The upper portion of an inclined plane of inclination $\alpha $ is smooth and the lower portion is rough. A particle slides down from rest from the top and just comes to rest at the foot. If the ratio of the smooth length to rough length is $m : n$ , the coefficient of friction is
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On smooth surface $a_{1}=g \sin \alpha$
$\therefore v^{2}=u^{2}+2 a_{1} s_{1}$
$=0+2 g \sin \alpha m$
On rough surface
$a_{2}=g \sin \alpha-\mu g \cos \alpha$
$\therefore \quad v^{\prime 2}=v^{2}+2 a_{2} s_{2}$
$\mathrm{O}=2 \mathrm{mg} \sin \alpha+2 \mathrm{g}(\sin \alpha-\mu \cos \alpha) \mathrm{n}$
$\Rightarrow \mu=\left[\frac{\mathrm{m}+\mathrm{n}}{\mathrm{n}}\right] \tan \alpha$
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