Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_4

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Specify the -, -, and -coordinates of the mass center of the quadrant of the homogeneous solid cylinder. The length of the cylinder is and the radius of its cross-section is . Problem 5/4 (a) The x-coordinate of the mass center () 1. Formula: For a homogeneous body with a constant cross-section oriented along the -axis, the centroidal -coordinate is located at the geometric center of its length : 2. Substitution: From the problem diagram, the length along the -axis is : 3. Calculation: 4. Result: ● Final Conclusion: The -coordinate of the mass center is . (b) The y-coordinate of the mass center () 1. Formula: The centroid of a quarter-circular area of radius is located at a distance from the center of curvature defined by: xyz 240 mm 120 mm xˉ xxL =xˉ 2 L xL= 240 mm =xˉ 2 240 mm =xˉ120 mm 120 mm x120 mm yˉ r =yˉ 3π 4r 2. Substitution: The radius of the cylinder is given as : 3. Calculation: Numerator: Denominator: Result: 4. Result: ● Final Conclusion: The -coordinate of the mass center is (or ). (c) The z-coordinate of the mass center () 1. Formula: Due to the symmetry of the quarter-circular cross-section in the - plane, the formula for the -coordinate is identical to that for the -coordinate: 2. Substitution: Using the radius : 3. Calculation: 4. Result: ● Final Conclusion: The -coordinate of the mass center is (or ). ✨ Final Answer Summary (a) (b) r=120 mm =yˉ 3π 4(120 mm) 4×120=480 3π = 3π 480 ≈ π 160 50.9296 mm 50.9 mm y50.9