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# A smooth block is released at rest on $45 ^{\circ}$ incline and then slides a distance 'd'. The time taken to slide is 'x' times as much to slide on rough incline than on smooth incline.The coefficient of friction is

$(a)\;\mu_k=1-\frac{1}{x^2} \quad (b)\;\mu_k=\sqrt {1-\frac{1}{x^2}} \quad (c)\;\mu_s=1-\frac{1}{x^2} \quad (d)\;\mu_s=\sqrt{1-\frac{1}{x^2}}$

On a rough surface
$a=(g \sin \theta-\mu g \cos \theta)$
$time =xt$
$S=ut+\large\frac{1}{2} $$at^2 \quad=0+\large\frac{1}{2}$$[g \sin \theta -\mu g \cos \theta](xt)^2$-----(1)
On plane surface
$a=g \sin \theta$
$time=t$
$S=0+\large\frac{1}{2}$$(g \sin \theta)t^2-----(2) Solving (1) and (2) We get \mu =1-\large\frac{1}{x^2}\quad$$[\mu=\mu_k$ as the body is moving$]$
Hence a is the correct answer
edited May 27, 2014