Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_8

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the - and -coordinates of the centroid of the trapezoidal area. Problem 5/8 (a) x-coordinate of the centroid 1. Formula: The -coordinate of the centroid for a composite area is given by: where is the total area, and is the first moment of area with respect to the -axis. We decompose the trapezoid into a rectangle (Area 1) and a right triangle (Area 2). Area 1 (Rectangle): , with centroid Area 2 (Triangle): , with centroid 2. Substitution: Substitute the properties of the sub-areas into the expression for and the total area : 3. Calculation: Simplify the numerator : Now, calculate : xy xxˉ =xˉ = A ∑ i A ∑ i xˉ i A Q y AQ y y A = 1 ah =xˉ 1 2 a A = 2 (b− 2 1 a)h =xˉ 2 a+ (b− 3 1 a)= 3 2a+b Q y A A=A + 1 A = 2 ah+ (b− 2 1 a)h= (a+ 2 1 b)h Q = y (ah) + ( 2 a ) (b−a)h [ 2 1 ]( 3 2a+b ) Q y Q = y + 2 ah 2 6 h(b−a)(2a+b) Q = y = 6 3ah+h(2ab+b−2a−ab) 222 6 h(a+ab+b) 22 xˉ =xˉ = 2 (a+b)h 6 h(a+ab+b) 22 ⋅ 6 h(a+ab+b) 22 (a+b)h 2 4. Result: ● Final Conclusion: The -coordinate of the centroid is . (b) y-coordinate of the centroid 1. Formula: The -coordinate of the centroid is given by: where is the first moment of area with respect to the -axis. Area 1 (Rectangle): , with centroid Area 2 (Triangle): , with centroid 2. Substitution: Substitute the properties of the sub-areas into the expression for : 3. Calculation: Simplify the expression for : Now, calculate using the total area : 4. Result: ● Final Conclusion: The -coordinate of t