Meriam, Kraige & Bolton — Internal Forces and Moments: Problem 7_46

⚡ Mecademy AIENG정역학 · ch7  Problem Statement The uniform link has a mass , and its left end travels freely in the fixed horizontal slot. End is attached to the vertical plunger, which compresses the spring as falls. The spring would be uncompressed at the position . Determine the angle for equilibrium (other than the impossible position corresponding to ) and designate the condition which will ensure stability. Problem 7/46 (a) Determine the equilibrium angle 1. Formula: We use the principle of potential energy. The total potential energy of the system is the sum of the gravitational potential energy of the link and the elastic potential energy of the spring. Equilibrium occurs where . 2. Substitution: Let the horizontal slot define the datum . The vertical position of end is . The center of mass of the uniform link is at its midpoint, so its vertical position is . The spring is uncompressed at , so its compression is . Total potential energy: 3. Calculation: Differentiate with respect to and set to zero for equilibrium: ABmA BB θ=0θ θ=90 ∘ θ V V g V e = dθ dV 0 V=V + g V e y=0B y = B −lsinθ GAB y = G y = 2 1 B − sin θ 2 l θ=0x=−y = B lsinθ V = g mgy = G − mglsinθ 2 1 V = e kx= 2 1 2 k(lsinθ)= 2 1 2 klsinθ 2 1 2 2 V= klsinθ− 2 1 2 2 mglsinθ 2 1 Vθ = dθ dV klsinθ− mglsinθ = dθ d ( 2 1 22 2 1 )0 klsinθcosθ− 2 m gl cos θ = 2 1 0 Factor out the common terms: Since the problem specifies , we have . Therefore: 4. Result: ● Final Conclusion: The equilibrium angle is . (b) Designate