Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_192

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the x-, y-, and z-coordinates of the mass center of the body constructed of uniform slender rod which is bent into circular arcs of radius r. Problem 5/192 (a) Determination of coordinates of the mass center ● Calculation Process 1. Identify Components and their Properties: The body consists of three identical quarter-circular arcs of radius . For a uniform slender rod of length , the mass center (centroid) is determined by its geometric properties. Arc 1 (in the -plane): Length: Centroid: , , Arc 2 (in the -plane): Length: Centroid: , , Arc 3 (in the -plane): Length: Centroid: , , 2. Present Final Formula for Composite Centroid: The coordinates of the mass center for the composite body are given by: 3. Substitute Values and Perform Operations: First, calculate the total length of the rods: Calculate the sum of the moments of length for the -coordinate: rL xyL = 1 2 πr =xˉ 1 π 2r =yˉ 1 π 2r =zˉ 1 0 yzL = 2 2 πr =xˉ 2 0 =yˉ 2 π 2r =zˉ 2 π 2r zxL = 3 2 πr =xˉ 3 π 2r =yˉ 3 0 =zˉ 3 π 2r =X ˉ ,= L ∑ i L ∑ i xˉ i Y ˉ ,= L ∑ i L ∑ i yˉ i Z ˉ L ∑ i L ∑ i zˉ i L = total L + 1 L + 2 L = 3 3 = ( 2 πr ) 2 3πr x L = ∑ i xˉ i + ( 2 πr )( π 2r )0+ = ( 2 πr )( π 2r )r+ 2 0+r= 2 2r 2 Substitute into the formula for : Due to the symmetry of the body with respect to the , , and axes (the body remains unchanged if the axes are rotated), we have: 4. Final Calculation: ● Final Conclusion: The coordinates of the mass center are . ✨ Final Answe