Meriam, Kraige & Bolton — Analysis of Structures: Problem 6_87

⚡ Mecademy AIENG정역학 · ch6  Problem Statement For a certain coefficient of friction \mu and a certain angle \alpha, the force P required to raise m is 4 kN, and that required to lower m at a constant slow speed is 1.6 kN. Calculate the mass m. Problem 6/87 (a) Calculation of the mass 1. Formula: The relationship between tensions on either side of a fixed drum with belt friction is defined by the equation: where is the larger tension, is the smaller tension, is the coefficient of friction, and is the contact angle (wrap angle) in radians. For this problem: Case 1 (Raising ): The applied force must overcome both the weight and the friction, so . Case 2 (Lowering ): The weight is the driving force and must overcome the resisting force and friction, so . 2. Substitution: We are given the following values: Force to raise: Force to lower: From the "Lowering" equation, we have . Substituting this into the "Raising" equation: 3. Calculation: Calculate the weight : m T = 2 T e 1 μβ T 2 T 1 μ β mP P = raise We μα mW PW=P e lower μα P = raise 4 kN P = lower 1.6 kN e= μα P lower W P = raise W ⟹ ( P lower W )W= 2 P ⋅ raise P lower W W= =4 kN⋅1.6 kN kN 6.4 Calculate the mass using standard gravity : 4. Result: ● Final Conclusion: By relating the belt friction equations for both raising and lowering the mass, we find that the weight of the object is the geometric mean of the two applied forces. Dividing the resulting weight by the acceleration due to gravity, the mass is determined to be 25