When a body slides down from rest along a smooth inclined plane...

On smooth incline,

$a = gsin30 °$

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

$S = \dfrac{1}{2}\dfrac{g}{2}{T}^{2} = \dfrac{g}{4}{T}^{2} . . . ( i )$

On rough incline

$a = gsin30 ° - \mu gcos30 °$

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

$S = \dfrac{1}{4}g ( 1 - \sqrt{3}\mu ) ( \alphaT{)}^{2} . . . ( ii )$

By $( i )$ and $( ii )$

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

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

$⇒ x = 3 . 00$