Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_186

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the area of the curved surface of the solid of revolution shown. Problem 5/186 (a) Determination of the curved surface area The area of a surface of revolution is determined using the Theorem of Pappus-Guldinus. The surface is generated by revolving the quarter-circular arc of radius about the -axis through an angle of . 1. Formula: The formula for the surface area generated by revolving a curve is: where: is the angle of revolution in radians. is the radial distance from the axis of revolution to the centroid of the generating curve. is the length of the generating curve. 2. Substitution: From the geometric properties shown in the diagram: The angle of revolution is . The length of the generating quarter-circular arc of radius is . The generating arc has its inner endpoint at a distance from the -axis. Given its radius is , its center of curvature is also at a distance from the -axis. The centroid of a quarter-circular arc relative to its center is located at . Thus, the distance from the -axis to the centroid is: 3. Calculation: ABCD ABCDABa z90 ∘ A A=θLrˉ θ rˉ L θ=90= ∘ rad 2 π ABaL= (2πa)= 4 1 2 πa ABAaz aaz π 2a z =rˉa+ π 2a Substitute the values into the area formula: Multiply the terms: Distribute : Factor out common terms: 4. Result: ● Final Conclusion: The area of the curved surface is . ✨ Final Answer Summary (a) The area of the curved surface is . Mecademy AI Solution · ENGProblem 5/186 A= a+ ( 2 π )( π 2a )(