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(B) For pure rolling,the point of contact $P$ must be at rest relative to the surface.Let $a_{cm}$ be the acceleration of the center of mass and $\alp

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(B) For pure rolling,the point of contact $P$ must be at rest relative to the surface.Let $a_{cm}$ be the acceleration of the center of mass and $\alpha$ be the angular acceleration.The condition for pure rolling is $a_{cm} = R\alpha$.Applying Newton's second law for linear motion: $F - f = M a_{cm} \Rightarrow a_{cm} = \frac{F - f}{M}$.Applying Newton's second law for rotational motion about the center of mass: $	au = I\alpha \Rightarrow fR = I\alpha \Rightarrow \alpha = \frac{fR}{I}$.Substituting these into the rolling condition: $\frac{F - f}{M} = R \left( \frac{fR}{I} ight) = \frac{fR^2}{I}$.$F - f = \frac{M R^2}{I} f \Rightarrow F = f \left( 1 + \frac{M R^2}{I} ight)$.Since $F$ is positive,$f$ must be positive,meaning it acts in the same direction as $F$ (forward direction).

In the following problem,indicate the correct direction of the friction force acting on a cylinder of mass $M$ and radius $R$,which is pulled on a rough surface by a constant horizontal force $F$ applied at its center. The friction force can be represented by which of the following diagrams?

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