Meriam, Kraige & Bolton — Rigid Bodies: Equivalent Systems of Forces: Problem 3_40

⚡ Mecademy AIENG정역학 · ch3  Problem Statement The uniform bar OC of length L pivots freely about a horizontal axis through O. If the spring of modulus k is unstretched when C is coincident with A, determine the tension T required to hold the bar in the 45° position shown. The diameter of the small pulley at D is negligible. Problem 3/40 (a) Tension T required for equilibrium 1. Formula: To determine the tension , we apply the equilibrium of moments about the pivot point . where is the moment of the tension , is the moment of the bar's weight, and is the moment of the spring force. 2. Substitution: Let the pivot be at the origin . The bar of length is at from the vertical. Coordinates of points: , , . Spring properties: The unstretched length is the distance when is at (vertical position), so . The current spring length is . Forces: Weight acts downwards at . Spring force acts from toward . Tension force acts at along the cable at an angle of with the horizontal. Moment components (counter-clockwise is positive): T O M = ∑ O M + T M + W M = s 0 M T TM W M s O(0,0)OCL θ=45 ∘ A=(0,−L)B=( sin45,− cos45) 2 L ∘ 2 L ∘ C=(Lsin45,−Lcos45) ∘∘ l 0 ABCA l = 0 L− = 2 L 2 L l= ∣∣= AB = ( sin45)+(L− cos45) 2 L ∘2 2 L ∘2 L = −cos45 4 5 ∘ 2 L 5−2 2 W=mgBF = s k(l− l ) 0 BATC 30 ∘ M = W −m g sin45 = ( 2 L ∘ ) − 4 mgL 2 3. Calculation: Substitute terms into the equilibrium equation: Simplify by dividing by : Using numerical approximations: 4. Result: ● Final Conclusion: The tension required to