Meriam, Kraige & Bolton — Rigid Bodies: Equivalent Systems of Forces: Problem 3_25

⚡ Mecademy AIENG정역학 · ch3  Problem Statement Find the angle of tilt with the horizontal so that the contact force at will be one-half that at for the smooth cylinder. Problem 3/25 (a) Determination of the tilt angle ● Calculation Process 1. Present Final Formula: We establish a coordinate system where the -axis is aligned with the centerline (angle bisector) of the V-notch and the -axis is parallel to its base. The notch is tilted at an angle relative to the horizontal, meaning the weight vector makes an angle with the -axis. Since the cylinder is smooth, contact forces and are normal to the surfaces, each acting at an angle of to the - axis. The equilibrium equations are: From these, we derive the relation: 2. Substitute Values: The problem specifies that the contact force at is one-half that at , so we set . Also, . 3. Partial Operations: Simplify the expression for : Numerator: Denominator: Ratio: So, . θB A θ y x θ mgθy R A R B 45 ∘ y F = ∑ x R sin45− A ∘ R sin45− B ∘ mgsinθ=0 F = ∑ y Rcos45+ A ∘ R cos45− B ∘ mgcosθ=0 tanθ= tan45 R +R AB R −R AB ∘ B AR = B 0.5R A tan45= ∘ 1 tanθ= (1) R +0.5R AA R −0.5R AA tanθ R − A 0.5R = A 0.5R A R + A 0.5R = A 1.5R A = 1.5R A 0.5R A 3 1 tanθ= 3 1 4. Final Calculation: Calculate the angle: ● Final Conclusion: The required angle of tilt is . ✨ Final Answer Summary (a) Mecademy AI Solution · ENGProblem 3/25 θ= arctan ≈ ( 3 1 )18.4349 ∘ θ=18.43 ∘ θ=18.43 ∘