Disclaimer: I may seem to be using strange terms because although I know English rather well, it is not the language I learn physics in.
Consider a cylinder (with mass $m$ and radius $r$) on an inclined plane (incline is $\alpha$, an angle in radians). As far as I know this is a valid way to calculate its acceleration ($a$):
$\epsilon$ - angular acceleration, $\tau$ - moment of force, $I$ - moment of inertia, $g$ - gravitational acceleration
$\epsilon=\tau/I,\tau =F_f *r, a/r=\epsilon \rightarrow a/r=F_f*r/I $
$F_f=F-am $ (this is the part I am asking about)
I am asking when does the Newton's second law for progressive movement ($F=am$) apply to rigid-bodies (with torque) and why is that the case? Does it only apply when the movement is without sliding or always etc.? I am uncertain because it seems to me that the force of friction does not affect the entire body evenly. I would appreciate an answer with an explanation why can this law be applied.
As another example consider that the cylinder would have a string attached to it. Its movement would differ were the string be attached to its center of mass or its bottom, so it seems that the point where the force is applied does matter.
To be clear I asking when and why $F=am$ applies for rigid-bodies (especially ones that are rotating). I know that the angular laws such as $\tau=\epsilon*I$ apply.