Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_15

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the -coordinate of the centroid of the shaded area. Problem 5/15 (a) Determine the x-coordinate of the centroid 1. Formula: For an area defined in polar coordinates by a curve , the area and the -coordinate of the centroid are given by: 2. Substitution: From the diagram, the curve is . The boundary conditions are: At , (the point on the -axis). At , (the point on the -axis). Substituting these into the equation for : Thus, the equation of the curve is . 3. Calculation: Step 3.1: Calculate Area Let , then . The limits for are from to . x r=f(θ)A xxˉ A= r d θ∫ θ 1 θ 2 2 1 2 =xˉ = A Q y = A xdA∫ A r cos θ d θ∫ θ 1 θ 2 3 13 r=k + 1 k θ 2 θ=0r=ax θ=π/2r=2ay r a=k + 1 k (0)⟹ 2 k = 1 a 2a=a+k (π/2)⟹ 2 k = 2 π 2a r=a+ θ= π 2a a 1+ ( π 2θ ) Au=1+ π 2θ dθ= du 2 π u12 A= a1+ d θ = ∫ 0 π/2 2 1 [( π 2θ )] 2 u du 2 a 2 ∫ 1 2 2 ( 2 π ) A= = 4 πa 2 [ 3 u 3 ] 1 2 − = 4 πa 2 ( 3 8 3 1 ) 12 7πa 2 Step 3.2: Calculate First Moment of Area Using the integration by parts formula , where : At the upper limit , : At the lower limit , : The integral evaluates to: Substituting : Thus, . Step 3.3: Final Division 4. Result: ● Final Conclusion: The -coordinate of the centroid for the given shaded area is , which numerically evaluates to approximately . ✨ Final Answer Summary (a) Mecademy AI Solution · ENGProblem 5/15 Q y Q = y r cos θ d θ = 3 1 ∫ 0 π/2 3 a1+ cos θ d θ 3 1 ∫ 0 π/2 3 ( π 2θ ) 3 (a+∫bθ)cosθdθ= 3 (a+ bθ)sinθ+ 3 3b(a+bθ)cosθ− 2 6b(a+ 2 bθ