12-01 Chap Gere - 12 Review of Centroids and Moments of...

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Centroids of Plane Areas The problems for Section 12.2 are to be solved by integration. Problem 12.2-1 Determine the distances x w and y w to the centroid C of a right triangle having base b and altitude h (see Case 6, Appendix D). Solution 12.2-1 Centroid of a right triangle 12 Review of Centroids and Moments of Inertia dA 5 xdy 5 b (1 2 y / h ) dy Similarly, x 5 b 3 y 5 Q x A 5 h 3 5 bh 2 6 Q x 5 # y dA 5 # h 0 yb (1 2 y / h ) dy 5 bh 2 A 5 # dA 5 # h 0 b 2 y / h ) dy y y d y x h b O C x y x 5 b (1 2 y ) h Problem 12.2-2 Determine the distance y w to the centroid C of a trapezoid having bases a and b and altitude h (see Case 8, Appendix D). Solution 12.2-2 Centroid of a trapezoid Width of element 5 b 1 ( a 2 b ) y / h dA 5 [ b 1 ( a 2 b ) y / h ] dy 5 h ( a 1 b ) 2 A 5 # dA 5 # h 0 [ b 1 ( a 2 b ) y / h ] dy y 5 Q x A 5 h (2 a 1 b ) 3( a 1 b ) 5 h 2 6 a 1 b ) Q x 5 # y dA 5 # h 0 y [ b 1 ( a 2 b ) y / h ] dy y y d y x b a O h C y 727
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728 CHAPTER 12 Review of Centroids and Moments of Inertia Problem 12.2-3 Determine the distance y w to the centroid C of a semicircle of radius r (see Case 10, Appendix D). Solution 12.2-3 Centroid of a semicircle Q x 5 # y dA 5 # r 0 2 y Ï r 2 2 y 2 dy 5 2 r 3 3 5 p r 2 2 A 5 # dA 5 # r 0 2 Ï r 2 2 y 2 dy dA 5 2 Ï r 2 2 y 2 dy y 5 Q x A 5 4 r 3 p y y dy x O C y r 2 r 2 2 y 2 Problem 12.2-4 Determine the distances x w and y w to the centroid C of a parabolic spandrel of base b and height h (see Case 18, Appendix D). Solution 12.2-4 Centroid of a parabolic spandrel dA 5 ydx 5 hx 2 dx b 2 y 5 Q x A 5 3 h 10 Q x 5 # y / 2 dA 5 # b 0 1 2 ¢ hx 2 b 2 ¢ hx 2 b 2 dx 5 bh 2 10 x 5 Q y A 5 3 b 4 5 b 2 h 4 5 # b 0 hx 3 b 2 dx Q y 5 # x dA A 5 # dA 5 # b 0 hx 2 b 2 dx 5 bh 3 y d x x O C y y 5 hx 2 b 2 h x x b Problem 12.2-5 Determine the distances x w and y w to the centroid C of a semisegment of n th degree having base b and height h (see Case 19, Appendix D). Solution 12.2-5 Centroid of a semisegment of n th degree 5 bh 2 B n 2 ( n 1 1)(2 n 1 1) R Q x 5 # y 2 dA 5 # b 0 1 2 h ¢ 1 2 x n b n ( h ) ¢ 1 2 x n b n dx x 5 Q y A 5 b ( n 1 2( n 1 2) 5 hb 2 2 ¢ n n 1 2 Q y 5 # x dA 5 # b 0 xh ¢ 1 2 x n b n dx 5 bh ¢ n n 1 1 A 5 # dA 5 # b 0 h ¢ 1 2 x n b n dx dA 5 y dx 5 h ¢ 1 2 x n b n dx y 5 Q x A 5 hn 2 n 1 1 y dx O y 5 h (1 2 x n b n ) x b C h y x x n . 0
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SECTION 12.3 Centroids of Composite Areas 729 Centroids of Composite Areas The problems for Section 12.3 are to be solved by using the formulas for composite areas. Problem 12.3-1 Determine the distance y w to the centroid C of a trapezoid having bases a and b and altitude h (see Case 8, Appendix D) by dividing the trapezoid into two triangles. Solution 12.3-1 Centroid of a trapezoid y 5 Q x A 5 h (2 a 1 b ) 3( a 1 b ) Q x 5 a y i A i 5 2 h 3 ¢ ah 2 1 h 3 ¢ bh 2 5 h 2 6 a 1 b ) A 5 a A i 5 ah 2 1 bh 2 5 h 2 ( a 1 b ) y 2 5 h 3 A 2 5 bh 2 y 1 5 2 h 3 A 1 5 ah 2 y b a O h C C 2 C 1 y x A 1 A 2 Problem 12.3-2 One quarter of a square of side a is removed (see figure).
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12-01 Chap Gere - 12 Review of Centroids and Moments of...

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