Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_33

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the -coordinate of the mass center of the thin homogeneous parabolic shell. Evaluate your results for and . Problem 5/33 (a) General expression for the -coordinate of the mass center 1. Formula: For a thin homogeneous shell of revolution about the -axis, the - coordinate of the mass center is equivalent to the centroidal -coordinate of its surface area: where is the area of a differential ring element, is the radius at height , and is the differential arc length along the parabola. 2. Substitution: The equation of the parabolic shell with vertex at the origin and base radius at height is: The differential arc length is given by: Let . Then . The area element is . The integrals become: 3. Calculation: Perform a substitution , which implies and . The upper limit is y h=200 mmr=70 mm y yy yˉy =yˉ dA∫ ydA∫ dA=2πρdsρ yds rh ρ= 2 y ⟹ h r 2 y= ρ r 2 h 2 ds ds= dρ= 1+ ( dρ dy ) 2 dρ 1+ ( r 2 2hρ ) 2 k= r 2 2h ds= dρ 1+kρ 22 dA= 2 π ρ dρ 1+kρ 22 A= 2 π ρ dρ ∫ 0 r 1+kρ 22 M = xz ρ 2 π ρ dρ= ∫ 0 r ( r 2 h 2 )1+kρ 22 ρ dρ r 2 2πh ∫ 0 r 3 1+kρ 22 u= 2 1+kρ 22 2udu= 2kρdρ⟹ 2 ρdρ= k 2 udu ρ= 2 k 2 u −1 2 U= . Total area : Moment : Substituting back and simplifying: 4. Result: ● Final Conclusion: The general expression for the -coordinate of the mass center is with . (b) Numerical evaluation for and 1. Formula: Use the derived formula from part (a): 2. Substitution: Substitute and : Ratio 3. Calculation: Numerator Denominator Prefactor