question_answer

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

A) \[2\,\tan \,\phi \]

B) \[\,\tan \,\phi \]

C) \[2\,sin\,\phi \]  

D) \[2\,\cos \,\phi \]

Correct Answer: A

Solution :

[a] For first half acceleration \[=gsin\phi \] Therefore, velocity after travelling half distance \[{{v}^{2}}=2(gsin\phi )l\]                                (i) For second half, acceleration \[=g(sin\phi -{{\mu }_{k}}cos\phi )\] So \[{{0}^{2}}={{v}^{2}}+2g(sin\phi -{{\mu }_{k}}cos\phi )l\]                       (ii) Solving (i) and (ii), we get \[{{\mu }_{k}}=2\tan \phi \]