Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_47

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the -coordinate of the centroid of the shaded area. Problem 5/47 (a) y-coordinate of the centroid 1. Formula: By considering the shaded area as a composite of a large triangle with a circular sector removed, the -coordinate of its centroid is given by: where: and are the area and centroid -coordinate of the triangle bounded by the horizontal line and the slanted sides. and are the area and centroid -coordinate of the circular sector removed from the bottom. 2. Substitution: From the diagram, the total vertex angle is ( on each side of the -axis), and the height of the triangle is . For the triangle (Area 1): Base width Area Centroid (distance from vertex at the origin) For the circular sector (Area 2): Radius , and half-angle Area Centroid y y =yˉ = A ∑ i A ∑ i yˉ i A −A 12 A −A 1 yˉ 12 yˉ 2 A 1 yˉ 1 y A 2 yˉ 2 y 120 ∘ 60 ∘ yh w=2⋅htan60= ∘ 2 h 3 A = 1 wh= 2 1 (2 h)h= 2 1 3 h3 2 =yˉ 1 h 3 2 r=aα=60= ∘ rad 3 π A = 2 rα= 2 a = 2 ( 3 π ) 3 πa 2 =yˉ 2 = 3α 2rsinα = 3(π/3) 2asin(60) ∘ = π 2a( /2)3 π a 3 3. Calculation: First, calculate the first moment of area about the -axis (): Next, calculate the total shaded area (): Finally, divide to find : 4. Result: ● Final Conclusion: The -coordinate of the centroid for the shaded area is . By symmetry, the -coordinate . ✨ Final Answer Summary (a) Mecademy AI Solution · ENGProblem 5/47 xA ∑yˉ A = ∑yˉA − 1 yˉ 1 A = 2 yˉ 2 ( h ) − 3 2 ( 3 2h ) = ( 3 πa 2 )( π a 3 ) 3 2 h3 3 A∑ A= ∑A −