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Question

A given object takes $$n$$ times as much time to slide down at $$45^{\circ}$$ rough incline as it takes to slide down a perfectly smooth $$45^{\circ}$$ incline. The coefficient of kinetic friction between the object and the incline is given by :


A
(11n2)
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B
11n2
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C
(11n2)
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D
11n2
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Solution

The correct option is A $$\left (1 - \dfrac {1}{n^{2}}\right )$$
For rough surface,
$$mg \sin \theta -\mu mg \cos \theta =ma$$ 
$$\to \dfrac {g}{\sqrt 2}- \dfrac {\mu g}{\sqrt 2}=a......(1)$$

or $$'a=\dfrac {g}{\sqrt 2} [1-\mu], T=nt $$ [Acc. to question]

For smooth surface
$$mg\sin \theta =ma'$$
$$\dfrac {g}{\sqrt 2}=a'\quad T=t$$

since both starts From $$O$$ i.e. from rest and the distance $$(S)$$ is same then
$$S_1 =S_2$$
$$ \dfrac {1}{2} at^2 =\dfrac {1}{2} a(nt)^2$$

$$\to \dfrac {g}{\sqrt 2} t^2 =\dfrac {g}{\sqrt 2} (1-\mu) n^2 t^2$$

$$\to n^2 [1-\mu]=1$$

$$\to 1-\mu =\dfrac {1}{n^2}$$

$$\to \mu =\left (1-\dfrac {1}{n^2}\right)$$

 (A) Ans.

1457940_1113252_ans_7b4b415e1a37443ba43670c3e1d1ec8b.PNG

Physics

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