Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_17

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Find the distance from the vertex of the right-circular cone to the centroid of its volume. Problem 5/17 (a) Determination of the centroid distance from the vertex ● Calculation Process 1. Present Final Formula: The -coordinate of the centroid of a volume is defined by the ratio of the first moment of the volume to the total volume: where is measured from the vertex. 2. Substitute Values: Consider a differential volume element consisting of a thin circular disk of thickness at a distance from the vertex. Let be the radius of the cone's base and be its total height. By similar triangles, the radius of the disk at distance is: The differential volume element is: 3. Partial Operations: Calculate the total volume and the first moment of volume : Total Volume: First Moment of Volume about the vertex: zˉ z =zˉ V zdV∫ z dV dzzR h rzr= z h R dV=πrdz= 2 π z d z = ( h R ) 2 z d z h 2 πR 2 2 V zdV∫ V= d V = ∫ 0 h z d z = ∫ 0 h h 2 πR 2 2 = h 2 πR 2 [ 3 z 3 ] 0 h = h 2 πR 2 ( 3 h 3 ) πR 3 1 2 z d V = ∫ z z d z = ∫ 0 h ( h 2 πR 2 2 ) z d z = h 2 πR 2 ∫ 0 h 3 = h 2 πR 2 [ 4 z 4 ] 0 h h 2 πR 4. Final Calculation: Substitute the results into the centroid formula: ● Final Conclusion: The distance from the vertex of the right-circular cone to the centroid of its volume is . This result is independent of the base radius . ✨ Final Answer Summary (a) Mecademy AI Solution · ENGProblem 5/17 =zˉ = πRh 3 1 2 πRh 4 122 h 4 3 zˉ h 4 3 R =zˉ h 4 3