Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_37

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the -coordinate of the mass center of the cylindrical shell of small uniform thickness. Problem 5/37 (a) Determination of the -coordinate of the mass center 1. Formula: For a homogeneous cylindrical shell of uniform small thickness, the - coordinate of the mass center () is equivalent to the -coordinate of the centroid of its surface area. The formula for the -coordinate of the centroid is: where is a differential area element and is the -coordinate of the centroid of that element. 2. Substitution: We define the length of the cylinder as a function of the angle around the -axis. Let at the top (where length is ) and at the bottom (where length is ). Since the cut is a plane, varies linearly with : The average length is . Thus, A differential area element at angle is a strip of length and width . Its centroid is at . The substitution into the centroid formula gives: 3. Calculation: First, calculate the denominator (which is related to the total surface area): x x x xˉx x =xˉ dA∫ x dA∫ c dAx c x Lφ xφ=02Rφ=π 4RL(φ)cosφ L(φ)=L − avg Rcosφ L = avg = 2 2R+4R 3R L(φ)=3R−Rcosφ dAφL(φ)Rdφ x = c 2 L(φ) dA=L(φ)⋅Rdφ=R(3R−Rcosφ)dφ=R(3− 2 cosφ)dφ =xˉ = L(φ) R dφ ∫ 0 2π ⋅[L(φ) R dφ] ∫ 0 2π 2 L(φ) 2 1 L(φ)dφ ∫ 0 2π L(φ)dφ ∫ 0 2π 2 Next, calculate the integral in the numerator: Using the standard integrals , , and : Finally, substitute these back into the expression for : 4. Result: ● Final Conclusion: The -coordinate of the mass center