Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_16

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the -coordinate of the centroid of the shaded area. Problem 5/16 (a) Determination of the -coordinate of the centroid 1. Formula: The -coordinate of the centroid of an area defined between two curves and is given by: 2. Substitution: From the geometry of the shaded area: The top boundary is a straight line passing through and . Its equation is: The bottom boundary is the curve . Since it passes through the point , we have . Thus, the equation is: Substitute these into the area integral: And the moment integral: 3. Calculation: y y y y (x) top y (x) bottom =yˉ = A dA∫y ~ (y −y )dx∫ 0 a topbottom (y −y )dx∫ 0 a 2 1 top 2 bottom 2 (0,b)(a,2b) y = top b+ x= a 2b−b b 1+ ( a x ) x=ky 2 (a,b)a=k(b)⟹ 2 k= b 2 a x= y ⟹ b 2 a 2 y = bottom b a x A= b1+ −b dx ∫ 0 a [( a x ) a x ] Q = x dA= ∫y ~ b1+ −b dx 2 1 ∫ 0 a [ 2 ( a x ) 2 ( a x ) 2 ] Total Area : First Moment of Area : Centroid : 4. Result: ● Final Conclusion: The -coordinate of the centroid of the shaded area is . ✨ Final Answer Summary (a) Mecademy AI Solution · ENGProblem 5/16 A A= b 1+ − dx= ∫ 0 a ( a x a 1/2 x 1/2 ) b x+ − [ 2a x 2 3a 1/2 2 x 3/2 ] 0 a A= b a+ − = [ 2a a 2 3a 1/2 2 a 3/2 ] b a+ − = ( 2 a 3 2a ) ab = ( 6 6+3−4 ) Q x Q = x 1+ + − dx= 2 b 2 ∫ 0 a ( a 2x a 2 x 2 a x ) 1+ + dx 2 b 2 ∫ 0 a ( a x a 2 x 2 ) Q = x x+ + = 2 b 2 [ 2a x 2 3a 2 x 3 ] 0 a a+ + = 2 b 2 ( 2 a 3 a ) 2 ba 2 ( 6 6+3+2 ) yˉ =yˉ = A Q x ab 6 5 ab 12 112 =yˉ ⋅ 12 11 b= 5 6 b= 10 11 1.1b y =yˉ1