Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_68

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Calculate the volume of the solid generated by revolving the 60-mm right-triangular area about the -axis through . Problem 5/68 (a) Volume of the solid of revolution 1. Formula: The volume of a solid generated by revolving a plane area through an angle about an axis is given by the second theorem of Pappus-Guldinus: where: is the angle of revolution in radians. is the perpendicular distance from the axis of revolution to the centroid of the area. is the area of the generating shape. 2. Substitution: From the given diagram for Problem 5/68: The angle of revolution is . The area is a right triangle with legs of length and . The right-angled vertex of the triangle is at a distance of from the - axis. The centroid of a right triangle is located at a distance of of the leg length from the right angle. Thus, the radial distance to the centroid is: Substituting these into the volume formula: V z180 ∘ VA θ V=θArˉ θ rˉ A θ=180= ∘ π rad Ab=60 mmh= 60 mm A= bh= 2 1 (60 mm)(60 mm)= 2 1 1800 mm 2 30 mmz 1/3 rˉ =rˉ30 mm+ (60 mm)= 3 1 30 mm+20 mm=50 mm V=(π rad)(50 mm)(1800 mm) 2 3. Calculation: 4. Result: ● Final Conclusion: The volume of the solid generated by revolving the triangular area through is (or ). ✨ Final Answer Summary (a) Mecademy AI Solution · ENGProblem 5/68 V=π×50×1800 V=90,000π mm 3 V≈90,000×3.14159265 mm 3 V≈282,743.3 mm 3 V=2.83×10 mm 53 180 ∘ 282,743 mm 3 2.83×10 mm 53 V=2.83×10 mm 53