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The Fundamental Equation of Dynamics

Problem 1.110

A sleeve AA can slide freely along a smooth rod bent in the shape of a half-circle of radius RR (Fig. 1.28 ). The system is set in rotation with a constant angular velocity ω\omega about a vertical axis OO′O O^{\prime}. Find the angle θ\theta corresponding to the steady position of the sleeve.

Reveal Answer
1.110. When ω2R>g\omega^{2} R>g, there are two steady equilibrium positions: θ1=0\theta_{1}=0 and θ2=arccos⁡(g/ω2R).\theta_{2}=\arccos \left(g / \omega^{2} R\right) . When ω2R^<g\omega^{2} \hat{R}<g, there is only one equilibrium position: θ1=0.\theta_{1}=0 . As long as there is only one lower equilibrium position, it is steady. Whenever the second equilibrium position appears (which is permanently steady) the lower one becomes unsteady.
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