Circular Motion

Concept:The normal reaction provides centripetal force, and friction balances the weight to prevent sliding.Explanation:For the mass $M$ to remain stuck, vertical forces must be balanced: $f = Mg$, where $f$ is the friction force.The normal reaction $N$ provides the centripetal force: $N = M R \omega^{2}$.Friction is self-adjusting up to its maximum value: $f \le \mu N$.Substituting $f = Mg$ and $N = M R \omega^{2}$: $Mg \le \mu (M R \omega^{2})$.Cancel $M$: $g \le \mu R \omega^{2}$.Rearrange: $\omega^{2} \ge \frac{g}{\mu R}$.Thus minimum angular speed: $\omega_{\text{min}} = \sqrt{\frac{g}{\mu R}}$.Answer:Option D: $\sqrt{\frac{g}{\mu R}}$.