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

Answer

C

Explanation

When fiction is absent
`a_(1)=g sin theta`
`:. S_(1)=1/2 a_(1)t_(1)^(2)`......(i)
When friction is present
`a_(2)= g sin theta-mu g cos theta`
`:. S_(2)=1/2 a_(2)t_(2)^(2).....(ii)`
From eq.(i) and (ii)
`1/2 a_(1)t_(1)^(2)=1/2a_(2)t_(2)^(2)`
or `a_(1)t_(1)^(2)=a_(2)(nt_(1))^(2) ( :. t_(2)=nt_(1))`
or `a_(1)=n^(2)a_(2)`
or `(a_(2))/(a_(1))=(g sin theta-mu g cos theta)/(g sin theta) =1/(n^(2))`
or `(g sin 45^(@)-mu g cos 45^(@))/(g sin 45^(@))=1/(n^(2))`
or `1-mu_(k)=1/(n^(2)) rArr ` or `mu_(k)=1-1/(n^(2))`