A smooth block is released at rest on a $45^{\circ}$ incline and then slides a distance d. The time taken to…

A smooth block is released at rest on a $45^{\circ}$ incline and then slides a distance d. The time taken to slide is $\mathrm{n}$ times as much to slide on rough incline than on a smooth incline. The coefficient of friction is
  1. $\mu_{\mathrm{k}}=1-\frac{1}{\mathrm{n}^2}$
  2. $\mu_{\mathrm{k}}=\sqrt{1-\frac{1}{\mathrm{n}^2}}$
  3. $\mu_s=1-\frac{1}{n^2}$
  4. $\mu_s=\sqrt{1-\frac{1}{n^2}}$

Solution

$\mathrm{d}=\frac{1}{2} \frac{\mathrm{g}}{\sqrt{2}} \mathrm{t}_1^2$ $\mathrm{~d}=\frac{1}{2} \frac{\mathrm{g}}{\sqrt{2}}\left(1-\mu_{\mathrm{k}}\right) \mathrm{t}_2^2$ $\frac{\mathrm{t}_2^2}{\mathrm{t}_1^2}=\mathrm{n}^2=\frac{1}{1-\mu_{\mathrm{k}}}$

Asked in: JEE Main 2005

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