A body of mass ' $m$ ' is launched up on a rough inclined plane making an angle of $30^{\circ}$ with the horizontal. The coefficient of friction between the body and plane is $\frac{\sqrt{x}}{5}$ if the time of ascent is half of the time of descent. The value of $x$ is_________.
$\mathrm{t}_{\mathrm{a}}=\frac{1}{2} \mathrm{t}_{\mathrm{d}}$
$\sqrt{\frac{2 \mathrm{~s}}{\mathrm{a}_{\mathrm{a}}}}=\frac{1}{2} \sqrt{\frac{2 \mathrm{~s}}{\mathrm{a}_{\mathrm{d}}}}$ .............(i)
$\mathrm{a}_{\mathrm{a}}=\mathrm{g} \sin \theta+\mu \mathrm{g} \cos \theta$
$=\frac{g}{2}+\frac{\sqrt{3}}{2} \mu g$
$\mathrm{a}_{\mathrm{d}}=\mathrm{g} \sin \theta-\mu \mathrm{g} \cos \theta$
$=\frac{\mathrm{g}}{2}-\frac{\sqrt{3}}{2} \mu \mathrm{g}$
using the above values of $\mathrm{a}_{\mathrm{a}}$ and $\mathrm{a}_{\mathrm{d}}$ and putting in eqution (i) we will gate $\mu=\frac{\sqrt{3}}{5}$
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