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
d=12(g sinθ)t1,
d=12(g sinθ−μg cosθ)t2
t1=√2dg sinθ
t2=√2dg sinθ−μg cosθ
According to question, t2=nt1
n√2dg sinθ=√2dg sinθ−μg cosθ
μ, applicable here, is kinetic friction as the block moves over the inclined plane.
n=1√1−μk(∵cos 45∘=sin 45∘=1√2)
n2=11−μk or 1−μk=1n2
Or μk=1−1n2