The upper half of an inclined plane of inclination $\theta$ is perfectly smooth while lower half is rough. A block starting from rest at the top of the plane will again come to rest at the bottom, if the coefficient of friction between the block and lower half of the plane is given by
AIPMT 2013, Medium
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Let $m$ be mass of the block and $L$ be lenght of the inclined plane.
According to work - energy theorem $W=\Delta K=0 ($Initial and final speeds are zero$)$
$\therefore$ Wrok done by friction + Work done by gravity $=0$
$-\mu m g \cos \theta \frac{L}{2}+m g \sin \theta L=0$
$ \frac{\mu}{2} \cos \theta=\sin \theta$
$\mu=\frac{2 \sin \theta}{\cos \theta}=2 \tan \theta$
According to work - energy theorem $W=\Delta K=0 ($Initial and final speeds are zero$)$
$\therefore$ Wrok done by friction + Work done by gravity $=0$
$-\mu m g \cos \theta \frac{L}{2}+m g \sin \theta L=0$
$ \frac{\mu}{2} \cos \theta=\sin \theta$
$\mu=\frac{2 \sin \theta}{\cos \theta}=2 \tan \theta$
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