Meriam, Kraige & Bolton — Rigid Bodies: Equivalent Systems of Forces: Problem 3_26

⚡ Mecademy AIENG정역학 · ch3  Problem Statement The rack has a mass . What moment must be exerted on the gear wheel by the motor in order to lower the rack at a slow steady speed down the incline? Neglect all friction. The fixed motor which drives the gear wheel via the shaft at is not shown. Problem 3/26 (a) Calculation of the required motor moment 1. Formula: Since the rack is being lowered at a slow steady speed, it is in a state of dynamic equilibrium (acceleration is zero). We first analyze the forces acting on the rack along the direction of the incline. Let be the tangential force exerted by the gear on the rack to resist its downward motion. The equilibrium equation for the rack along the incline is: The moment required at the center of the gear to produce this force at the pitch radius is given by: Combining these, the final formula for the required moment is: 2. Substitution: Using the given values: , , , and the pitch diameter . 3. Calculation: Calculate the component of the rack's weight acting down the incline: . Calculate the pitch radius of the gear: . m=75 kgM 60 ∘ O F F = ∑ incline 0⟹F−mgsinθ=0 MOF r M=F⋅r=F⋅ 2 d M= (m g sin θ ) 2 d m=75 kgg= 9.81 m/s 2 θ=60 ∘ d=150 mm=0.15 m M=(75⋅9.81⋅sin60)⋅ ∘ 2 0.15 W = parallel 75⋅9.81⋅sin60≈ ∘ 735.75⋅0.8660=637.18 N r= = 2 0.15 0.075 m Calculate the required moment: . 4. Result: ● Final Conclusion: To lower the rack at a slow steady speed, the motor must exert a moment on the gear wheel. ✨ Final Answer Summary (a) Mecadem