Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_185

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the - and -coordinates of the centroid of the shaded area. Problem 5/185 (a) Determination of the -coordinate () of the centroid 1. Formula: The -coordinate of the centroid for a composite area is given by: The area is decomposed into three basic shapes: Shape 1: A rectangle of width and height (). Shape 2: A semi-circle on the right with radius . Shape 3: A semi-circular cutout on the left with radius . 2. Substitution: Areas: Individual Centroids (-coordinates): 3. Calculation: Total Area: xy xxˉ x =xˉ A ∑ i Ax ∑ ii 240 mm210 mm105×2 R=105 mm r=65 mm A = 1 240×210=50400 mm 2 A = 2 πR= 2 12 π(105)= 2 12 5512.5π mm 2 A = 3 − πr= 2 12 − π(65)= 2 12 −2112.5π mm 2 x x = 1 = 2 240 120 mm x = 2 240+ = 3π 4R 240+ = 3π 4(105) 240+ mm π 140 x = 3 0+ = 3π 4r = 3π 4(65) mm 3π 260 A= ∑50400+5512.5π−2112.5π=50400+3400π≈61081.42 Sum of First Moments (): Compute : 4. Result: ● Final Conclusion: The -coordinate of the centroid is . (b) Determination of the -coordinate () of the centroid 1. Formula: By inspection of the geometry, the shaded area is symmetric about a horizontal axis passing through its center. 2. Substitution: The total height of the object is . Each sub-component (rectangle, right semi-circle, and left semi-circular cutout) is centered vertically at the same height relative to the origin . 3. Calculation: The individual centroids are all located at . Therefore, . 4. Result: ● Final Conclusion: Due to geometric symmetry