Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_55

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the -, -, and -coordinates of the mass center of the homogeneous body shown. The hole in the upper surface is drilled completely through the object. Problem 5/55 (a) Determination of the coordinates of the mass center ● Calculation Process 1. Present Final Formula: Since the body is homogeneous, the mass center coincides with the centroid of the volume. We treat the body as a composite of three simpler volumes: a solid rectangular prism (Part 1), a negative vertical cylinder representing the hole (Part 2), and a negative horizontal semi-cylinder representing the cutout at the end (Part 3). The coordinates of the centroid are given by: 2. Substitute Values: From the diagram, the overall dimensions are , , and height (derived from the radius and the top/bottom offsets). Part 1 (Solid block): ; centroid . Part 2 (Vertical hole): . ; centroid . Part 3 (Semi-cylindrical cutout): , length (along ). The axis is at . The centroid of a semi-cylinder relative to its diameter is . ; centroid . 3. Partial Operations: Sum of volumes: xyz =xˉ , = V ∑ i V ∑ i xˉ i yˉ ,= V ∑ i V ∑ i yˉ i zˉ V ∑ i V ∑ i zˉ i W = x 50 mm L = y 25+20+30=75 mmH = z 5+10+10+5= 30 mm V = 1 50×75×30=112500 mm 3 (25,37.5,15) mm r=10 mm,h=30 mmV = 2 −π(10)(30)= 2 −3000π mm 3 (25,25+ ,15)= 2 20 (25,35,15) mm r=10 mmL=50 mm xy=75 mm,z=15 mm 3π 4r V = 3 − π(10)(50)= 2 1 2 −2500π mm 3 (25,75− ,15) mm 3π 4×10 V = ∑112500−3000π−2500π=112500−5500π≈95221.2 mm 3 Sum of m