Meriam, Kraige & Bolton — Equilibrium of Rigid Bodies: Problem 4_122

⚡ Mecademy AIENG정역학 · ch4  Problem Statement The tree feller cuts off large trees near ground level and then continues to grasp the trunk. Determine the force in hydraulic cylinder for the position shown if the tree weighs 6000 lb. Determine the required pressure on the 4.72-in.-diameter piston of the cylinder. Problem 4/122 (a) Force in hydraulic cylinder ● Calculation Process 1. Present Final Formula: First, establish a coordinate system with the origin at the boom pivot . To find the force in cylinder , we apply the principle of moment equilibrium for the boom about pivot : where is the moment due to the tree's weight and is the moment due to the hydraulic cylinder force acting at point . The moment of a force acting at point about the origin is . 2. Substitute Values: Based on the geometric data from the diagram: Tree weight: Horizontal distance from to tree center of gravity: (to the left, so ). Coordinates of cylinder pivot on the boom relative to : , . Coordinates of cylinder pivot on the frame relative to : , . Calculate the cylinder geometry and force vector components: Vector . Length of cylinder: . AB AB OAB O M = ∑ O M + W M = cyl 0 M W M cyl BF=F i+ x F j y (x,y)M=xF − y yF x W=6000 lb Od = W 7.5+ ′ 6.8= ′ 14.3 ftx = tree −14.3 ft BOx = B −0.6 ft y = B 3.6 ft AOx = A −2.4 ft y = A −3.6 ft = AB(x − B x )i+ A (y − B y )j= A [−0.6−(−2.4)]i+ [3.6−(−3.6)]j=1.8i+7.2j ft L = AB = 1.8+7.2 22 ≈ 55.087.4216 ft Cylinder force components at : , . Establish the moment equili