# Ideal Rolling Motion on Surface with Friction

I have a question concerning ideal rolling motion on a surface containing friction.

By ideal rolling motion, I mean the tangental velocity of the rolling object is the same as the velocity of the rolling object's center of mass.

If an object is rolling on a flat surface with friction, the only force that produces a net force is the force of friction, acting in the direction opposite to the motion. Thus, the net force on the rolling object is in the direction opposite to motion, so it would seem that the center of mass would accelerate in the direction opposite to that of motion.

On the other hand, the work done by the friction is 0, since the friction is applied over no distance (a point of the rolling wheel only touches the ground for an infinitesimally small amount of time).

Is there any way to reconciliate the seemingly different results that dynamics and work analysis produce? Any guidance would be appreciated.

"a point of the rolling wheel only touches the ground for an infinitesimally small amount of time", so in that amount of time you only make a differential of work (that is, infinitesimal) $dW=Fdt$, now add the infinite number of differential time intervals into a finite time interval and you get a finite amount of work: $W=\int{Fdt}=F\Delta t$ (I am not sure how familiar are you with calculus)