Meriam, Kraige & Bolton — Analysis of Structures: Problem 6_104

⚡ Mecademy AIENG정역학 · ch6  Problem Statement A light cable is connected to the ends of the uniform bar and goes over a fixed peg . Beginning from the horizontal position shown in part of the figure, a length of cable is moved from the right to the left side of the peg as shown in part of the figure. If the cable first slips on the peg at this position, determine the coefficient of static friction between the peg and the cable. Neglect the effects of the diameter of the peg. Problem 6/104 (a) Determination of cable geometry and slipping condition 1. Formula: Cable lengths in part (a): Cable lengths in part (b): , Geometric constraint for the bar of length : Belt friction formula at the verge of slipping: , where is the wrap angle. Equilibrium of the uniform bar: and . For a uniform bar, these combine with geometry to show . 2. Substitution: Initial cable segment length: Part (b) lengths: , Geometric parameters: , 3. Calculation: Calculate : ABC ad=0.15 m b L = AC L = BC h+(L/2) 22 L = 1 L + AC dL = 2 L − BC d LL + 1 2 L + 2 2 2L L cos(α+ 12 γ)=L 2 = T 2 T 1 e μ β s β=α+γ F =∑ x 0⟹T cosα= 1 T cosγ 2 M =∑ G 0⟹T sin(α− 1 θ)=T sin(γ+ 2 θ) = T 2 T 1 L 2 L 1 L = AC = 0.3+0.3 22 0.3 m≈ 2 0.42426 m L = 1 0.3 + 20.15≈0.57426 mL = 2 0.3 − 2 0.15≈0.27426 m L=0.6 md=0.15 m L + 1 2 L 2 2 (0.3 + 20.15)+ 2 (0.3 − 20.15)= 2 2(0.3⋅ 2 2+0.15)= 2 2(0.18+0.0225)=0.405 Calculate : Find wrap angle : Find tension ratio: Apply friction formula: 4. Result: ● Final Conclusion: The minimum coefficient