Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_46

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the coordinates of the mass center of the body which is constructed of three pieces of uniform thin plate welded together. Problem 5/46 (a) Determination of the Mass Center Coordinates 1. Formula: For a body composed of several uniform thin plates of the same material and thickness, the coordinates of the mass center are calculated using the principle of composite areas: 2. Substitution: Define the three plates based on the provided coordinate system (origin at the junction of the three plates): Plate 1 (x-z plane): Dimensions . Area . Centroid . Plate 2 (y-z plane): Dimensions . Area . Centroid . Plate 3 (x-y plane): Dimensions . Area . Centroid . Total Area: . 3. Calculation: Compute the first moments of area for each axis: Solve for the mass center coordinates: (,,)X ˉ Y ˉ Z ˉ =X ˉ ,= A ∑ i A ∑ i xˉ i Y ˉ ,= A ∑ i A ∑ i yˉ i Z ˉ A ∑ i A ∑ i zˉ i b×bA = 1 b 2 ( , , )=xˉ 1 yˉ 1 zˉ 1 ( ,0, ) 2 b 2 b b×2bA = 2 2b 2 ( , , )=xˉ 2 yˉ 2 zˉ 2 (0,b, ) 2 b b×2bA = 3 2b 2 ( , , )=xˉ 3 yˉ 3 zˉ 3 ( ,b,0) 2 b A=∑A + 1 A + 2 A = 3 b+ 2 2b+ 2 2b= 2 5b 2 A=∑xˉA + 1 xˉ 1 A + 2 xˉ 2 A = 3 xˉ 3 ( b )( )+ 2 2 b (2b)(0)+ 2 (2 b )( )= 2 2 b 0.5b+ 3 0+b= 3 1.5b 3 A =∑yˉA + 1 yˉ 1 A + 2 yˉ 2 A = 3 yˉ 3 (b)(0)+ 2 (2b)(b)+ 2 (2b)(b)= 2 0+2b+ 3 2b= 3 4b 3 A=∑zˉA + 1 zˉ 1 A + 2 zˉ 2 A = 3 zˉ 3 ( b )( )+ 2 2 b (2 b )( )+ 2 2 b (2b)(0)= 2 0.5b+ 3 b+ 3 0=1.5b 3 3 4. Result: ● Final Conclusion: The mass center of the welded body is located at the coor