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

This preview shows pages 1–4. Sign up to view the full content.

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 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 Qx 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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2 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 d y 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 d x O y 5 h (1 2 x n b n ) x b C h y x x n . 0
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 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).

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 03/30/2008 for the course CVEN 205 taught by Professor Muliana during the Spring '08 term at Texas A&M.

### Page1 / 14

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online