Meriam, Kraige & Bolton — Rigid Bodies: Equivalent Systems of Forces: Problem 3_70

⚡ Mecademy AIENG정역학 · ch3  Problem Statement The 800-lb tree trunk is known to have insect damage near point , so the winch arrangement shown is used to fell the tree with no cutting. If winch is tightened to and winch to , determine the force and moment reactions at . If the tree ultimately falls at this point because of the moment at , determine the angle which characterizes the line of impact . Assume that the base of the tree is equally strong in all directions. Problem 3/70 (a) Force and moment reactions at O 1. Formula: Establishing equilibrium at point : Coordinates: , , , , . 2. Substitution: Weight: Tension : Direction . Magnitude . Tension : Direction . Magnitude . 3. Calculation: O W 1 200 lb W 2 300 lbO Oθ OE O F= ∑R + O T + 1 T + 2 W=0⟹R = O −(T + 1 T + 2 W) M = ∑ O M + O r × OA T + 1 r × OB T = 2 0⟹M = O −(r × OA T O(0,0,0)A(0,0,10) ftB(0,0,16) ftC(25,0,0) ft D(20sin30,20cos30,0)= ∘∘ (10,17.32,0) ft W=−800k lb T 1 r = AC (25−0)i+(0−0)j+(0−10)k= 25i−10kL = AC = 25+(−10) 22 ≈ 72526.93 ft T = 1 (25i− 26.93 200 10k)=185.70i−74.28k lb T 2 r = BD (10−0)i+(17.32−0)j+(0− 16)k=10i+17.32j−16kL = BD = 10+17.32+(−16) 222 ≈ 65625.61 ft T = 2 (10i+ 25.61 300 17.32j−16k)=117.13i+202.88j−187.41k lb Reaction Force : Reaction Moment : 4. Result: , ● Final Conclusion: The reaction force at the base is and the reaction moment is . The magnitudes are and . (b) Angle for the line of impact OE 1. Formula: The tree falls in the direction of the resultant horizontal tipping force. A mo