Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_184

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the -coordinate of the mass center of the homogeneous parabolic plate of varying thickness. Take , , , and . Problem 5/184 (a) Determination of the -coordinate of the mass center 1. Formula: The -coordinate of the mass center for a body with varying cross- sectional area and thickness along the -axis is given by: where the width is and the thickness is . At , , so . Thus, . 2. Substitution: Let , then and . The limits change from to . The area element's volume is . 3. Calculation: Denominator integral (): Numerator integral (): z b=750 mmh=400 mmt = 0 35 mmt = 1 7 mm z z z =zˉ = dm∫ zdm∫ x( z )⋅t( z )d z∫ 0 h z ⋅x( z )⋅t( z )d z∫ 0 h x(z)= b 1− ( h 2 z 2 )t(z)=t − 0 kz 2 z=ht=t 1 t = 1 t − 0 kh⟹ 2 k= h 2 t −t 01 t(z)=t − 0 (t − 0 t ) 1 h 2 z 2 u= h z z=hudz=hdu [0,h][0,1]dV=x(u)t(u)hdu=b(1− u)[t − 2 0 (t − 0 t )u]hdu 1 2 =zˉ = b(1− u )[ t −( t − t ) u ]hdu ∫ 0 1 2 001 2 (hu)⋅b(1− u )[ t −( t − t ) u ]hdu ∫ 0 1 2 001 2 h (1− u )[ t −( t − ∫ 0 1 2 00 (u− u )[ t −( t − ∫ 0 1 3 00 I D I = D [ t − ∫ 0 1 0 (2t − 0 t )u+ 1 2 (t − 0 t )u]du= 1 4 t u− u+[ 0 3 2t −t 01 3 I N I = N [ t u− ∫ 0 1 0 (2t − 0 t )u+ 1 3 (t − 0 t )u]du= 1 5 u− u[ 2 t 0 2 4 2t −t 01 4 Combine results: Plug in numerical values (, , ): 4. Result: ● Final Conclusion: The -coordinate of the mass center is . ✨ Final Answer Summary (a) Mecademy AI Solution · ENGProblem 5/184 =zˉh⋅ = 15 2(4t +t ) 01 12 2t +t 01 h⋅ ⋅ 12 2t +t 01 = 2(4t +t ) 01 15 8(4t +t ) 01 5h(2