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

We know that when there is no friction acceleration down the incline is
`a = g sin theta`
And when friction is present , acceleration down the incline is
` a = g (sin theta - mu cos theta)`
In each case distance covered is
`d = (1)/(2) at^(2) = (1)/(2) a t^(2)`
`(a)/(a) = t^(2)/(t^(2)) = (nt)^(2)/(t^(2) ) = n^(2)`
`(g sin theta)/(g (sin theta - mu cos theta ))=n^(2) `
`n^(2) sin theta - mu n^(2) cos theta = sin theta `
or ` (n^(2) - 1 ) sin theta = mu n^(2) cos theta `
or `mu = ((n^(2)-1))/(n^(2)) tan theta `
` mu = (1 - (1)/(n^(2))) tan 45^(@) `
` mu = 1 - (1)/(n^(2) ` .