )dx a a xel x y el 1 1 b b ( y 2 y1 ) ( x 3 x 3 ) 2 2 a a b a x3 b x2 x4 a 1 ab x dx a 0 a 2 4a 2 0 4 a 2 A dA a b b 3 b a 2 x4 b x3 x5 a 2 2 x dA x x x dx x dx a b el 0 a a 3 a 0 a 3 5a 2 0 15 a 2 yel dA a 0 b2 2a 2 1b b 3 b b 3 b2 x 3 x x 3 x dx 2 2a a a 2a a 2 x6 b2 x3 x7 a 2 x dx ab 2 2 4 0 a 4 2a 3 7a 0 21 a 1 2 XA xel dx X ab a 2 b 4 15 1 2 Y A y el dx Y ab ab 2 4 21 8 a 15 8 Y b 21 X a 0 x4 x 2 1 2 a dx