Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_29

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the -coordinate of the centroid of the solid spherical segment. Evaluate your expression for and . Problem 5/29 (a) General expression for the -coordinate of the centroid 1. Present Final Formula: For a solid of revolution about the -axis, the -coordinate of the centroid is given by: where is the differential volume element. 2. Substitute Values: We use a circular disk of thickness as our volume element. The radius of the disk at any position is from the equation of a circle . Differential volume: Limits of integration: From to The total volume is: The first moment of volume is: 3. Partial Operations: Calculate the volume : x h=R/4h=0 x xx =xˉ dV∫ xdV∫ dV dx xr= R−x 22 x+ 2 r= 2 R 2 dV=πrdx= 2 π(R− 2 x)dx 2 x=hx=R V V= π ( R − ∫ h R 2 x)dx 2 Q x Q = x x⋅ ∫ h R π(R− 2 x)dx 2 V V=πRx− x = [ 2 3 1 3 ] h R π(R− R)−(Rh− h)( 3 3 1 32 3 1 3 ) V=π R−Rh+ h = ( 3 2 32 3 1 3 ) (2 R − 3 π 3 3Rh+ 2 h) 3 Using polynomial division, we can factor the term as: . Calculate the first moment : This can be factored as: . 4. Final Calculation: Divide the first moment by the volume to find : 5. Result: ● Final Conclusion: The general expression for the -coordinate of the centroid of a solid spherical segment defined from to is . (b) Evaluation for 1. Formula: 2. Substitution: Substitute into the general formula. 3. Calculation: Numerator part: Denominator part: Combine: Simplify the fraction by dividing by 3: 4. Result: 2R− 3 3Rh+ 2 h= 3 (R− h