A given object takes n times as much time to slide down a $$45$$° rough incline as it takes to slide down a perfectly smooth $$45$$° incline. The coefficient of kinetic friction between the object and the incline is given by

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Infinity Learn · Accepted Answer

On smooth inclined plane, the acceleration of the body,$$a=gsin$$⁡θIf s be the distance traveled, $$thens=12gsin$$⁡θ$$t12$$   …(1)On$$ rough inclined plane, the acceleration is given $$bya=gsin$$⁡θ−μgcos⁡θ∴$$s=12(gsin$$⁡θ−μgcos⁡θ$$)t22$$   ...$$(2)From$$ eqs. $$(1) and (2),$$t22t12=$$$$sin$$⁡θ$$sin$$⁡θ−μ$$cos$$⁡θBut $$t2=nt1ort22/t12=n2$$∴$$n2=$$$$sin$$⁡θ$$sin$$⁡θ−μ$$cos$$⁡θSolving, we getμ$$=n2$$−$$1n2$$×$$sin$$⁡θ$$cos$$⁡θ$$=n2$$−$$1n2$$×$$sin$$⁡$$45$$∘$$cos$$⁡$$45$$∘ $$=(n2$$−$$1)n2$$×(1/2)(1/2)=n2$$−$$1n2=(1$$−$$1n2).

A given object takes n times as much time to slide down a 45° rough incline as it takes to slide down a perfectly smooth 45° incline. The coefficient of kinetic friction between the object and the incline is given by

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