Meriam, Kraige & Bolton — Internal Forces and Moments: Problem 7_10

⚡ Mecademy AIENG정역학 · ch7  Problem Statement The spring of constant is unstretched when . Derive an expression for the force required to deflect the system to an angle . The mass of the bars is negligible. Problem 7/10 (a) Expression for Force 1. Formula: The principle of virtual work states that for a system in equilibrium, the total virtual work done by all external forces and internal elastic forces is zero: where is the elastic potential energy of the spring. 2. Substitution: Let the fixed pivot be the origin . The coordinates of point where force is applied and the sliding collar are: The spring is unstretched at , where . The stretch in the spring at angle is: The elastic potential energy is: 3. Calculation: Determine the virtual displacements and the variation in potential energy: Virtual displacement of in the -direction: Variation of potential energy: Substitute into the virtual work equation: kθ=0P θ P δW=P δr − ∑ ii δV = e 0 V e C(0,0)B PA x = B lsinθ,y = A 2lcosθ θ=0y = A 2ls θ s=2l−2lcosθ=2l(1−cosθ) V = e ks = 2 1 2 k [2 l (1− 2 1 cosθ)]= 2 2kl(1− 2 cosθ) 2 Bxδx = B ( l sin θ ) δθ = dθ d lcosθδθ δV = e δθ = dθ dV e [2 kl (1− dθ d2 cosθ)]δθ 2 δV = e 4kl(1− 2 cosθ)(sinθ)δθ Pδx − B δV = e 0 Divide by (assuming ): 4. Result: ● Final Conclusion: The horizontal force required to maintain equilibrium at angle is . ✨ Final Answer Summary (a) Mecademy AI Solution · ENGProblem 7/10 P(lcosθδθ)−4kl(1− 2 cosθ)sinθδθ=0 lcosθδθθ=π/2 P= lcosθ 4kl(1−cosθ)sinθ 2 P=4kltanθ(1−cosθ