Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_69

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the volume generated by revolving the quarter-circular area about the -axis through an angle of . Problem 5/69 (a) Calculation of Volume of Revolution To find the volume of the solid generated by revolving an area about an axis, we use the second Theorem of Pappus-Guldinus. 1. Formula: The volume is given by: where: is the angle of revolution in radians. is the distance from the axis of revolution to the centroid of the area . is the area of the plane figure being revolved. 2. Substitution: Based on the geometric properties of a quarter-circular area of radius : The area is . The centroid of a quarter circle is located at a distance of from its straight edges. Since one edge lies on the -axis (the axis of revolution), . The angle of revolution is , which is radians. Substituting these into the formula: 3. Calculation: Simplify the algebraic expression: Cancel in the numerator and denominator: Group the constants: Vz 90 ∘ V V=θ⋅⋅rˉA θ rˉA A a A= πa 4 12 3π 4a z=rˉ 3π 4a 90 ∘ θ= 2 π V= ⋅ ( 2 π ) ⋅ ( 3π 4a ) πa( 4 1 2 ) πV= ⋅ 2 1 ⋅ 3 4a πa 4 1 2 V= πa⋅( 2⋅3⋅4 1⋅4⋅1 )a 2 Simplify the fraction: Final expression: 4. Result: ● Final Conclusion: The volume generated by revolving the quarter-circular area through an angle of about the -axis is . This result is consistent with the fact that it represents one-quarter of a hemisphere's volume (). ✨ Final Answer Summary (a) Mecademy AI Solution · ENGProblem 5/69 = 24 4 6 1 V= πa 6 1