Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_12

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the coordinates of the centroid of the shaded area. Problem 5/12 (a) x-coordinate of the centroid 1. Present Final Formula: The -coordinate of the centroid for an area is given by: where is a differential area element and is the -coordinate of its centroid. 2. Substitute Values: From the given diagram, the area is bounded by the -axis, the line , and the curve . At the point , we have: Thus, the equation of the curve is , or . Choosing a vertical differential element of width : The centroid of this element is at . 3. Partial Operations: First, calculate the total area : Next, calculate the first moment of area about the -axis: 4. Final Calculation: xxˉ =xˉ dA∫ x dA∫ c dAx c x x x=bx=ky 2 (b,a) b=k(a)⟹ 2 k= a 2 b x= y a 2 b 2 y=a b x dx dA=ydx= a dx b x x = c x A A= a dx= ∫ 0 b b x x dx= b a ∫ 0 b 1/2 x = b a [ 3 2 3/2 ] 0 b b = b a ( 3 2 3/2 ) y x dA= ∫ c x a dx= ∫ 0 b ( b x ) x dx= b a ∫ 0 b 3/2 x = b a [ 5 2 5/2 ] 0 b b a =xˉ = ab 3 2 ab 5 22 ⋅ 5 2 b= 2 3 b 5 3 ● Final Conclusion: The -coordinate of the centroid is . (b) y-coordinate of the centroid 1. Present Final Formula: The -coordinate of the centroid for an area is given by: where is the -coordinate of the centroid of the differential area element. 2. Substitute Values: Using the same vertical element , its centroid is at its midpoint: 3. Partial Operations: Calculate the first moment of area about the -axis: Substitute : The total area was found in part (a) to b