ch12-1 - 12 Review of Centroids and Moments of Inertia...

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Differential Equations of the Deflection Curve The problems for Section 12.2 are to be solved by integration. Problem 12.2-1 Determine the distances x and y 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 x dy b (1 y h ) dy Similarly, x b 3 y Q x A h 3 bh 2 6 Q x y dA h 0 yb (1 y h ) dy bh 2 A dA h 0 b (1 y h ) dy y y d y x h b O C x y x b (1 y ) h Problem 12.2-2 Determine the distance y 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 b ( a b ) y h dA [ b ( a b ) y h ] dy h ( a b ) 2 A dA h 0 [ b ( a b ) y h ] dy y Qx A h (2 a b ) 3( a b ) h 2 6 (2 a b ) Q x y dA h 0 y [ b ( a b ) y h ] dy y y d y x b a O h C y
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2 CHAPTER 12 Review of Centroids and Moments of Inertia Problem 12.2-3 Determine the distance y 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 y dA r 0 2 y r 2 y 2 dy 2 r 3 3 r 2 2 A dA r 0 2 r 2 y 2 dy dA 2 r 2 y 2 dy y Q x A 4 r 3 y y d y x O C y r 2 r 2 y 2 Problem 12.2-4 Determine the distances x and y 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 ydx hx 2 dx b 2 y Q x A 3 h 10 Q x y 2 dA b 0 1 2 ¢ hx 2 b 2 ¢ hx 2 b 2 dx bh 2 10 x Q y A 3 b 4 b 2 h 4 b 0 hx 3 b 2 dx Q y x dA A dA b 0 hx 2 b 2 dx bh 3 y d x x O C y y hx 2 b 2 h x x b Problem 12.2-5 Determine the distances x and y 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 bh 2 B n 2 ( n 1)(2 n 1) R Q x y 2 dA b 0 1 2 h ¢ 1 x n b n ( h ) ¢ 1 x n b n dx x Q y A b ( n 1) 2( n 2) hb 2 2 ¢ n n 2 Q y x dA b 0 xh ¢ 1 x n b n dx bh ¢ n n 1 A dA b 0 h ¢ 1 x n b n dx dA y dx h ¢ 1 x n b n dx y Q x A hn 2 n 1 y d x O y h (1 x n b n ) x b C h y x x n 0
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SECTION 12.3 Centroids of Composite Areas 3 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 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 Q x A h (2 a b ) 3( a b ) Q x a y i A i 2 h 3 ¢ ah 2 h 3 ¢ bh 2 h 2 6 (2 a b ) A a A i ah 2 bh 2 h 2 ( a b ) y 2 h 3 A 2 bh 2 y 1 2 h 3 A 1 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). What are the coordinates x and y of the centroid C of the remaining area? Solution 12.3-2 Centroid of a composite area y y x O C a 2 a 2 a 2 a 2 x PROBS. 12.3-2 and 12.5-2 x y Qx A 5 a 12 Q x a y i A i 3 a 4 ¢ a 2 4 a 4 ¢ a 2 2 5 a 3 16 A a A i 3 a 2 4 y 2 a 4 A 2 a 2 2 y 1 3 a 4 A 1 a 2 4 y a O x A 2 A 1 a 2 a 2 a 2 a 2
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4 CHAPTER 12 Review of Centroids and Moments of Inertia Solution 12.3-3 Centroid of a channel section a 6 in. b 1 in.
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