Meriam, Kraige & Bolton — Analysis of Structures: Problem 6_83

⚡ Mecademy AIENG정역학 · ch6  Problem Statement What is the minimum coefficient of friction between the rope and the fixed shaft which will prevent the unbalanced cylinders from moving? Problem 6/83 (a) Determination of the minimum coefficient of static friction 1. Formula: To prevent the unbalanced cylinders from moving, the friction between the rope and the fixed shaft must be sufficient to balance the difference in weights. This is a belt friction problem where the relationship between the tensions on either side of the shaft at the point of impending slip is defined by: Solving for the coefficient of static friction : 2. Substitution: Based on the problem diagram, the larger tension is and the smaller tension is . The rope is draped over the top half of the shaft, making the contact angle (angle of wrap) . Converting this to radians: Substituting the values into the formula: 3. Calculation: Calculate the tension ratio: Calculate the natural logarithm: Divide by : 4. Result: μ T = 2 T e 1 μβ μ μ= β ln(T /T ) 21 T = 2 100 lb T = 1 50 lb β=180 ∘ β=180× ∘ = ( 180 ∘ π rad )π rad μ= π rad ln(100 lb/50 lb) = T 1 T 2 = 50 100 2 ln(2)≈0.69315 πμ= ≈ 3.14159 0.69315 0.22063 μ≈0.221 ● Final Conclusion: The minimum coefficient of static friction required to prevent the 100- lb cylinder from moving downward is . ✨ Final Answer Summary (a) Mecademy AI Solution · ENGProblem 6/83 μ=0.221 μ=0.221