Meriam, Kraige & Bolton — Analysis of Structures: Problem 6_114

⚡ Mecademy AIENG정역학 · ch6  Problem Statement Determine the range of mass over which the system is in equilibrium (a) if the coefficient of static friction is 0.20 at all three fixed shafts and (b) if the coefficient of static friction associated with shaft is increased to 0.50. Problem 6/114 (a) Range of mass for at all shafts 1. Formula: The relationship between tensions and for a flexible belt or rope passing over a series of fixed shafts with friction is given by the belt friction formula: where , is the coefficient of static friction, and is the total angle of contact in radians. For the three shafts and , the total contact angle is the sum of individual contact angles: . 2. Substitution: From the diagram, each wrap is a half-circle (), so rad. Total contact angle: rad. Coefficient of friction: . Fixed mass at one end: , so tension . Suspended mass at other end: , so tension . 3. Calculation: Maximum mass (Impending motion towards ): Minimum mass (Impending motion towards 2-kg mass): m B mμ=0.20 T 1 T 2 T = 2 T e 1 μβ total T > 2 T 1 μβ total A,B,C β = total β + A β + B β C 180 ∘ β = A β = B β = C π β = total π+π+π=3π μ=0.20 m = 0 2 kgT = 2kg 2g mT = m mg m max m mg=2g⋅e ⟹ μβ total m = max 2 e (0.20)(3π) m = max 2e≈ 0.6π 2×6.5864=13.1728 kg m min 2g=mg⋅e ⟹ μβ total m = min 2 e −μβ total m = min 2e≈ −0.6π 2/6.5864=0.3037 kg 4. Result: The mass must be in the range . ● Final Conclusion: The range of mass for which the system is in equilibrium is . (b) Range of mass for and