Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_14

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the -coordinate of the centroid of the shaded area. Problem 5/14 (a) Determination of the -coordinate of the centroid 1. Formula: The -coordinate of the centroid for an area bounded by two curves and is given by: where and . 2. Substitution: From the diagram, the upper boundary is the curve , and the lower boundary is a dashed straight line passing through and . Upper boundary: Lower boundary: Integration limits: ranges from to . Substituting into the formulas: 3. Calculation: Calculate Area : y y y y (x) 1 y (x) 2 =yˉ = A M x dA∫ dA∫y ~ dA=(y − upper y )dx lower =y ~ 2 y +y upperlower x=y/b 2 (0,0)(b,b/2) y = upper bx y = lower x= b b/2 2 x x0b dA= − dx (bx 2 x ) dM = x − dx= 2 +x/2 bx (bx 2 x ) ( )− dx= 2 1 [bx 2 ( 2 x ) 2 ] b 2 1 ( A A= x− x dx= ∫ 0 b (b 1/2 2 1 ) x− [ 3 2 b 3/2 4 x 2 ] 0 b A= ( b )− 3 2 b 3/2 = 4 b 2 b − 3 2 2 b = 4 1 2 b = 12 8−3 2 b 12 5 2 Calculate First Moment : Calculate Centroid : 4. Result: ● Final Conclusion: The -coordinate of the centroid of the shaded area is . ✨ Final Answer Summary (a) Mecademy AI Solution · ENGProblem 5/14 M x M = x bx− dx= ∫ 0 b 2 1 ( 4 x 2 ) − 2 1 [ 2 bx 2 12 x 3 ] 0 b M = x − = 2 1 ( 2 b 3 12 b 3 ) b = 2 1 ( 12 6−1 3 ) b 24 5 3 yˉ =yˉ = A M x b 12 5 2 b 24 53 =yˉ ⋅ 24 5 b= 5 12 b= 24 12 2 b =yˉ 2 b y =yˉ 2 b =yˉ 2 b