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) `
`(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 0 = `45^(@)`)
` mu prop 1- (1)/(n^2)`