Meriam, Kraige & Bolton — Analysis of Structures: Problem 6_15

⚡ Mecademy AIENG정역학 · ch6  Problem Statement A uniform ladder is positioned as shown for the purpose of maintaining the light fixture suspended from the cathedral ceiling. Determine the minimum coefficient of static friction required at ends A and B to prevent slipping. Assume that the coefficient is the same at A and B. Problem 6/15 (a) Determination of the minimum coefficient of static friction ● Calculation Process 1. Present Final Formula: To solve for the minimum coefficient of static friction , we establish the equilibrium equations for the ladder as a rigid body. Let be the length of the ladder and be its weight acting at the center. The ladder is at an angle with the horizontal. The roof (ceiling) has a slope defined by a vertical rise of and a horizontal run of . Let the ceiling angle be : Based on the free-body diagram (FBD) at the point of impending slip: Where forces at (floor) are (up) and (to the right, opposing the ladder's tendency to slide left as slides down the roof). Forces at (ceiling) are the normal force (directed from the ceiling into the ladder: down and to the left) and friction (directed up along the ceiling). 2. Substitute Values: Force components at B: Equilibrium equations: μ s L W θ=60 ∘ 32 ′ 24 ′ α tanα= = 24 32 ⟹ 3 4 sinα=0.8,cosα=0.6 F = ∑ x 0, F = ∑ y 0, M = ∑ A 0 AN A F = A μ N sA BB N B F = B μ N sB N = B N (−sinα,−cosα)= B N (−0.8,−0.6) B F = B F (−cosα,sinα)= B μ N (−0.6,0.8) sB F = ∑ x F − A 0.8N − B 0.6μ N = sB 0⟹μ N = sA (0.8+0.6μ )N