Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_64

⚡ Mecademy AIENG정역학 · ch5  Problem Statement An opening is formed in the thin cylindrical shell. Determine the -, -, and -coordinates of the mass center of the homogeneous body. Problem 5/64 (a) -coordinate of the mass center 1. Formula: The body is a composite thin shell. The -coordinate of the mass center is determined using the area centroid formula for composite bodies: where subscript 1 refers to the complete cylindrical shell and subscript 2 refers to the removed opening (sector). 2. Substitution: For the complete shell: , . For the opening: Area . Centroid of the opening's arc: . Net area . Substitute into the formula: 3. Calculation: 4. Result: xyz x x =X ˉ A −A 12 A −A 1 xˉ 12 xˉ 2 A = 1 2πRL =xˉ 1 0 A = 2 R =( 4 π )( 8 3L ) 32 3πRL =xˉ 2 R sin θ d θ = π/4 1 ∫ 0 π/4 (1− π 4R cos )= 4 π (1− π 4R )= 2 2 π 2R(2− )2 A=A − 1 A = 2 2πRL− = 32 3πRL 32 61πRL =X ˉ 32 61πRL (2 πR L)(0)− ( 32 3πRL )( π 2R(2− )2 ) =X ˉ = 32 61πRL − ⋅ 32 3πRL π 2R(2−)2 − = 61πRL 3 R L⋅2 R (2− ) 2 − 61π 6 R (2− ) 2 =xˉ − 61π 6R(2−)2 ● Final Conclusion: The -coordinate of the mass center is . The negative sign indicates a shift away from the quadrant where the opening was removed. (b) -coordinate of the mass center 1. Formula: Similar to part (a), use the composite area centroid formula for the -axis: 2. Substitution: Full shell centroid: . Opening centroid: . Substitute the known areas and centroids: 3. Calculation: Numerator: Denominator: 4. Result: ● Final Conclusion: The -coordinate of the mas