\({a}={g} \sin 30^{\circ}\)

by \({S}={ut}+\frac{1}{2} {at}^{2}\)

\({S}=\frac{1}{2} \frac{{g}}{2} {T}^{2}=\frac{{g}}{4} {T}^{2} \ldots \ldots (i)\)

On rough incline

\({a}={g} \sin 30^{\circ}-\mu {g} \cos 30^{\circ}\)

\({by} {S}={ut}+\frac{1}{2} {at}^{2}\)

\({S}=\frac{1}{4} {g}(1-\sqrt{3} \mu)(\alpha {T})^{2} \ldots \text { (ii) }\)

\(\text { By (i) and (ii) }\)

\(\frac{1}{4} {g} {T}^{2}=\frac{1}{4} {g}(1-\sqrt{3} \mu) \alpha^{2} {T}^{2}\)

\(\Rightarrow 1-\sqrt{3} {g}=\frac{1}{\alpha^{2}} \Rightarrow {g}=\left(\frac{\alpha^{2}-1}{\alpha^{2}}\right) \cdot \frac{1}{\sqrt{3}}\)

\(\Rightarrow {x}=3.00\)