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

Answer

C

Explanation

When friction is absent
The acceleration of block, `a_(1)=gsintheta`
:. `s_(1)=(1)/(2)a_(1)t_(1)^(2)` (i)
When friction is present the acceleration of block,
`a_(2)=gsintheta-mu_(k)gcostheta`
:. `s_(2)=(1)/(2)a_(2)t_(2)^(2)` (ii)
From Eqs. (i) and (ii), `(1)/(2)a_(1)a_(1)^(2)=(1)/(2)a_(2)t_(1)^(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))=(gsintheta-mu_(k)gcostheta)/(gsin45^(@))=(1)/(n^(2))`
or `=(gsin45^(@)-mu_(k)gcos45^(@))/(gsin45^(@))=(1)/(n^(2))`
or `1-mu_(k)=(1)/(n^(2)` implies `mu_(k)=1-(1)/(n^(2))` .