Meriam, Kraige & Bolton — Analysis of Structures: Problem 6_86

⚡ Mecademy AIENG정역학 · ch6  Problem Statement A 50-kg package is attached to a rope which passes over an irregularly shaped boulder with uniform surface texture. If a downward force is required to lower the package at a constant rate, (a) determine the coefficient of friction between the rope and the boulder. (b) What force would be required to raise the package at a constant rate? Problem 6/86 (a) Coefficient of friction 1. Formula: The relationship between tensions in a belt/rope passing over a fixed curved surface is given by the belt friction formula: where is the larger tension, is the smaller tension, is the coefficient of friction, and is the angle of wrap in radians. First, calculate the weight of the package: From the problem diagram, both ends of the rope are vertical, so the angle of wrap is: 2. Substitution: Mass , acceleration due to gravity . When lowering the package, motion is towards the package side. Friction opposes this, acting towards the force side. Thus, and . 3. Calculation: Rearrange for the exponential term: Take the natural logarithm of both sides: P=70 N μ P ′ μ T = 2 T e 1 μβ T 2 T 1 μ β W=mg β=180= ∘ π rad m=50 kgg= 9.81 m/s 2 W=50×9.81=490.5 N PT = 2 WT = 1 P= 70 N 490.5=70e μπ e= μπ = 70 490.5 7.0071 μπ=ln(7.0071)=1.9469 Solve for : 4. Result: ● Final Conclusion: The coefficient of friction between the rope and the boulder is . (b) Force required to raise the package 1. Formula: When raising the package, motion is towards the pulling force side