Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_57

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the -coordinate of the centroid of the rectangular solid with the hemispherical hole. The center of the hemisphere is centered on the upper face of the solid, and is measured upward from the lower face. Problem 5/57 (a) Determination of the z-coordinate of the centroid 1. Formula: The -coordinate of the centroid for a composite volume is given by the weighted average of the centroids of its components: where: are the volume and centroid of the solid rectangular block. are the volume and centroid of the hemispherical hole (subtracted). 2. Substitution: From the geometric properties shown in the diagram: For the rectangular block: Dimensions: Volume: Centroid: For the hemispherical hole (radius ): Volume: Centroid: The centroid of a hemisphere is located at a distance of from its base. Since the base is at the top surface () and the hole extends downward: z z z =zˉ = V ∑ i V z ∑ ii V −V 12 V z −V z 1122 V ,z 11 V ,z 22 2.5R×2.5R×1.5R V = 1 (2.5R)(2.5R)(1.5R)=9.375R 3 z = 1 = 2 1.5R 0.75R R V = 2 πR= 2 1 ( 3 43 ) πR≈ 3 23 2.0944R 3 8 3R z=1.5R z = 2 1.5R− = 8 3R 1.125R Substituting these into the main formula: 3. Calculation: Numerator: Denominator (Total Volume): Final division: 4. Result: ● Final Conclusion: The -coordinate of the centroid of the composite solid is measured from the base. ✨ Final Answer Summary (a) Mecademy AI Solution · ENGProblem 5/57 =zˉ 9.375R− πR 3 3 2 3 (9.375R)(0.75R)− πR(1.125R) 3 ( 3 23 ) (9.375)