Meriam, Kraige & Bolton — Rigid Bodies: Equivalent Systems of Forces: Problem 3_59

⚡ Mecademy AIENG정역학 · ch3  Problem Statement The vertical mast supports the 4-kN force and is constrained by the two fixed cables BC and BD and by a ball-and-socket connection at A. Calculate the tension in BD. Can this be accomplished by using only one equation of equilibrium? Problem 3/59 (a) Calculate the tension in BD 1. Formula: To find the tension while eliminating other unknowns (reactions at and tension in cable ), we use the moment equilibrium equation about an axis passing through points and . Since is at the origin and lies on the -axis, this is the equation. 2. Substitution: Coordinates: , m, m, m. External force: kN applied at point m. Tension : Acts from to . Substitute into the moment equation: 3. Calculation: Moment of force : . The -component is . T 1 T 1 T 1 A T 2 BC ACACy M =∑ y 0 M = ∑ y (r × F F)⋅j+(r × B T )⋅ 1 j=0 A=(0,0,0)B=(0,0,10)C=(0,−5,0)D= (4,4,2) F=−4iE=(0,0,5) T 1 BD = BD(4−0)i+(4−0)j+(2−10)k=4i+4j−8k ∣∣= BD = 4+4+(−8) 222 = 16+16+64 = 96 4 m 6 T = 1 T = 1 4 6 4i+4j−8k (i+ 6 T 1 j−2k) (5k)×(−4i) +[] y (10k)× (i+j−2k) = [ 6 T 1 ] y 0 F(5k)×(−4i)=−20j kN⋅my −20 Moment of tension : . The -component is . Combine: 4. Result: ● Final Conclusion: The tension in cable is . (b) Can this be accomplished by using only one equation of equilibrium? 1. Formula: Analyze the forces that have moments about the chosen axis. 2. Substitution: The axis through points and is the -axis. 3. Calculation: The reaction force components at the ball-and-socket joint () all