Meriam, Kraige & Bolton — Internal Forces and Moments: Problem 7_8

⚡ Mecademy AIENG정역학 · ch7  Problem Statement The uniform platform of mass is supported in the position shown by uniform supports of mass and length . If a couple holds the platform and supports in the equilibrium position shown, determine the orientation . Problem 7/8 (a) Determination of the equilibrium orientation ● Calculation Process 1. Formula: We use the principle of virtual work for a system with one degree of freedom . For equilibrium, the total virtual work must be zero: Alternatively, using the potential energy method where is the total gravitational potential energy: 2. Substitution: First, we define the potential energy of the system with respect to the datum at the base pivots (). The platform of mass is at a height . Each of the uniform supports of mass has its center of mass at a height . 3. Calculation: Differentiate the potential energy with respect to to find the virtual work done by gravity: m 0 n mbM θ θ θδU δU=δU + couple δU = gravity 0 V δU=Mδθ−δV=0⇒M= dθ dV V y=0 m 0 y = p bsinθ nm y = s sin θ 2 b V=m g(bsinθ)+ 0 n⋅ m g sinθ( 2 b ) V=m + g bsin θ( 0 2 nm ) θ = dθ dV m + gbsinθ = dθ d [( 0 2 nm )]m + g bcos θ( 0 2 nm ) Set up the equilibrium equation: 4. Result: Solve for and then : ● Final Conclusion: The equilibrium orientation is given by the expression . This angle defines the configuration where the work done by the applied couple balances the change in gravitational potential energy of the platform and its supports. ✨ Final Answer Summary (a) Meca