Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_10

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the - and -coordinates of the centroid of the shaded area. Problem 5/10 (a) Determination of the -coordinate of the centroid, 1. Formula: The -coordinate of the centroid for a 2D area is defined as . For a vertical differential strip of width , the area is and its centroid is at . Thus, . 2. Substitution: From the given diagram, the shaded area is bounded by and on the -axis, and bounded above by the curve . Area: Moment about -axis: 3. Calculation: Calculate total area : Calculate first moment about the -axis : Compute : 4. Result: ● Final Conclusion: The -coordinate of the centroid of the shaded area is . xy xxˉ x=xˉ xdA A 1 ∫dxdA=ydx =x ~ x=xˉ ydx∫ xydx∫ x=1 x=2xy=1+ 8 x 3 A= 1+ dx ∫ 1 2 ( 8 x 3 ) yM = y x 1+ dx ∫ 1 2 ( 8 x 3 ) A A=x+ = [ 32 x 4 ] 1 2 2+ − ( 32 2 4 )1+ = ( 32 1 4 )(2+0.5)−(1+0.03 yM y M = y x+ dx= ∫ 1 2 ( 8 x 4 ) + = [ 2 x 2 40 x 5 ] 1 2 + − ( 2 2 2 40 2 5 ) +( 2 1 2 40 1 5 M = y (2+0.8)−(0.5+0.025)=2.8−0.525=2.275 xˉ =xˉ = A M y ≈ 1.46875 2.275 1.5489 ≈xˉ1.549 x=xˉ1.549 (b) Determination of the -coordinate of the centroid, 1. Formula: The -coordinate of the centroid is . For the same vertical strip, the strip's centroid is at its midpoint . Thus, . 2. Substitution: 3. Calculation: Calculate first moment about the -axis : Compute : 4. Result: ● Final Conclusion: The -coordinate of the centroid of the shaded area is . ✨ Final Answer Summary (a) (b) Mecademy AI Solution · ENGProblem 5/10 y yˉ y =yˉ dA A