Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_36

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the - and -coordinates of the centroid of the volume generated by rotating the shaded area about the -axis through . Problem 5/36 (a) x-coordinate of the centroid 1. Formula: The -coordinate of the centroid of a volume is defined by: 2. Substitution: The shaded area is a quarter-circle of radius in the - plane. Rotating it about the -axis generates a spherical octant. In spherical coordinates : Coordinates: , , Differential volume: Limits for the octant: , , Total Volume: Substitute these into the integral: 3. Calculation: Evaluate the triple integral by separating the variables: Radial part: Polar part: Azimuthal part: xy z90 ∘ xV =xˉ V xdV∫ axz 90 ∘ z (ρ,φ,θ) x=ρsinφcosθy=ρsinφsinθz=ρcosφ dV=ρsinφdρdφdθ 2 0≤ρ≤a0≤φ≤ 2 π 0≤θ≤ 2 π V= πa= 8 1 ( 3 4 3 ) 6 πa 3 xd V = ∫ (ρsinφcos θ )( ρ sinφ)dρdφd θ∫ 0 π/2 ∫ 0 π/2 ∫ 0 a 2 xd V = ∫ ρdρ sinφdφ cosθdθ(∫ 0 a 3 )(∫ 0 π/2 2 )(∫ 0 π/2 ) ρdρ=∫ 0 a 3 = [ 4 ρ 4 ] 0 a 4 a 4 sinφdφ= ∫ 0 π/2 2 dφ= ∫ 0 π/2 2 1−cos2φ − = [ 2 φ 4 sin2φ ] 0 π/2 4 π cos θ d θ = ∫ 0 π/2 sinθ = [] 0 π/2 1 Combine the results: Now, calculate : 4. Result: ● Final Conclusion: The -coordinate of the centroid of the spherical octant is . (b) y-coordinate of the centroid 1. Formula: Due to the geometric symmetry of the spherical octant with respect to the plane , the -coordinate must be equal to the -coordinate: 2. Substitution: Using spherical coordinates where : 3. Calculation: Radial and Polar parts are identical