What is the minimum value of coefficient of friction between the cylinder and inclined plane for rolling without slipping ?

Options

A. `(1)/(3) sin theta`
B. `(1)/(3) tan theta`
C. `(2)/(3) sin theta`
D. `(2)/(3) tan theta`

Answer

B

Explanation

(b) Equation of motion
`Mg sin theta - f = Ma`…(i)
Also `fR = tau = I prop = Mk^2 (a)/( R)` …(ii)
But `a = (g sin theta)/(1 + (k^2)/(R^2))`….(iii)
Putting value of `a` in equation (ii)
`f = ((Mk^2)/(R^2))(g sin theta)/(1 + (k^2)/(R^2))`
For cylinder : `Mk^2 = I = (1)/(2) MR^2`
`k^2 = (1)/(2)R^2 , f = (M((1)/(2)) g sin theta)/((1 + (1)/(2))) = (1)/(3) Mg sin theta `
In case of static friction, `f_2 = mu N = mu Mg cos theta`
`(1)/(3) Mg sin theta = mu Mg cos theta rArr mu = (1)/(3) tan theta`.