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# The upper half of an inclined plane with inclination of $\phi$ is perfectly smooth while the lower half is rough. A body starting from rest at top will again come to rest at bottom if coefficient of friction for lower half is given by

$(a)\;2 \sin \phi \quad (b)\;2 \cos \phi \quad (c)\;\tan \phi \quad (d)\;2 \tan \phi$

By work energy theorem
$W_g=W_f$
$-mg(2S \sin \phi)=(\mu mg \cos \phi)S$
$2 \sin \phi=\mu \cos \phi$
$\mu=2 \tan \phi$
Hence d is the correct answer.
edited May 28, 2014

+1 vote