State the necessary condition for a solid cyl | vedclass

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(B) For a body of mass $M$ and radius $R$ rolling down an inclined plane of angle $	heta$,the forces acting are gravity ($Mg \sin 	heta$ down the pl

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$\mu_{S} \geq \frac{1}{3} 	an 	heta$

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(B) For a body of mass $M$ and radius $R$ rolling down an inclined plane of angle $	heta$,the forces acting are gravity ($Mg \sin 	heta$ down the plane),normal force $(N = Mg \cos 	heta)$,and static friction $(f)$.For pure rolling,the acceleration $a = \frac{Mg \sin 	heta}{M + I/R^2}$.For a solid cylinder,the moment of inertia $I = \frac{1}{2} MR^2$.Substituting $I$,we get $a = \frac{Mg \sin 	heta}{M + (1/2)MR^2/R^2} = \frac{Mg \sin 	heta}{1.5M} = \frac{2}{3} g \sin 	heta$.The frictional force is $f = Ma - Mg \sin 	heta$ (in magnitude,$f = Mg \sin 	heta - Ma$).$f = Mg \sin 	heta - M(\frac{2}{3} g \sin 	heta) = \frac{1}{3} Mg \sin 	heta$.For rolling without slipping,$f \leq \mu_{S} N$.$\frac{1}{3} Mg \sin 	heta \leq \mu_{S} Mg \cos 	heta$.Therefore,$\mu_{S} \geq \frac{1}{3} 	an 	heta$.

State the necessary condition for a solid cylinder to roll without slipping down an inclined plane with friction.

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