Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_39

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the coordinates of the centroid of the trapezoidal area shown. Problem 5/39 (a) x-coordinate of the centroid () ● Calculation Process 1. Present Final Formula: We decompose the trapezoid into a rectangle (Part 1) and a triangle (Part 2). The x-coordinate of the centroid for a composite area is given by: 2. Substitute Values: Based on the diagram, the vertices are , , , and . Part 1 (Rectangle): Width , Height . Part 2 (Triangle): Base , Height . Vertices are . Substituting into the centroid formula: xˉ =xˉ = A ∑ i A ∑ i xˉ i A +A 12 A +A 1 xˉ 12 xˉ 2 (0,0)(300,0) (600,600)(0,600) =300 mm=600 mm A = 1 300×600=180000 mm 2 =xˉ 1 = 2 300 150 mm =600−300=300 mm=600 mm (300,0),(600,600),(300,600) A = 2 × 2 1 300×600=90000 mm 2 =xˉ 2 = 3 300+600+300 400 mm =xˉ 180000+90000 180000(150)+90000(400) 3. Partial Operations: 4. Final Calculation: ● Final Conclusion: The x-coordinate of the centroid is . (b) y-coordinate of the centroid () ● Calculation Process 1. Present Final Formula: Using the same decomposition: 2. Substitute Values: Part 1 (Rectangle): Part 2 (Triangle): Substituting values: 3. Partial Operations: 4. Final Calculation: A= ∑xˉ27000000+36000000=63000000 mm 3 A= ∑270000 mm 2 =xˉ = 270000 63000000 ≈ 27 6300 233.33 mm =xˉ233 mm yˉ =yˉ = A ∑ i A ∑ i yˉ i A +A 12 A +A 1 yˉ 12 yˉ 2 =yˉ 1 = 2 600 300 mm =yˉ 2 = 3 0+600+600 400 mm =yˉ 180000+90000 180000(300)+90000(400) A = ∑yˉ54000000+36000000=90000000 mm 3 =yˉ = 270000 90