The upper half of an inclined plane of inclination $\theta$ is perfectly smooth while the lower half is rough. A body starting from the rest at top comes back to rest at the bottom if the coefficient of friction for the lower half is given by
AIEEE 2005, Difficult
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(d) For upper half
${v^2} = {u^2} + 2al/2 = 2(g\sin \theta )l/2 = gl\sin \theta $
For lower half
$⇒$ $0 = {u^2} + 2g(\sin \theta - \mu \cos \theta )\frac{l}{2}$
$⇒$ $ - gl\sin \theta = gl(\sin \theta - \mu \cos \theta )$
$⇒$ $\mu \cos \theta = 2\sin \theta \Rightarrow \mu = 2\tan \theta $
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