A smooth block is released at rest on a 45° incline and then slides a distance d. The time taken to slide is n times as much to slide on rough incline than on a smooth incline. The coefficient of friction is

See fig.Let t be the time taken by the block to cover a distance d.Then using the formula, s=ut+(1/2)g′t2, we get d=12g sin⁡45∘t2   …(1)∵u=0  and  g′=gsin⁡45∘Let t' be the time taken for rough plane.Then d=12gsin⁡45∘−μkcos⁡45∘t′2 or d=12gsin⁡45∘−μkcos⁡45∘(nt)2    …(2)Equating equations (1) and (2), we getsin⁡45∘=sin⁡45∘−μkcos⁡45∘n2or 12×1n2=12−μk2  or  μk=1−1n2.