Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_183

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the -coordinate of the centroid of the shaded area shown. (Carefully observe the proper sign of the radical involved.) Problem 5/183 (a) y-coordinate of the centroid 1. Formula: The -coordinate of the centroid is defined as . For a symmetric area, it is most efficient to use horizontal strips of width and thickness , where . 2. Substitution: The geometry consists of two circles of radius with centers at and . The intersection region's upper half is bounded below by the -axis () and above by two circular arcs meeting at the -axis at height . Right arc (part of circle 1 centered at ): Left arc (part of circle 2 centered at ): Strip width: 3. Calculation: Total Area : y y =yˉ = A Q x dA∫ ydA∫ w(y) dydA=w(y)dy a C (−a/2,0) 1 C (a/2,0) 2 xy=0y h= 2 a 3 −a/2(x+a/2)+ 2 y= 2 a⟹ 2 x = R −a/2+ a−y 22 a/2(x−a/2)+ 2 y= 2 a⟹ 2 x = L a/2− a−y 22 w(y)=x − R x = L (−a/2+ )− a−y 22 (a/2− )= a−y 22 2 − a−y 22 a A A= (2 − ∫ 0 a /23 a−y 22 a)dy=y +aarcsin −ay[a−y 22 2 a y ] A= ⋅ +aarcsin −a⋅ − ( 2 a 3 2 a 2 2 3 2 a 3 )(0)= + 4 a3 2 − 3 πa 2 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 5/183 Q x Q = x y (2 − ∫ 0 a /23 a−y 22 a)dy=− (a−y)− ay[ 3 2 223/2 2 1 2 ] 0 a /3 Q = x − − ⋅ − ( 3 2 ( 4 a 2 ) 3/2 2 a 4 3a 2 )− a−0 = ( 3 2 3 ) − − 12 a 3 + 8 3a 3 yˉ =yˉ = a ( π /3− /4) 2 3 5a/24 3 = 24(4 π −3 )/12 3 5a 8