Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_67

⚡ Mecademy AIENG정역학 · ch5  Problem Statement The area of the circular sector is rotated through about the -axis. Determine the volume of the resulting body, which is a portion of a sphere. Problem 5/67 (a) Determine the volume of the resulting body ● Calculation Process 1. Present Final Formula: According to the second theorem of Pappus-Guldinus, the volume of a body of revolution is equal to the product of the generating area , the angle of rotation (in radians), and the distance from the axis of rotation to the centroid of the generating area: 2. Substitute Values: The generating area is a circular sector of radius and total angle (since it spans from to relative to the -axis). Thus, rad. The area of the sector is . The centroid of a circular sector is located on its axis of symmetry at a distance from the vertex. Here, the axis of symmetry is the -axis, so : The rotation is about the y$-axis, so rad. Substituting these into the volume formula: 3. Partial Operations: Multiply and : 180 ∘ y V VA θxˉ V=θAxˉ a2α=60 ∘ −30 ∘ +30 ∘ xα=30= ∘ 6 π A=aα= 2 a = 2 ( 6 π ) 6 πa 2 =rˉ 3α 2asinα x=xˉrˉ =xˉ = 3(π/6) 2asin(30) ∘ = π/2 2a(1/2) = π/2 a π 2a 180 ∘ θ=180= ∘ π V= ( π ) ( π 2a )( 6 πa 2 ) θxˉπ⋅ = π 2a 2a 23 Multiply the result by the area : 4. Final Calculation: Simplify the fraction: ● Final Conclusion: The volume of the resulting body is . ✨ Final Answer Summary (a) Mecademy AI Solution · ENGProblem 5/67 A2a⋅ = 6 πa 2 6 2πa 3 V= 3 πa 3 V= 3 πa 3 V= 3 πa 3