A block takes n times as much time to slide down a rough incline of `45^(@)` as it takes to slide down a perfectly smooth `45^(@)` incline. Coefficient of kinetic friction is :

Options

A. `(1)/(1-n^(2))`
B. `1-(1)/(n^(2))`
C. `(1)/sqrt(1-n^(2))`
D. `sqrt(1-(1)/(n^(2)))`

Answer

b

Explanation

Time taken to slide smooth inclined plane of length `l=(1)/(2)gsintheta.t^(2)impliesl=(1)/(2)g(1)/(sqrt2)t6(2)..(i)`
Time taken to slide rough inclined plane is
`l=(1)/(2)g(sintheta-mucostheta)t^(2)`
`l=(1)/(2)g(1)/(sqrt2)-muxx(1)/(sqrt2)t^(2)...(2)`
From (1) and (2),
`(1)/(2)g((1)/(2))((1)/((sqrt(2)-muxx(1))/(sqrt(2))))t^(2)=(1)/(2)gxx(1)/(sqrt(2))t^(2)`
As t. = nt
` implies(1 - mu) n^(2).t^(2) = t^(2)
n^(2) - n^(2) mu = 1`
`n^(2)mu=n^(2)-1`
`mu=1-(1)/(n^(2))`
Hence correct choice is (b).