When a body slides down from rest along a smooth inclined plane making an angle of 30 ° with the…

On smooth incline,

$a=g\sin 30°$

by $S=ut+\frac{1}{2}a{t}^2$

$S=\frac{1}{2}\frac{g}{2}{T}^2=\frac{g}{4}{T}^2 ...\left(\mathrm{i}\right)$

On rough incline

$a=g\sin 30°-\mu g\cos 30°$

by $S=ut+\frac{1}{2}a{t}^2$

$S=\frac{1}{4} g(1-\sqrt{3}\mu )(\alpha T{)}^2 ...\left(\mathrm{ii}\right)$

By $\left(\mathrm{i}\right)$ and $\left(\mathrm{ii}\right)$

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

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

$\Rightarrow x=3.00$