A cylinder of mass m, radius R and moment of inertia I is placed on a rough inclined plane. What should be the minimum coefficient of friction between cylinder and inclined plane so that cylinder rolls down the inclined plane without slipping?

Consider a cylinder placed on the inclined plane inclined at angle θ the with the horizontal.
The various forces acting on the cylinder are:

(i) Weight mg acting vertically downward.

(ii) The friction F between the cylinder and surface of inclined plane.
                 
On resolving the components of weight mg, along and perpendicular to the inclined plane we have mg sinθand mg cosθ respectively.
Let a be the acceleration with which cylinder rolls down the inclined plane.

Equation of linear motion down the inclined plane is,

Since the cylinder rolls due to the force of friction,

where  is angular acceleration of cylinder. 
We know,
 
Substituting the value of F in equation (1), we get
 
Substituting a  in (4),  we get,

There will be no slipping if and only if

Therefore minimum coefficient of friction  is,