Smooth block is released at rest on a `45^@` incline and then slides a distance d. If the time taken of slide on rough incline is n times as large as that to slide than on a smooth incline. Show that coefficient of friction. `mu = (1 - (1)/(n^2))`

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 - mu 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)`