question_answer

The upper half of an inclined plane of inclination q 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                                                 [Pb. PMT 2000]

A)             \[\mu =\sin \theta \]

B)               \[\mu =cot\theta \]

C)             \[\mu =2\cos \theta \]

D)               \[\mu =\text{ }2\text{ }tan\theta \]

Correct Answer: D

Solution :

                For upper half             \[{{v}^{2}}={{u}^{2}}+2al/2=2(g\sin \theta )l/2=gl\sin \theta \]             For lower half             Þ\[0={{u}^{2}}+2g(\sin \theta -\mu \cos \theta )\frac{l}{2}\]             Þ \[-gl\sin \theta =gl(\sin \theta -\mu \cos \theta )\]             Þ\[\mu \cos \theta =2\sin \theta \Rightarrow \mu =2\tan \theta \]