Meriam, Kraige & Bolton — Internal Forces and Moments: Problem 7_23

⚡ Mecademy AIENG정역학 · ch7  Problem Statement The postal scale consists of a sector of mass hinged at and with center of mass at . The pan and vertical link have a mass and are hinged to the sector at . End is hinged to the uniform link of mass , which in turn is hinged to the fixed frame. The figure forms a parallelogram, and the angle is a right angle. Determine the relation between the mass to be measured and the angle , assuming that when . Problem 7/23 (a) Relation between mass and angle 1. Formula: The principle of virtual work for a system in equilibrium is given by , where is the total potential energy. Alternatively, using the balance of moments about the hinge : Taking the horizontal through as the reference for potential energy (), the total potential energy is: where are the vertical coordinates of the centers of mass of the sector, the pan/link assembly, and link , respectively. 2. Substitution: Let and . Let be the angle of from the vertical (pointing downward). Since , link makes an angle with the horizontal. Vertical position of : Vertical position of : (assuming is below the horizontal) Vertical position of the center of mass of link : The total potential energy is: m 0 OG ABm 1 BA ACm 2 OBACGOB mθθ=θ 0 m=0 mθ δV=0V O M = ∑ O 0 Oy=0 V V=mgy + 0G (m+m )gy + 1B m gy 2AC y ,y ,y GBAC ABAC a=OGb=OB=ACθOG ∠GOB=90 ∘ OBθ Gy = G −acosθ By = B −bsinθOB ACy = AC y − C sin θ 2 b V=−m gacosθ− 0 (m+m )gbsinθ− 1 m g sin θ + 2 2 b constant 3. Calculation: Differentiating wi