A body at rest slides down a 30o inclined plane. The time taken by it to slide down is twice the time it takes when it slides down the same distance in the absence of friction. The coefficient of friction between the body and the inclined plane is
A
0.43
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B
0.37
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C
0.64
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D
0.75
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Solution
The correct option is B0.43 When a plane is inclined to the horizontal at an angle θ, which is greater than the angle of repose, then the body placed on the inclined plane slides down with an acceleration a. From figure, R=mgcosθ ....(i) Net force on the body down the inclined plane f=mgsinθ−F ....(ii) i.e., f=ma=mgsinθ−μR[∵μ=FR] ∴ma=mgsinθ−μmgcosθ [from equation (i)] =mg(sinθ−μcosθ) ⇒a=g(sinθ−μcosθ) Time taken by the body to slide down the plane t1=√2sa=√2s(gsinθ−μgcosθ) t2=√2sgsinθ [in absence of friction] ∵t1=2t2 ⇒t21=4t22 [given] ∣∣
∣∣√2s(gsinθ−μgcosθ)∣∣
∣∣2=4×∣∣
∣∣√2sgsinθ∣∣
∣∣2 ⇒2sg(sinθ−μcosθ)=4×2sgsinθ ⇒sinθ=4sinθ−4μcosθ ⇒μ=34tanθ=34tan30o=0.43