Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_35

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the coordinates of the centroid of the volume obtained by revolving the shaded area about the -axis through the angle. The shaded area is a quarter-circle of radius located in the -plane, bounded by the -axis, the -axis, and the arc . Problem 5/35 (a) Determination of the -coordinate of the centroid, 1. Formula: The volume is obtained by revolving a quarter-circle area through ( radians). Using spherical coordinates , the -coordinate is given by: where and the differential volume is . 2. Substitution: The limits of integration for the first octant of a sphere (radius , angle ) are , , and . 3. Calculation: Evaluate individual integrals: Combine results: Calculate : 4. Result: z90 ∘ a xzxzx+ 2 z= 2 a 2 xxˉ 90 ∘ π/2(ρ,φ,θ)x =xˉ xd V V 1 ∫ V x=ρsinφcosθdV=ρsinφdρdφdθ 2 a π/20≤ρ≤a0≤φ≤π/20≤θ≤π/2 V= d θ sinφdφ ρ dρ= ∫ 0 π/2 ∫ 0 π/2 ∫ 0 a 2 6 πa 3 xd V = ∫ V cos θ d θ sinφdφ ρ dρ ∫ 0 π/2 ∫ 0 π/2 2 ∫ 0 a 3 cos θ d θ = ∫ 0 π/2 [sin θ ] = 0 π/2 1 sinφdφ= ∫ 0 π/2 2 dφ= ∫ 0 π/2 2 1−cos2φ [ − 2 φ ] = 4 sin2φ 0 π/2 4 π ρdρ=∫ 0 a 3 4 a 4 xdV=∫ V 1⋅ ⋅ 4 π = 4 a 4 16 πa 4 xˉ=xˉ = πa/6 3 πa/16 4 = 16 6a a 8 3 =xˉ a 8 3 ● Final Conclusion: The -coordinate of the centroid of the volume is . (b) Determination of the -coordinate of the centroid, 1. Formula: Due to the symmetry of the spherical octant with respect to the plane , the -coordinate must equal the -coordinate . where . 2. Substitution: 3. Calculation: The and integrals are identica