Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_81

⚡ Mecademy AIENG정역학 · ch5  Problem Statement A surface is generated by revolving the circular arc of 0.8-m radius and subtended angle of 120° completely about the z-axis. The diameter of the neck is 0.6 m. Determine the outside area A generated. Problem 5/81 (a) Calculation of the generated surface area A 1. Formula: According to the first Theorem of Pappus-Guldinus, the area of a surface of revolution is given by: where: is the angle of revolution in radians ( for a complete revolution). is the length of the generating curve. is the perpendicular distance from the axis of revolution to the centroid of the generating curve. 2. Substitution: First, we determine the geometric properties of the generating circular arc: Radius of the arc, Subtended angle of the arc, The length of the arc is The "neck" radius (distance from the -axis to the nearest point on the arc) is The distance of the center of curvature from the -axis is A A=θ⋅⋅rˉL θθ=2π L rˉ R=0.8 m α=120= ∘ rad 3 2π L=Rα=0.8 =( 3 2π ) m 3 1.6π z r = min = 2 0.6 0.3 m Czd = C r + min R=0.3+0.8=1.1 m The distance of the centroid of the arc from its own center of curvature is given by : The distance of the arc's centroid from the -axis is Now, substitute these into the Pappus formula: 3. Calculation: Calculate numerical values: Calculate the centroid distance : Final area calculation: 4. Result: ● Final Conclusion: The outside surface area generated by the revolution of the circular arc is approximately . ✨ Final Answer Summa