A given object taken n time more time to slide down `45^(@)` rough inclined plane as it taken to slide down a perfectly smooth `45^(@)` incline The coefficient of kintic friction between the object and the incline is .

Options

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

Answer

b

Explanation

From `s = ut + (1)/(2)at^(2) = 0+ (1)/(2)at^(2)`
`t =sqrt((2s)/(a)`
For smooth plane, `a = g sin theta`
For rough plane `a = g (sin theta - mu cos theta)`
`:. t' = sqrt((2s)/(g (sin theta -mu cos theta)))`
`=nt = nsqrt((2s)/(g sin theta))`
`:. n^(2)g (sin theta-mu cos theta) = g sin theta`
When `theta = 45^(@), sin theta = cos theta = 1//sqrt2`
Solving we get `mu = (1-(1)/(n^(2)))` .