Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_77

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the surface area of one side of the bell-shaped shell of uniform but negligible thickness. Problem 5/77 (a) Determination of Surface Area ● Calculation Process 1. Present Final Formula: Using the second theorem of Pappus-Guldinus, the surface area generated by revolving a planar curve about a non-intersecting axis is: where is the length of the generating curve, is the distance from the axis of revolution to the centroid of the curve, and is the angle of revolution (in radians). For a full revolution, . Alternatively, using direct integration: 2. Identify Geometry and Substitute Values: Based on the diagram, the bell is generated by revolving a quarter-circular arc of radius about the -axis. The "neck" of the bell starts at a radius from the -axis (at ). The profile flares outwards away from the -axis, ending at . To maintain a radius of for the arc and meet these conditions, the center of the arc is at . The profile curve is parametrized by: The radius of revolution at any point is . The differential arc length is . A A=L⋅⋅rˉθ Lrˉ θ θ=2π A= 2 πr ds ∫ az azz=0,x=a zz=a a (z,x)=(a,a) z=a−acosφ,x=a+asinφfor 0≤φ≤ 2 π r=x=a(1+sinφ) ds=adφ 3. Partial Operations: Set up the integral for the surface area: Perform the integration: 4. Final Calculation: ● Final Conclusion: The surface area of one side of the bell-shaped shell is . ✨ Final Answer Summary (a) Mecademy AI Solution · ENGProblem 5/77 A= 2 π (a+ ∫ 0 π/2 asinφ)(adφ) A=