Meriam, Kraige & Bolton — Analysis of Structures: Problem 6_123

⚡ Mecademy AIENG정역학 · ch6  Problem Statement The small roller on the upper end of the uniform rod rests against the vertical surface at A, while the rounded end B rests on the platform which is slowly pivoted downward beginning at the horizontal position shown. For a coefficient of static friction at B, determine the angle of the platform at which slipping will occur. Neglect the size and friction of the roller and the small thickness of the platform. Problem 6/123 (a) Calculation of the critical angle for slipping 1. Formula: First, we establish the geometric relationship. Let be the length of the rod and be the angle the rod makes with the horizontal. In the initial horizontal position (), the rod makes an angle of with the vertical, so . The distance from the pivot to contact point is . If the rod has not yet slipped at platform angle , point remains at distance from : From static equilibrium of the uniform rod, the normal force at the vertical wall is found by summing moments about : Resolving forces along and normal to the platform at angle : Slipping impends when: 2. Substitution: μ = s 0.40 θ θ Lβ θ=0 30 ∘ β = 0 60 ∘ OBd=Lsin30= ∘ 0.5L θBdO x = B dcosθ=0.5Lcosθ cosβ= = L x B 0.5cosθ N A B M = ∑ B 0⇒N (Lsinβ)− A m g cosβ = ( 2 L )0⇒N = A 0.5mgcot θ F = B mgsinθ+N cosθ(Friction force) A N = B mgcosθ−N sinθ(Normal force) A F = B μ N sB Substitute the expressions for , , and into the slip condition: Using , , and : 3. Calculation: Rearrange the equation to solve for : Let