Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_78

⚡ Mecademy AIENG정역학 · ch5  Problem Statement A thin shell, shown in section, has the form generated by revolving the arc about the z-axis through 360°. Determine the surface area A of one of the two sides of the shell. Problem 5/78 (a) Determination of the surface area A ● Calculation Process 1. Identify Properties of the Generating Curve: The generating curve is a circular arc of radius that subtends a total angle of . Length of the arc: . Distance from the center of curvature to the centroid of the arc: . 2. Determine the Centroidal Distance from the Axis of Revolution: From the given diagram, the center of the arc's circle is located at a distance from the -axis. The arc itself is positioned to the left of its center (closer to the -axis), forming a spool-like shape. Therefore, the distance from the -axis to the centroid of the arc, , is: 3. Apply the Second Theorem of Pappus-Guldinus: The surface area of a body of revolution is the product of the length of the generating curve and the distance traveled by its centroid during the revolution. For a full revolution of ( radians), the formula is: 4. Substitution and Final Calculation: Substitute the expressions for , , and : r2α L=r(2α)=2rα =xˉ c α rsinα Rz z zrˉ =rˉR− =xˉ c R− α rsinα A L 360 ∘ θ=2π A=L⋅θ⋅rˉ Lθrˉ A=(2rα)⋅(2π)⋅R− ( α rsinα ) Distribute the terms: Where must be expressed in radians. ● Final Conclusion: The surface area of one side of the shell is given by the expression . ✨ Final Answer Summary (a) Mecademy A