Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_80

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the volume and total surface area of the solid generated by revolving the area shown through about the -axis. Problem 5/80 (a) Volume of the generated solid 1. Formula: According to the second theorem of Pappus-Guldinus, the volume of a solid of revolution is given by: where is the angle of revolution in radians, is the distance from the axis of revolution to the centroid of the generating area, and is the area of the generating cross-section. 2. Substitution: From the given diagram for Problem 5/80: Angle of revolution: Generating area is a circular segment with radius and height . The distance from the center of the circle to the chord is . Half-angle of the segment: . Area of the cross-section: . Distance to centroid: (The area is symmetric about a vertical line at this distance). 3. Calculation: VA 180 ∘ z V V V=θArˉ θrˉ A θ=180= ∘ π rad R=40 mm h=30 mm d=R−h=40− 30=10 mm α= cos = −1 ( R d )cos = −1 ( 40 10 ) cos(0.25)≈ −1 1.3181 rad A=Rα− 2 d = R−d 22 40(1.3181)− 2 10 40−10 22 =rˉ75 mm A≈1600(1.3181)−10(38.73)=2108.96−387.3= 1721.66 mm 2 4. Result: ● Final Conclusion: The volume of the generated solid is approximately . (b) Total surface area of the generated solid 1. Formula: The total surface area consists of the curved outer surface (), the flat horizontal base (), and the two cross-sectional end faces (). where is the arc length of the cross-section and is the chord length. 2. Substitution: Arc length: . Chord l