Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_82

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Find the volume of the solid generated by revolving the shaded area about the -axis through . Problem 5/82 (a) Calculation of the Volume 1. Formula: According to the second theorem of Pappus-Guldinus, the volume of a solid of revolution is given by the product of the revolving angle (in radians), the area of the generating shape, and the perpendicular distance from the axis of revolution to the centroid of the area: 2. Substitution: From the geometric properties of the shaded quarter-circular area: Revolving angle: Area of the quarter-circle of radius : The distance from the center of the quarter-circle to its centroid is . Since the center of the quarter-circle is located at distance from the -axis, the distance to the centroid is: Substitute these into the volume formula: 3. Calculation: Expand the expression: First, multiply the constants and the area term: Distribute the terms: Vz 90 ∘ V θ Arˉ V=θ⋅A⋅rˉ θ=90= ∘ rad 2 π rA= πr 4 12 3π 4r az =rˉa+ 3π 4r V= ⋅ ( 2 π ) πr ⋅ ( 4 1 2 )a+ ( 3π 4r ) V= a+ 8 πr 22 ( 3π 4r ) V= + 8 πra 22 ( 8 πr 22 )( 3π 4r ) Simplify the second term: 4. Result: ● Final Conclusion: The volume of the solid generated by revolving the quarter-circular area about the -axis through is , which can also be written as . ✨ Final Answer Summary (a) Mecademy AI Solution · ENGProblem 5/82 = 8⋅3π πr⋅4r 22 = 24π 4πr 23 6 πr 3 V= + 8 πar 22 6 πr 3 z90 ∘ V= + 8 πar 22 6 πr 3 V= (3πa+ 24 πr 2 4r) V= + 8 πar 22 6 πr 3