Meriam, Kraige & Bolton — Internal Forces and Moments: Problem 7_53

⚡ Mecademy AIENG정역학 · ch7  Problem Statement A uniform rectangular block of height and mass is centered in a horizontal position on the fixed circular surface of radius . Determine the limiting value of for stability. Problem 7/53 (a) Determination of the limiting value of for stability 1. Formula: The stability of equilibrium for a rigid body can be determined using the potential energy method. A position of equilibrium is stable if the potential energy is a minimum, which requires the second derivative of with respect to the displacement coordinate to be positive: Alternatively, using the formula for a body with a flat base () resting on a fixed circular surface of radius : where is the height of the center of gravity above the point of contact. 2. Substitution: For a uniform rectangular block of height , the center of gravity is at the geometric center. In the centered horizontal position, the height of above the contact point is: The radius of curvature of the block's base is infinite since it is flat: The radius of the fixed circular surface is . hm rh h V V θ V=mgy G Stability Condition: > dθ 2 dV 2 0 at θ=0 ρ=∞ r > h G 1 + ρ 1 r 1 h G hG G h = G 2 h ρ=∞⟹ = ρ 1 0 r 3. Calculation: Substituting the known values into the stability condition: Solve for : The equilibrium is stable as long as . The limiting value occurs when the second derivative equals zero, marking the transition from stable to unstable equilibrium. 4. Result: ● Final Conclusion: The limiting value of the