question_answer

A smooth block is released at rest on a \[45{}^\circ \] incline and then slides a distanced. The time taken to slide is n times as much to slide on rough incline, than on a smooth incline. The coefficient of friction is:

A) \[{{\mu }_{\operatorname{k}}}=1-\frac{1}{{{\operatorname{n}}^{2}}}\]          

B) \[{{\mu }_{\operatorname{k}}}=\sqrt{1-\frac{1}{{{\operatorname{n}}^{2}}}}\]

C) \[{{\mu }_{\operatorname{s}}}=1-\frac{1}{{{\operatorname{n}}^{2}}}\]                      

D) \[{{\mu }_{\operatorname{s}}}=\sqrt{1-\frac{1}{{{\operatorname{n}}^{2}}}}\]

Correct Answer: C

Solution :

\[\operatorname{S}=\frac{1}{2}g\,Sin45{{t}^{2}}\]                                           ?.(1) For rough inclined plane \[\operatorname{S}=\frac{1}{2}\left[ g\,Sin45{}^\circ -\mu g\,Cos45{}^\circ  \right]{{t}^{2}}{{n}^{2}}\]           ?.(2) \[\left[ a = g Sin\theta - \mu g Cos\theta  where \theta  = 45{}^\circ   \right]\] By (1) and (2) \[\frac{g}{2}\times \frac{1}{\sqrt{2}}{{t}^{2}}=\frac{g}{2}\left[ \frac{1}{\sqrt{2}}-\frac{\mu }{\sqrt{2}} \right]{{\operatorname{n}}^{2}}{{t}^{2}}\] \[\frac{1}{\sqrt{2}}=\frac{\left( 1-\mu  \right)}{\sqrt{2}}{{\operatorname{n}}^{2}}\] \[\frac{1}{{{n}^{2}}}=1-\mu \] \[\mu =1-\frac{1}{{{n}^{2}}}\]