Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_34

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the -coordinate of the centroid of the plane area shown. Set in your result and compare with the result for a full semicircular area (see Sample Problem 5/3 and Table D/3). Also evaluate your result for the conditions and . Problem 5/34 (a) General derivation of the -coordinate of the centroid 1. Formula: The -coordinate of the centroid is given by , where is the total area and is the first moment of area about the -axis. 2. Substitution: For the circular segment of radius bounded below by , we use horizontal differential strips: Width of strip: Differential area: First moment of area: 3. Calculation: Total Area : Using the substitution (where is the angle from the -axis): At . At . First Moment of Area : yh=0 =yˉ 3π 4a h= 4 a h= 2 a y y =yˉ A ydA∫ A= dA∫ydA∫x ay=h 2x= 2 a−y 22 dA= 2 d ya−y 22 dQ = x ydA= 2 y d ya−y 22 A A= 2 d y∫ h a a−y 22 y=acosθ⟹dy=−asinθdθθ yy=h,θ=β=cos(h/a) −1 y=a,θ=0 A= 2asin θ (−asin θ )d θ = ∫ β 0 2 a sin θ d θ = 2 ∫ 0 β 2 a[θ− 2 sin θ cos θ A=a cos − = 2−1 ( a h ) a h 1− ( a h ) 2 a cos − 2−1 ( a h ) h a− 2 Q x Q = x 2 y d y∫ h a a−y 22 Let . 4. Result: ● Final Conclusion: The general expression for the -coordinate of the centroid is . (b) Comparison with a full semicircular area () 1. Formula: Substitute into the general centroid formula. 2. Substitution: 3. Calculation: 4. Result: ● Final Conclusion: Setting yields , which perfectly matches the standard result for a semicircular area. (c) Eva