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

Explanation

When there is no friction, the block slides down the inclined plane with acceleration.
`a = g sin theta`
when there is friction, the downward acceleration of the block is `a'=g (sin theta-a cos theta)`
As the block slides a distance d in each case so `d=1/2 at^(2)=1/2 a't^(2)`
`a/(a')=(t'^(2))/t^(2)=((nt)^(2))/(t^(2))=n^(2)`
`or (g sin theta)/(g(sin theta-mu cos theta))=n^(2)`
Solving, we get `("Using "theta-45^(@))`
`mu=1 -1/n^(2)`