Meriam, Kraige & Bolton — Analysis of Structures: Problem 6_45

⚡ Mecademy AIENG정역학 · ch6  Problem Statement If the coefficient of friction between the steel wedge and the moist fibers of the newly cut stump is 0.20, determine the maximum angle which the wedge may have and not pop out of the wood after being driven by the sledge. Problem 6/45 (a) Determination of the maximum wedge angle To prevent the wedge from popping out of the wood after the driving force is removed, the system must be in a state of self-locking equilibrium. This occurs when the friction forces acting downward along the faces of the wedge are sufficient to counteract the upward components of the normal forces exerted by the wood. 1. Formula: From the equilibrium of forces in the vertical direction for the impending motion (popping out), we have: Where . Substituting this gives: 2. Substitution: Substitute the given coefficient of static friction into the formula: 3. Calculation: Solve for the half-angle and then the full angle : 4. Result: α α F = ∑ y 0⟹ 2 F cos − ( 2 α ) 2 N sin = ( 2 α )0 F=μ N s μ N cos = s ( 2 α )N sin ⟹ ( 2 α )μ = s tan ( 2 α ) μ = s 0.20 tan = ( 2 α )0.20 2 α α = 2 α arctan(0.20) ≈ 2 α 11.31 ∘ α=2×11.31 ∘ α≈22.6 ∘ ● Final Conclusion: The maximum angle the wedge can have without popping out is . For any angle larger than this, the component of the normal force pushing the wedge out would exceed the maximum available friction force. ✨ Final Answer Summary (a) Mecademy AI Solution · ENGProblem 6/45 α 22.6 ∘ α=22.6 ∘