A given object takes n times as much time to slide down a `45^(@)` rough incline as it takes to slide down a perfectly smooth `45^(@)` incline. The coefficeint of kinetic friction between the object and the incline is given by:

Options

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

Answer

B

Explanation

`s= u t+ (1)/(2) a t^(2), a = g sin theta, u= 0`
Then, `s= 0+(1)/(2) a t^(2)`
or `t= sqrt((2s)/(a))= sqrt((2s)/(g sin theta))` [for smooth plane]

For the rough plane, the effective value of acceleration along the incline is `a.= g sin theta - mu_(k) g cos theta and t.= sqrt((2s)/(a)) = sqrt((2s)/(g sin theta- mu_(k) g co stheta))`
`(t.)/(t)= n = sqrt((g sin theta)/(g sin theta- mu_(k) g cos theta))`
`=sqrt((sin 45^(@))/(sin 45^(@)- mu_(k) cos 45^(@)))`
`=sqrt((1)/(1-mu_(k)))`
`rArr (1)/(n^(2)) =1 -mu_(k) rArr mu_(k)=1- (1)/(n^(2))`