Meriam, Kraige & Bolton — Internal Forces and Moments: Problem 7_52

⚡ Mecademy AIENG정역학 · ch7  Problem Statement The light bar is pivoted at and swings in the vertical plane. When , the spring of stiffness is unstretched. Determine the equilibrium angle corresponding to a given vertical force applied to the end of the bar. Neglect the mass of the bar and the diameter of the small pulleys. Problem 7/52 (a) Determination of the Equilibrium Equation 1. Formula: The equilibrium of the system can be analyzed using the principle of virtual work or by summing moments about the pivot . For equilibrium, the sum of moments must be zero: The total potential energy of the system is given by: where is the current length of the spring, is its unstretched length, and is the vertical position of the point of application of force relative to . 2. Substitution: Let be the origin at the pivot. From the diagram, the fixed pulley is at coordinates . Point on the bar is at a distance from , and point is at from . The coordinates of point at an angle (measured from the vertical) are: The length of the spring between and is: At , the spring is unstretched, so: OCOθ=0 k P O M = ∑ O 0 V=V + e V = g k ( l − 2 1 l )+ 0 2 Py C ll 0 y C PO O(0,0) B(a,a)AaO C2aOAθ x = A asinθ,y = A acosθ lB(a,a)A(asinθ,acosθ) l= = (a−asinθ)+(a−acosθ) 22 a (1−sinθ)+(1−cosθ) 22 l= a = 1−2sinθ+sinθ+1−2cosθ+cosθ 2 2 a 3−2(sinθ+c θ=0 l = 0 l(0)= a = 3−2(0+1) a = 1a The vertical position of point (where is applied) relative to is (taking upwards). 3. Calculation: We use the virtual work princip