A solid cylinder is relased from rest from the top of an inclined plane of inclination `60^@` where friction coefficient varies with distance `x` as `mi = (2 - 3x)/(sqrt(3))`. Find the distance travelled by the cylinder on incline before it starts slipping.

Answer

`1//3 m`

Explanation

`mu = (2- 3x)/(sqrt(3))`
For translation motion
`mg sin theta - f = ma` …(1)
For rotational motion
`f xx R = I alpha rArr f xx R = ((1)/(2) mR^(2)) (a)/( R)`
`f = (ma)/(2)`
Solving (1) & (2) `a = (2 g sin theta)/(3)` & `f = (g sin theta)/(3)`
when slipping starts max friction force will act so
`mu mg cos theta = (mg sin theta)/(3)`

`((2-3x)/(sqrt(3))) mg xx (1)/(2) = (mg)/(3) xx (sqrt(3))/(2)`
`2 - 3x = 1`
`x = (1)/(3) m`.