Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_13

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the coordinates of the centroid of the shaded area. The area is bounded by the - axis, the horizontal line , and the parabolic curve which passes through the point . Problem 5/13 (a) x-coordinate of the centroid () 1. Formula: The -coordinate of the centroid is given by: where is the -coordinate of the centroid of a differential area element . 2. Substitution: Using a horizontal differential strip at height with thickness : Width of the strip: At point : . Thus, . Area element: Centroid of the horizontal strip: Substituting into the formula with limits from to : 3. Calculation: Denominator (Total Area ): Numerator (): y y=ax=ky 2 (b,a) xˉ x =xˉ dA∫ dA∫x ~ x ~ xdA ydy x=ky 2 (b,a)b=k(a)⟹ 2 k= a 2 b x= y a 2 b 2 dA=xdy= y d y a 2 b 2 =x ~ = 2 x y 2a 2 b 2 y=0y=a =xˉ y d y∫ 0 a a 2 b 2 y y d y∫ 0 a ( 2a 2 b 2 )( a 2 b 2 ) A A= y d y = ∫ 0 a a 2 b 2 = a 2 b [ 3 y 3 ] 0 a = a 2 b ( 3 a 3 ) 3 ab dA∫x ~ dA= ∫x ~ y d y = ∫ 0 a 2a 4 b 2 4 = 2a 4 b 2 [ 5 y 5 ] 0 a = 2a 4 b 2 ( 5 a 5 ) 10 ab 2 Final calculation: 4. Result: ● Final Conclusion: The -coordinate of the centroid is . (b) y-coordinate of the centroid () 1. Formula: The -coordinate of the centroid is given by: 2. Substitution: Using the same horizontal strip: Centroid of the strip: Area element: Substituting into the formula: 3. Calculation: Numerator (): Final calculation: 4. Result: ● Final Conclusion: The -coordinate of the centroid is . ✨ Final Answer Summary (a) =xˉ