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

Question

Infinity Learn · Accepted Answer

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 $$=$$  $$3$$μmg $$=$$  $$3$$  ×  $$0.6$$  ×  $$3$$  ×  $$10$$  $$N=54$$  N

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