Meriam, Kraige & Bolton — Internal Forces and Moments: Problem 7_17

⚡ Mecademy AIENG정역학 · ch7  Problem Statement Determine the couple which must be applied at in order to support the mechanism in the position . The masses of the disk at , bar , and bar are , , and , respectively. Problem 7/17 (a) Determination of the supporting couple 1. Formula: The principle of virtual work for a system in equilibrium is . For a system with potential energy and an external couple , this is expressed as: where is the total gravitational potential energy of the system. 2. Substitution: First, we determine the vertical positions (heights ) of the center of gravity for each part relative to point . For bar (length , mass ): the center of mass is at its midpoint, so . For bar (total length , mass ): the diagram shows point is the midpoint (). Thus, the center of mass is at , so . For the disk at (mass ): since is the midpoint of and is on the horizontal axis through , point is at twice the height of , so . Substitute these into the potential energy equation: 3. Calculation: Simplify the potential energy expression: MO θ=30 ∘ COABCm 0 m2m M δU=0 VM δU=Mδθ−δV=0⟹M= dθ dV V=m gy ∑ ii y O OAlmy = 1 sin θ 2 l BC2l2mA BA=AC=lAy = 2 lsinθ Cm 0 ABCB OCAy = 3 2lsinθ V= m g sinθ + ( 2 l )2mg(lsinθ)+m g(2lsinθ) 0 V= m+2m+2m gl sin θ = ( 2 1 0 ) m+2m gl sin θ( 2 5 0 ) Differentiate with respect to to find the equilibrium couple : Evaluate at : 4. Result: ● Final Conclusion: The couple required to support the mechanism in equilibrium at is , or approximately . ✨ Final Answer