Direction of friction on a rolling body

[Happiness]

A cylinder rolls on a horizontal, flat surface without sliding towards the left, so it must be rolling anticlockwise about its center of mass (CM). Suppose it slows down to a stop due to friction. What is the direction of the friction at the point of contact?

Since it slows down, friction must be acting to the right. But this rightward friction produces an anti-clockwise torque $\tau$ about the CM. Since $\tau=I\alpha$, this anti-clockwise $\tau$ produces an anti-clockwise angular acceleration $\alpha$ about the CM. Since the cylinder does not slide, a faster rotation means its CM moves faster. This contradicts the premise that the cylinder comes to a stop. What's wrong?

I am guessing any cylinder that comes to a stop must slide. For a cylinder that is not observed to be sliding, it is still sliding but not noticeably. Is this true?

 

[Doc Al]

Since it slows down, friction must be acting to the right. But this rightward friction produces an anti-clockwise torque $\tau$ about the CM. Since $\tau=I\alpha$, this anti-clockwise $\tau$ produces an anti-clockwise angular acceleration $\alpha$ about the CM. Since the cylinder does not slide, a faster rotation means its CM moves faster. This contradicts the premise that the cylinder comes to a stop. What's wrong?

It's not simple friction that causes the rolling cylinder to slow and stop, but deformation of the surface. (This effect is called "rolling resistance".) The effect of surface deformation ends up with the reaction force of the surface acting ahead of the center of the cylinder, thus creating a clockwise angular acceleration.

 

[A.T.]

It's not simple friction that causes the rolling cylinder to slow and stop, but deformation of the surface...

... and deformation of the cylinder, and aerodynamic drag.