Meriam, Kraige & Bolton — Analysis of Structures: Problem 6_6

⚡ Mecademy AIENG정역학 · ch6  Problem Statement Determine the minimum coefficient of static friction which will allow the drum with fixed inner hub to be rolled up the incline at a steady speed without slipping. What are the corresponding values of the force and the friction force ? Problem 6/6 (a) Value of the force 1. Formula: To find the force required to move the drum up the incline at a steady speed, we apply the equilibrium of moments about the contact point . 2. Substitution: Let the center of the drum be at distance from the incline. The force is applied at the top of the hub of radius , which is at a distance of from the contact point . The weight acts at . Taking counter- clockwise as positive: 3. Calculation: Divide both sides by : Solve for : Substitute trigonometric values (, ): 4. Result: ● Final Conclusion: The magnitude of the required force is . (b) Value of the friction force μ s 15 ∘ PF P P C M = ∑ C 0 Gr Pr/2r+ r/2=1.5rCmgG (Pcos30)(1.5r)− ∘ (mgsin15)(r)= ∘ 0 r1.5Pcos30= ∘ mgsin15 ∘ PP= = 1.5cos30 ∘ mgsin15 ∘ 3cos30 ∘ 2mgsin15 ∘ sin15≈ ∘ 0.2588cos30≈ ∘ 0.8660 P= ≈ 3⋅0.8660 2⋅0.2588⋅mg m g ≈ 2.598 0.5176 0.1992mg P=0.1992mg P=0.1992mg F 1. Formula: From the force equilibrium parallel to the incline (-direction): 2. Substitution: Substitute the expression for from the previous step. 3. Calculation: 4. Result: ● Final Conclusion: The magnitude of the friction force acting up the incline is . (c) Minimum coefficient of static friction 1. Formula: The minimum coe