Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_182

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Locate the centroid of the shaded area. Problem 5/182 (a) x-coordinate of the centroid 1. Formula: First, we define the equations of the boundaries. The curve is given by . Since it passes through the point , we have: Thus, the upper boundary is . The lower boundary is a straight line passing through and , with equation . The area is calculated as: The x-coordinate of the centroid is given by: 2. Substitution: Substitute the expressions for and : 3. Calculation: Perform the integrations: Area: xˉ x=ky 2 (2b,b) 2b=k(b)⇒ 2 k= b 2 y = 1 2 xb (0,0)(2b,b)y = 2 x= 2b b 2 x A A= ( y − ∫ 0 2b 1 y )dx 2 xˉ =xˉ x( y − A 1 ∫ 0 2b 1 y )dx 2 y 1 y 2 A= x− x dx ∫ 0 2b ( 2 b 1/2 2 1 ) A=xˉ x x− x dx= ∫ 0 2b ( 2 b 1/2 2 1 ) x− x dx ∫ 0 2b ( 2 b 3/2 2 1 2 ) A= x− x = [ 2 b 3 2 3/2 4 1 2 ] 0 2b ⋅ ⋅(2b) − ( 2 b 3 2 2b) ⋅4b = ( 4 1 2 ) Moment about y-axis: 4. Result: ● Final Conclusion: The x-coordinate of the centroid is . (b) y-coordinate of the centroid 1. Formula: The y-coordinate of the centroid for the area between two curves is: 2. Substitution: Substitute and : 3. Calculation: Evaluate the integral: 4. Result: ● Final Conclusion: The y-coordinate of the centroid is . ✨ Final Answer Summary (a) A=xˉ x− x = [ 2 b 5 2 5/2 6 1 3 ] 0 2b ⋅ ⋅(2b) − ( 2 b 5 2 2 2b) ⋅8b( 6 1 3 ) =xˉ = b 3 1 2 b 15 4 3 b= 5 4 0.8b =xˉ0.8b yˉ yˉ =yˉ ( y − A 1 ∫ 0 2b 2 1 1 2 y )dx 2 2 y = 1 2 xb y = 2 2 x A =yˉ − dx ∫ 0 2b 2 1 ( 2 xb 4 x 2 ) A =yˉ − = 2 1 [ 4 bx 2 12 x 3