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

Options

A. `mu_(s) = sqrt(1 - (1)/(n^(2)))`
B. `mu_(s) = 1 - (1)/(n^(2))`
C. `mu_(k) = sqrt(1 - (1)/(n^(2)))`
D. `mu_(k) = 1 - (1)/(n^(2))`

Answer

D

Explanation

The acceleration of the block in case of smooth surface, `a_(1) = (mg sin theta)/(m) = g sin theta`
For smooth plane, using `s = ut + (1)/(2) at^(2)`, we get

`d = (1)/(2) (g sin theta) t^(2)`
For rough plane, the acceleration of the block
`a_(2) = ((m g sin theta - mu_(k) mg cos theta))/(m) g (sin theta - mu_(k) cos theta)`
Again using `s = ut + (1)/(2) at^(2)`, we get `d = (1)/(2) (g sin theta - mu_(k) cos theta) (nt)^(2)`
`implies (1)/(2) (g sin theta) t^(2) = (1)/(2) (g sin theta - mu_(k) g cos theta) n^(2) t^(2) implies sin theta = n^(2) (sin theta = mu_(k) cos t+N60heta)`
Putting `theta = 45^(@)`, we get `sin 45^(@) = n^(2) (sin 45^(@) - mu_(k) cos 45^(@))`
`(1)/(sqrt(2)) = (n^(2))/(sqrt(2)) (1 - mu_(k))`or `mu_(k) = 1 - (1)/(n^(2))`