question_answer

A given object takes \[n\]times as much time to slide down a \[45{}^\circ \] rough incline as it takes to slide down a perfectly smooth \[45{}^\circ \] incline. The coefficient of friction between the object and the incline is

A) \[(1-1/{{n}^{2}})\]                  

B) \[1/(1-{{n}^{2}})\]

C) \[\sqrt{(1-1/{{n}^{2}})}\]                     

D) \[1/\sqrt{(1-{{n}^{2}})}\]

Correct Answer: A

Solution :

For smooth surface, \[s=\frac{1}{2}g\sin \theta t_{1}^{2}\]	..(i)
For rough surface,\[a=g(\sin \theta -\mu \cos \theta )_{t_{2}^{2}}^{{}}\]
\[\therefore s=\frac{1}{2}g{{(sin\theta -\mu cos\theta )}_{t_{2}^{2}}}\]   ..(ii)

From (i) and (ii)

\[\therefore s=\frac{1}{2}g\sin {{\theta }_{t_{2}^{2}}}=\frac{1}{2}g{{(\sin \theta -\mu \cos \theta )}_{t_{2}^{2}}}\]

Given, \[\theta =45{}^\circ \therefore t_{1}^{2}=(1-\mu )t_{2}^{2}\]

Also, given that, \[{{t}^{2}}=n{{t}_{1}}\therefore t_{1}^{2}=\left( 1-\mu  \right){{n}^{2}}t_{1}^{2}\]

\[\frac{1}{{{n}^{2}}}=1-\mu \therefore \mu =\left( 1-\frac{1}{{{n}^{2}}} \right)\]