Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_66

⚡ Mecademy AIENG정역학 · ch5  Problem Statement The circular arc is rotated through 360° about the y-axis. Determine the outer surface area S of the resulting body, which is a portion of a spherical shell. Problem 5/66 (a) Determination of the outer surface area S ● Calculation Process 1. Present Final Formula: We use the Second Theorem of Pappus-Guldinus to find the area of the surface of revolution: where: is the length of the generating curve (the circular arc). is the perpendicular distance from the centroid of the curve to the axis of revolution (-axis). is the angle of revolution in radians. 2. Substitute Values: From the given diagram: Radius of the arc, . The arc is symmetric about the -axis, spanning from to relative to the positive -axis. The total subtended angle is . Thus, the half-angle is . The angle of revolution is . First, calculate the arc length : Next, determine the distance (which corresponds to ) from the -axis to the centroid S=L⋅⋅rˉθ L rˉ y θ R=a x45 ∘ 135 ∘ y 2α=135− ∘ 45= ∘ 90= ∘ rad 2 π α=45= ∘ rad 4 π θ=360= ∘ 2π rad L L=R⋅(2α)=a⋅ 2 π xˉrˉy of the circular arc: 3. Partial Operations: Calculate the intermediate values for the centroid distance and arc length: Now, substitute these into the surface area formula: 4. Final Calculation: Simplify the expression: Evaluating numerically: ● Final Conclusion: The outer surface area of the resulting portion of the spherical shell is . ✨ Final Answer Summary (a) Mecademy AI Solution · ENGProblem 5/66 =xˉR = α si