Suppose we have a cylinder on an inclined plane of mass $m$ and radius $R$ moving without sliding (so that $\varepsilon = a/R$). Why is the friction $F$ causing the circular motion sometimes lower than $F' = f \cdot N$, where $N$ is a normal force and $f$ the friction coefficient? ($F \le f \cdot N$)
Draw a free body diagram for the object. You will see that this friction $F$ is the only force that attacks at the edge. This is the only force that gives a torque.
Since it isn't balanced by any other torques, this is the reason for the accelerating rotation.
The size of this force $F$ doesn't matter - as long as it exists, it will cause a torque.