Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_52

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the coordinates of the mass center of the welded assembly of uniform slender rods made from the same bar stock. Problem 5/52 (a) Calculation of the x-coordinate of the mass center () 1. Formula: The -coordinate of the centroid for a composite line is given by: where is the length of each segment and is its individual centroid. 2. Substitution: Identify the four rod segments and their properties: Segment 1 (Vertical rod on -axis): , Segment 2 (Rod on -axis): , Segment 3 (Rod parallel to -axis from to ): , Segment 4 (Quarter-circular arc in -plane from to ): , 3. Calculation: Numerator: Denominator: Result: 4. Result: ● Final Conclusion: The -coordinate of the mass center is . xˉ x =xˉ L ∑ i L ∑ i xˉ i L i xˉ i zL = 1 a =xˉ 1 0 yL = 2 a =xˉ 2 0 x(0,a,0)(a,a,0)L = 3 a =xˉ 3 2 a xz(0,0,a)(a,0,0) L = 4 2 πa =xˉ 4 π 2a =xˉ a+a+a+ 2 πa a(0)+a(0)+a( )+ ( ) 2 a 2 πa π 2a L =∑ i xˉ i 0+0+ + 2 a 2 a= 2 a= 2 3 2 1.5a 2 L =∑ i 3a+= 2 πa a(3+ ) 2 π =xˉ = a(3+0.5π) 1.5 a 2 6+π 3a =xˉ ≈ 6+π 3a 0.328a x=xˉ ≈ 6+π 3a 0.328a (b) Calculation of the y-coordinate of the mass center () 1. Formula: 2. Substitution: Properties for the -coordinate: Segment 1: Segment 2: Segment 3: Segment 4: 3. Calculation: Numerator: Denominator: Result: 4. Result: ● Final Conclusion: The -coordinate of the mass center is . (c) Calculation of the z-coordinate of the mass center () 1. Formula: 2. Substitution: Properties for the -coordinate: Segment 1: Segment 2: