Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_45

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Calculate the -coordinate of the centroid of the shaded area. Problem 5/45 (a) y-coordinate of the centroid ● Calculation Process 1. Present Final Formula: The -coordinate of the centroid for a composite area is determined by the formula: where is the area of each component part and is the -coordinate of the centroid of that part. For this problem, we treat the area as a solid semicircle (Part 1) minus a rectangular cutout (Part 2). 2. Substitute Values: Identify the geometric properties for each part based on the provided diagram: Part 1: Semicircle Radius: Area: Centroid: Part 2: Rectangular Cutout Width: Height: Area: Centroid: y y =yˉ A ∑ i A ∑ i yˉ i A i yˉ i y =yˉ A −A 12 A −A 1 yˉ 12 yˉ 2 r=74 mm A = 1 πr= 2 1 2 π(74) 2 1 2 =yˉ 1 = 3π 4r 3π 4(74) b=32+32=64 mm h=32 mm A = 2 b⋅h=64⋅32 =yˉ 2 = 2 h = 2 32 16 mm Substituting into the formula: 3. Partial Operations: Calculate individual components: First Moment of Area (Part 1): First Moment of Area (Part 2): Net Area: Net First Moment: 4. Final Calculation: ● Final Conclusion: The -coordinate of the centroid for the shaded area is . ✨ Final Answer Summary (a) Mecademy AI Solution · ENGProblem 5/45 =yˉ π(74)−(64⋅32) 2 1 2 π(74) − 64⋅32 16 [ 2 12 ][ 3π 4(74) ][][] A = 1 0.5⋅π⋅5476≈8601.50 mm 2 A = 1 yˉ 1 r= 3 23 (74)= 3 23 (405224)≈ 3 2 270149.33 mm 3 A = 2 2048 mm 2 A = 2 yˉ 2 2048⋅16=32768 mm 3 A=∑8601.50−2048=6553.50 mm 2 A =∑yˉ270149.33−32768=237381.33 mm 3 =yˉ ≈ 6553.50 2373