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

The acceleration of the block in case of smooth surface
`a_(1) = (mg sin theta)/(m) =g sin theta`
For smooth plane, using `s =ut + (1)/(2) at^(2)` we get
`d =(1)/(2) (g sin theta)t^(2)…(i)`
For rough plane, the acceleration of the block
`a_(2)=(mgsintheta-mu_(k)mgcostheta)/(m) =g(sintheta-mu_(k)costheta)`
Again using `d =ut + (1)/(2) at^(2)` we get
`a =(1)/(2) (g sin theta -mu_(k) cos theta) (nt)^(2)`
` rArr (1)/(2) (g sin theta)t^(2) = (1)/(2) (g sin theta -mu_(k) g cos theta)n^(2)t^(2)`
`rArrsin theta = n^(2) (sin theta - mu_(k) cos theta)`
Putting `theta =45^(@)` we get
`sin 45^(@) =n^(2) (sin 45^(@) -mu_(k) cos 45^(@))`
`(1)/(sqrt2) = (n^(2))/(sqrt2) (1-mu_(k))ormu_(k)=1-(1)/(n^(2))`
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