Meriam, Kraige & Bolton — Rigid Bodies: Equivalent Systems of Forces: Problem 3_87

⚡ Mecademy AIENG정역학 · ch3  Problem Statement Determine a general expression for the normal force exerted by the smooth vertical wall on the uniform slender bar of mass and length . The mass of the cylinder is , and all bearings are ideal. Determine the value of which makes (a) and (b) . Problem 3/87 (a) General Expression for the Normal Force 1. Formula: Apply the principle of moments about the pivot for the bar in equilibrium: Where counter-clockwise moments are considered positive. 2. Substitution: Consider the forces and their respective lever arms from . Let be the angle the bar makes with the horizontal (given as ). The weight acts at the midpoint . The normal force acts at the end (length ). The tension in the horizontal cable acts at point (distance from ). N A mLm 1 m 1 N = A mg/2 N = A 0 N A O M = ∑ O 0 Oθ 30 ∘ W=mg L/2N A AL T=m g 1 B2L/3O m gcosθ − ( 2 L )N (Lsinθ)− A m g sinθ = 1 ( 3 2L )0 3. Calculation: Simplify the equation by dividing through by : Solve for : Substituting the given angle (): 4. Result: For the position shown (): ● Final Conclusion: The general expression for the normal force at is . For the specific orientation shown, . (b) Value of for 1. Formula: Set the derived expression for equal to at : 2. Substitution: 3. Calculation: Isolate the term with : L cos θ − 2 mg N sinθ− A m g sin θ = 3 2 1 0 N A N sinθ= A cos θ − 2 mg m g sin θ 3 2 1 N = A cot θ − 2 mg m g 3 2 1 θ=30 ∘ cot30= ∘ 3 N = A m g − 2 3 m g 3 2 1 N = A cot θ − 2 mg m g 3 2 1 θ=30 ∘