A solid cylinder of mass 3 kg rests on a rough horizontal plane. A horizontal force F is applied to the top most point of the cylinder as shown in the figure. Coefficient of friction between the cylinder and the surface is 0.6. Then the maximum value of F for which the cylinder will roll without slipping is

Frictional force f is given byf=F 1−hrpWhere rP = rC + ICmrCrC= Distance of centre of mass from instantaneous axis of rotation O.For solid cylinder, rP = R+ 12  mR2m.R  =  3R2∴    f=F 1−hrP =  F 1−2R3/2R=−F3.Since ‘f’ is negative, it must act in the forward direction. Now fmax =  μmgNo slipping takes place if f <  μmgi.e., if F3 <  μmg∴    Fmax =  3μmg =  3  ×  0.6  ×  3  ×  10  N=54  N