A given object takes n times as much time to slide down a 45 degree rough incline plane as it takes to slide down a perfectly smooth 45 degree incline plane. Prove that the co efficient of kinetic friction between the object and the incline is given by [1-(1/n2)].

Suppose the length of the inclination is d. Now the acceleration at the smooth surface is a=gsin45°=g2so time taken is t=2da=2dg2Now if μ is the coefficient of kinetic firction then acceleration is a'=gsin45°-μgcos45°=g1-μ2so time taken is t'=2da'=2dg1-μ2Now we havet't=n2dg1-μ22dg2=ngg1-μ=ng=g1-μn2g1-n2=-μgn2μ=g1-n2-gn2=1-1n2 (proved)

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