Meriam, Kraige & Bolton — Analysis of Structures: Problem 6_120

⚡ Mecademy AIENG정역학 · ch6  Problem Statement Plot the force required to begin moving the 80-kg crate up the incline starting from rest at various values of from 1 to 10 m. Note that the coefficient of static friction increases with the distance down the incline according to , where and is in meters. Determine the minimum value of and the corresponding value of . Neglect the effects of the length of the crate along the incline. Problem 6/120 (a) Force Equilibrium and Expression for 1. Formula: From the free-body diagram of the crate at the point of impending motion up the incline, we establish the equilibrium equations along the axes parallel () and perpendicular () to the incline. where is the angle the cable makes with the incline, , and . 2. Substitution: From the geometry, and . Substituting from the first equation into the second: 3. Calculation: Simplify the expression for as a function of : 4. Result: P15 ∘ x xμ = s μ x 0 μ = 0 0.10x Px P x ′ y ′ F = ∑ y ′ N+Psinα−mgcosθ=0 F = ∑ x ′ Pcosα−μ N− s mgsinθ=0 αθ=15 ∘ μ = s 0.1x sinα= x+3 22 3 cosα= x+3 22 x N Pcosα−μ (mgcosθ− s Psinα)−mgsinθ=0 P(cosα+μ sinα)= s mg(sinθ+μ cosθ) s P= m g +(0.1x) x+9 2 x x+9 2 3 sin15+0.1xcos15 ∘∘ Px mg=80×9.81=784.8 N P= 784.8 x(1+0.3) (sin15+0.1xcos15) ∘∘ x+9 2 P= (sin15+ 1.3x 784.8∘ 0.1xcos15) ∘ x+9 2 P(x)= (sin15+ x 603.7∘ 0.1xcos15) N ∘ x+9 2 ● Final Conclusion: The required force as a function of distance is . (b) Determination of the Distance for Minimum Force 1. Formula: To find the mi