Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_31

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the coordinates of the mass center of the solid homogeneous body formed by revolving the shaded area 90° about the -axis. The curve bounding the area is defined by . Problem 5/31 (a) Total Volume of the Homogeneous Body 1. Formula: The volume of a body of revolution can be found using cylindrical coordinates . Since it is revolved (or radians), the volume is: 2. Substitution: Substitute into the integral: 3. Calculation: Evaluate inner integrals: and Combine: Integrate with respect to : Plug in limits: 4. Result: ● Final Conclusion: The total volume of the body is . z x= a 1− ( b 2 z 2 ) (r,θ,z)90 ∘ 2 π V= r d r d θ d z∫ 0 b ∫ 0 π/2 ∫ 0 x(z) x(z)= a 1− ( b 2 z 2 ) V= d z d θ r d r∫ 0 b ∫ 0 π/2 ∫ 0 a(1−z/b) 22 d θ = ∫ 0 π/2 2 π r d r = ∫ 0 a(1−z/b) 22 a1− 2 1 2 ( b 2 z 2 ) 2 V= ⋅ 2 π a1− + d z 2 1 ∫ 0 b 2 ( b 2 2 z 2 b 4 z 4 ) zV= z− + 4 πa 2 [ 3b 2 2 z 3 5b 4 z 5 ] 0 b V= b− + = 4 πa 2 ( 3 2b 5 b ) = 4 πab 2 ( 15 15−10+3 ) 4 πab 2 ( 15 8 ) V= 15 2πab 2 V= 15 2πab 2 (b) Vertical Coordinate of the Mass Center 1. Formula: For a homogeneous body, the -coordinate of the mass center is given by: 2. Substitution: 3. Calculation: Expand and integrate: Evaluate: Solve for : 4. Result: ● Final Conclusion: The -coordinate of the mass center is . (c) Horizontal Coordinates and of the Mass Center 1. Formula: In cylindrical coordinates, and . The - coordinate is: 2. Substitution: 3. Calculation: Evaluate integral: Evaluate and integra