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A smooth block is released at rest on a 45∘∘ incline and then slides a distance dd. The time taken to slide is nn times as much to slide on rough incline than on a smooth incline. The coefficient of friction is

(a) µs = 1 - 1/n2
(b) µs = √(1 - 1/n2)
(c) µk = 1 - 1/n2
(d) µk = √(1 - 1/n2)
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The correct answer is  a)  µk = 1 - 1/n

Solution::

On a smooth plane

d = ½ (g sinθ) t1

$$t_1=\sqrt{ 2d\over g\sin\theta}$$

On a rough surface

d = ½ (g sinθ – μg cosθ) t2

$$t_2=\sqrt{ 2d\over g\sin\theta - \mu g\cos\theta}$$

from the question , we know

t2=nt1

$$t_2=\sqrt{ 2d\over g\sin\theta - \mu g\cos\theta}= n\sqrt { 2d\over g\sin\theta}$$

$$n={1\over \sqrt{ 1-\mu k}}$$

$$n^2={ 1\over 1-\mu k}$$

$$\mu k = 1- {1\over n^2}$$

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