Question

[Key Concept: For rolling without slipping, a = g sin θ/(1 + I/mR²) and the friction needed is mg sin θ/(1 + mR²/I), limited by μN.]

$$a = \frac{gsin\theta}{1 + \frac{I}{{mR}^{2}}}$$

$$\begin{matrix} & a = \frac{g\left( \frac{1}{2} \right)}{1 + \frac{{mR}^{2}}{{mR}^{2}}} = \frac{g}{4} \\ & f = \frac{{mg}^{\sin\theta}}{1 + \frac{{mR}^{2}}{I}} \\ & f = \frac{mg\left( \frac{1}{2} \right)}{1 + 1} \\ & f = \frac{mg}{4} \rightarrow \text{~ans~}(b) \end{matrix}$$

$$\begin{matrix} & \frac{mg}{4} \leq \mu mg\frac{\sqrt{3}}{2} \\ & \mu_{\min} \geq \frac{1}{2\sqrt{3}} \end{matrix}$$