Meriam, Kraige & Bolton — Rigid Bodies: Equivalent Systems of Forces: Problem 3_33

⚡ Mecademy AIENG정역학 · ch3  Problem Statement Determine the force required to hold the uniform bar of mass and length in an arbitrary angular position . Plot your result over the range , and state the value of for . Problem 3/33 (a) General expression for the force 1. Formula: The system is in equilibrium, so the sum of moments about the pivot must be zero: 2. Substitution: Identify the forces and their respective moment arms from the geometry. The weight of the uniform bar is , acting at its center of mass , which is at a distance from . The moment arm for the weight is the horizontal distance from to , which is . The tension acts at point towards point . In the isosceles triangle (where ), the altitude from to the base represents the moment arm for . The length of this altitude is . The moment equilibrium equation is: 3. Calculation: Divide both sides by : TmL θ0≤θ≤90 ∘ T θ=40 ∘ T O M = ∑ O 0 W=mgG 2 L O OGd= W sinθ 2 L TABOAB OA=OB=LOAB Td = T Lcos(θ/2) T⋅(Lcos(θ/2))−mg⋅ sinθ = ( 2 L )0 TLcos(θ/2)= mgLsin θ 2 1 L Tcos(θ/2)= m g sin θ 2 1 Using the double-angle identity : 4. Result: Assuming (valid for ): ● Final Conclusion: The force required is . (b) Plotting the result over the range 1. Formula: Use the derived function . 2. Substitution: Evaluate the function at key points within the specified range. At : At : At : At : 3. Calculation: Analyze the derivative to determine the curve's shape. . Since this is positive for , increases with . . Since this is negative, the cur