Meriam, Kraige & Bolton — Analysis of Structures: Problem 6_38

⚡ Mecademy AIENG정역학 · ch6  Problem Statement The semicylindrical shell of mass and radius is rolled through an angle by the horizontal force applied to its rim. If the coefficient of friction is , determine the angle at which the shell slips on the horizontal surface as is gradually increased. What value of will permit to reach 90°? Problem 6/38 (a) Determination of the slip angle 1. Formula: Establish the equilibrium equations for the semicylindrical shell. Let the center of the diameter be and the contact point be . The distance from to the center of mass for a thin semicylindrical shell is . Sum of vertical forces: Sum of horizontal forces: Sum of moments about the contact point : Slip condition: 2. Substitution: From force equilibrium: and . The moment arm of force about is . The moment arm of weight about is . Moment equation: 3. Calculation: Substitute into the moment equation: Divide both sides by : Rearrange to solve for : mrθ Pμ s θ Pμ s θ θ OCO G=rˉ π 2r F =∑ y N−mg=0F =∑ x P−F=0 CM =∑ C 0F=μ N s N=mgF=P=μ mg s PCh = P r+rsinθ mgCd = mg sinθ=rˉ sinθ π 2r P⋅r(1+ sinθ)−mg⋅ sinθ=( π 2r )0 P=μ mg s (μ mg)⋅ s r(1+sinθ)=mg⋅ sin θ π 2r mgr μ (1+ s sinθ)= sin θ π 2 sinθ μ + s μ sinθ= s sin θ π 2 4. Result: ● Final Conclusion: The angle at which slipping occurs is . (b) Minimum coefficient of static friction for 1. Formula: Use the relation between and derived in part (a): 2. Substitution: Set . 3. Calculation: 4. Result: ● Final Conclusion: The coefficient of static fricti