Pipsqueakalchemist said:
So the reason it’s a variable is bc we don’t know if it’s static or kinetic friction?
Please excuse me the delayed response to your good question.
In this specific problem, they first assume no slipping occurs between belt and cylinders (like if the belt was a string wrapped around each cylinder several times).
By doing so, they calculate the resistive tangential force that the moments of inertia of both cylinders would induce as a function of the angular acceleration of the system (both cylinders acting as one).
Hence, the more force you pull the strings with, the more you overcome the inertial resistance of the system and the greater the angular accelerations of both cylinders get.
But in actuality, you don’t have wrapped strings but only the friction of both surfaces of the belt against the surfaces of the cylinders to overcome that resistance to accelerate.
Then, they calculate the maximum static friction that keeps the cylinders and belt from slipping.
If the belt pulls with a force greater than that value, slipping will occur and that will be the limit to obtain higher values of angular accelerations on both cylinders.
Once slipping begins, the belt pull can only achieve lower angular accelerations, because the dynamic friction will be of a lesser value than the static one.