Center of Gravity, Centroid (example continued)
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The element of area, dA,
= dx dy or dy dx. Select
integration in the y-direction first since it is the easiest
to evaluate. The limits of integration in the y-direction are
from ½ x to
2x – x2/6 .
Calculate the total area, A. x = 9 y =
2x – x2/6 x = 9 dA = dy dx so A
= ∫ ∫ dy dx = ∫
( 2x – x2/6 – ½ x ) dx x = 0
y = ½ x x = 0 The result is
A = 81/4
in2 . Next calculate
∫ ∫ y dA =
∫ ∫ y dy dx
x = 9 y =
2x – x2/6 x =
9 ycg A =
∫ ∫ y dy dx = ∫
[ ( 2x – x2/6 )2 – ( ½ x )2 ]/2 dx x = 0 y = ½ x x
= 0 x = 9 9 ycg A =
∫ [ 15/8 x2 + x4/72 - x3/3
] dx =
5/8 x3 + x5/360 - x4/12 | x = 0 0 ycg A =
[ (9)3 / 2 ][ 5/4 + 81/180 – 3/2 ] =
81(36)/40 in3 So ycg = [ 81 (36)/40 ]
/ (81/4) = 36/10
= 3.60 in. (result) |
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