Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_9

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the -coordinate of the mass center of the homogeneous paraboloid of revolution shown. Problem 5/9 (a) Determination of the -coordinate of the mass center 1. Formula: For a homogeneous body of revolution about the -axis, the -coordinate of the mass center (centroid) is given by: 2. Substitution: We select a differential volume element as a thin horizontal disk at height with radius and thickness . The volume of this element is . The cross-section is a parabola with its vertex at the origin . In the - plane, the equation of the parabola is . Given that the radius is at height , we substitute these values to find : Rearranging for : Substituting into the expression for : 3. Calculation: z z zz =zˉ dV∫ zdV∫ dV zxdz dV=πxdz 2 (0,0,0)xz z=kx 2 rhk h=kr⟹ 2 k= ⟹ r 2 h z= x r 2 h 2 x 2 x= 2 z h r 2 x 2 dV dV=π z d z( h r 2 ) First, calculate the total volume : Next, calculate the first moment of volume with respect to the -plane: Finally, compute the centroid coordinate : 4. Result: ● Final Conclusion: The -coordinate of the mass center of the homogeneous paraboloid of revolution, measured from its vertex at the origin, is . ✨ Final Answer Summary (a) Mecademy AI Solution · ENGProblem 5/9 V V= d V = ∫ π z d z = ∫ 0 h h r 2 = h πr 2 [ 2 z 2 ] 0 h = h πr 2 ( 2 h 2 ) πr h 2 1 2 xy z d V = ∫ zπ z d z = ∫ 0 h ( h r 2 ) z d z = h πr 2 ∫ 0 h 2 = h πr 2 [ 3 z 3 ] 0 h h πr 2 ( zˉ =zˉ = πrh 2 1 2 πrh 3 1 22 h= 1/2 1/3 h 3 2 =zˉ h 3 2 z =zˉ