Question

A solid cylinder of mass 6 kg lies on a rough horizontal surface .Find the maximum value of f if the cylinder rolls without slipping

Answer 1

This question is incomplete however we can suppose that:

F will be at maximum force when friction will just start acting

Thus, considering the free body we have,   F-f=ma ( f is friction ). ............. 1

here friction will act opposite to the direction of motion.    

now considering newtons 2nd law of rotation we have F x 2R/3-f x r =I x (angular accn.) ............. 2

I=mr2 /2 and due to rolling angle can be = a/R ............. 3

solving the equations 1, 2 and 3  

We have F= 9 µmg

And we substitute the values and get F = 108 N which is the maximum force that can be applied to the cylinder without it slipping.