A) \[\mu =\tan \theta \left( 1-\frac{1}{{{n}^{2}}} \right)\]
B) \[\mu =\cot \theta \left( 1-\frac{1}{{{n}^{2}}} \right)\]
C) \[\mu =tan\theta {{\left( 1-\frac{1}{{{n}^{2}}} \right)}^{1/2}}\]
D) \[\mu =\cot \theta {{\left( 1-\frac{1}{{{n}^{2}}} \right)}^{1/2}}\]
Correct Answer: A
Solution :
[a] Case I smooth inclined plane a = g sin 0 \[S=ut+\frac{1}{2}a{{t}^{2}}\] \[{{S}_{1}}=\frac{1}{2}g\sin \theta {{t}^{2}}\] (u = 0 start from rest) Case II Rough inclined plane \[a=g\sin \theta -\mu \cos \theta \] \[S=ut+\frac{1}{2}a{{t}^{2}}\] (u = 0 start from rest) \[{{S}_{2}}=\frac{1}{2}(g\,sin\theta -\mu g\,cos\theta ){{(nt)}^{2}}\] \[{{S}_{1}}={{S}_{2}}\] \[\frac{g}{2}\sin \theta {{t}^{2}}=\frac{g}{2}[sin\theta -\mu cos\theta ]{{n}^{2}}{{t}^{2}}\] \[\mu =\tan \theta \left[ 1-\frac{1}{{{n}^{2}}} \right]\]
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