Meriam, Kraige & Bolton — Analysis of Structures: Problem 6_27

⚡ Mecademy AIENG정역학 · ch6  Problem Statement The rack has a mass . What moment must be exerted by the gear wheel in order to (a) lower and (b) raise the rack at a slow steady speed on the greased rail? The coefficients of static and kinetic friction are and . The fixed motor which drives the gear wheel is not shown. Problem 6/27 (a) Moment to lower the rack 1. Formula: To lower the rack at a steady speed, the forces parallel to the incline must be in equilibrium. Since the component of weight down the incline () is greater than the kinetic friction (), the pinion must exert a resisting force up the incline. The equilibrium equation is: where is the pitch radius of the gear. 2. Substitution: Substitute the known values: , , , , and . 3. Calculation: Calculate total weight: Calculate geometric components: , Calculate force term: Final product: 4. Result: ● Final Conclusion: The moment required to lower the rack at a slow steady speed is . (The direction must be counter-clockwise to resist gravity). m=75 kgM 60 ∘ μ = s 0.10μ = k 0.05 M mgsinθ μ mgcosθ k M/r+F − k mgsinθ=0⟹M=mgr(sinθ−μ cosθ) k r m=75 kgg= 9.81 m/s 2 r= = 2 0.075 m 0.0375 mθ=60 ∘ μ = k 0.05 M=(75)(9.81)(0.0375)(sin60− ∘ 0.05cos60) ∘ W=75×9.81=735.75 N sin60≈ ∘ 0.8660cos60= ∘ 0.5 0.8660−(0.05×0.5)=0.8660−0.025= 0.8410 M=735.75×0.0375×0.8410≈23.204 N⋅m M=23.2 N⋅m 23.2 N⋅m (b) Moment to raise the rack 1. Formula: To raise the rack, the pinion must overcome both the downward component of weight and the kinetic fricti