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

From `s= ut+(1)/(2) at ^(2)` , on smooth incline,
`d=0+(1)/(2) ( g sin 45^(@))t^(2) ....(i)`
On rough incline,
`d=0+(1)/(2) g ( sin 45 ^(@)- mu_(k)cos 45^(@)) ( nt )^(2)` From `(i)` and `(ii)` ,
`(1)/(2) g ( sin 45 ^@- mu_(k)cos 45^(@))n^(2)t^(2) =(1)/(2) g sin 45^(@) t ^(2)`
`(1-mu_(k)) (n^(2))/(sqrt(2))=(1)/(sqrt(2))``mu_(k)=1-(1)/(n^(2))`