Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_181

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the x- and y-coordinates of the centroid of the shaded area. Problem 5/181 (a) x-coordinate of the centroid ● Calculation Process 1. Formula: The x-coordinate of the centroid for a composite area is given by: where is the area of the large rectangle and is the area of the triangular cutout. 2. Substitution: For the rectangle (Part 1): Width Height Area Centroid For the triangle (Part 2 - Cutout): Base Height Area Centroid Substituting these into the formula: 3. Calculation: Numerator: =X ˉ = A ∑ i A ∑ i xˉ i A −A 12 A −A 1 xˉ 12 xˉ 2 A 1 A 2 w=30 mm+270 mm=300 mm h=30 mm+120 mm=150 mm A = 1 300×150=45000 mm 2 =xˉ 1 = 2 300 150 mm b=270 mm h=120 mm A = 2 × 2 1 270×120=16200 mm 2 =xˉ 2 30+ (270)= 3 1 120 mm =X ˉ 45000−16200 45000(150)−16200(120) 6,750,000−1,944,000=4,806,000 mm 3 Denominator (Total Area): Final Division: 4. Result: ● Final Conclusion: The x-coordinate of the centroid is . (b) y-coordinate of the centroid ● Calculation Process 1. Formula: The y-coordinate of the centroid for a composite area is: 2. Substitution: For the rectangle (Part 1): Centroid For the triangle (Part 2 - Cutout): Centroid Substituting values: 3. Calculation: Numerator: Denominator (Total Area): Final Division: 4. Result: ● Final Conclusion: The y-coordinate of the centroid is . ✨ Final Answer Summary (a) 45000−16200=28800 mm 2 =X ˉ = 28800 4,806,000 166.875 mm =xˉ166.9 mm 166.9 mm =Y ˉ = A ∑ i A ∑ i yˉ i A −A 12 A −A 1 yˉ 12 yˉ 2 =yˉ 1