Problem 2 - Sliding sleeve on rotating half-circle

Question:

A sleeve A can slide freely along a smooth rod bent into the shape of a half-circle of radius R. The system is set in rotation with a constant angular velocity $\omega$ about a vertical axis OO’. Find the angle $\theta$ corresponding to the steady position of the sleeve

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Solution

Upon drawing a free-body diagram of the position of equilibrium, we can see that:

$$m{\omega }^{2}R\sin \theta \cos \theta =mg\sin \theta$$

Which simplifies to:

$$\theta =\arccos (\frac{g}{{\omega }^{2}R})$$