  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)  B
11n2  C
(11n2)  D
11n2  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. Physics

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