Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_5

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the -coordinate of the centroid of the shaded area. Problem 5/5 (a) Determination of the -coordinate of the centroid 1. Formula: The -coordinate of the centroid for a composite area is given by: where is the area of each component part and is the -coordinate of its centroid. 2. Substitution: We divide the trapezoid into two parts: a rectangle (Part 1) and a triangle (Part 2). Part 1 (Rectangle): Width = , Height = . Part 2 (Triangle): Base = , Height = . The centroid of the triangle relative to its own left vertical edge is . Since its left edge is at : 3. Calculation: First, calculate the total area : Next, calculate the sum of the moments of area about the -axis (): x x x =xˉ A ∑ i A ∑ i xˉ i A i xˉ i x ah A = 1 a⋅h =xˉ 1 2 a b−ah A = 2 (b− 2 1 a)h (b− 3 1 a) x=a =xˉ 2 a+ (b− 3 1 a)= 3 2a+b A A=A + 1 A = 2 ah+ (b− 2 1 a)h= = 2 h(2a+b−a) 2 h(a+b) yM = y A ∑ i xˉ i Expand the term : Substitute back into : Now, find : 4. Result: ● Final Conclusion: The -coordinate of the centroid of the trapezoidal area is . ✨ Final Answer Summary (a) Mecademy AI Solution · ENGProblem 5/5 M = y (ah) + ( 2 a ) (b−a)h [ 2 1 ]( 3 2a+b ) M = y + 2 ha 2 (b− 6 h a)(2a+b) (b−a)(2a+b) (b−a)(2a+b)=2ab+b− 2 2a− 2 ab=b+ 2 ab−2a 2 M y M = y 3a+(b+ab−2a) = 6 h [ 222 ] ( a + 6 h 2 ab+b) 2 xˉ =xˉ = A M y = (a+b) 2 h ( a +ab+ b ) 6 h 22 ⋅ 3 1 a+b a+ab+b 22 =xˉ 3(a+b) a+ab+b 22 x=xˉ 3(a+b) a+ab+b 22 =xˉ 3(a+b) a+ab+b 22