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  • When a body slides down from rest along a smooth inclined plane making an angle of with the horizontal, it takes time T. When the same body slides down from th

When a body slides down from rest along a smooth inclined plane making an angle of 30^{\circ} with the horizontal, it takes time T. When the same body slides down from the rest along a rough inclined plane making the same angle and through the same distance, it takes time \alpha T, where \alpha  is a constant greater than 1 . The co-efficient of friction between the body and the rough plane is \frac{1}{\sqrt{x}}\left(\frac{\alpha^{2}-1}{\alpha^{2}}\right) where x=____________.

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f= \mu mg\cos \theta
a_{1}\rightarrow acceleration on smooth surface a_{1}= g\sin \theta
a_{2}\rightarrowacceleration on rough surface
a_{2}= \left ( g\sin \theta -\mu g\cos \theta \right )
S= ut+\frac{1}{2}at^{2}
S_{1}=S_{2}   & u= 0
\therefore \frac{1}{2}a_{1}T^{2}= \frac{1}{2}a_{2}\left ( \alpha t \right )^{2}
g\sin \theta \cdot T^{2}= \left ( g\sin \theta -\mu g\cos \theta \right )\alpha ^{2}T^{2}
g\sin \theta = \alpha ^{2}g\sin \theta -\alpha ^{2}\mu g\cos \theta
\alpha ^{2}\mu \cos \theta = \left ( \alpha ^{2}-1 \right )\sin \theta
\mu = \frac{\left ( \alpha ^{2} -1\right )}{\alpha ^{2}}\tan \theta
\mu = \frac{1}{\sqrt{3}}\frac{\left ( \alpha ^{2} -1\right )}{\alpha ^{2}}
\therefore x= 3
 

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Deependra Verma

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