I haven’t even started thinking about this beyond just asking, so join in!
Say you have a half-atwoods setup with friction on the horizontal object. You start with a light enough hanging vertical mass such that the tension is not large enough to break static friction. Everything stays put, and so the tension is equal to the weight of the hanging mass.
Let’s say you then keep adding mass until it breaks static friction. The moment you break static friction, two things change:
- The tension in the string lessens, and
- and we switch from static to kinetic.
What are the different possibilities of things that can happen? Under what circumstances do they happen?
Teach. Brian. Teach. 
First thought: I want to say the concept of “jerk” comes into play. The acceleration accelerates. I’m picturing tug of war where the other side suddenly lets go.
Second thought: no – it’s just accelerating down – like when you release the brakes on the car and it starts to roll downhill.
What are you thinking?
In quickly thinking about it, I was wondering if a contradiction could arise. I had been thinking well the tension decreases AND the friction decreases, is it possible to have the right circumstances so that in the kinetic case, it doesn’t have enough tension force to actually be kinetic. But at the same time, has too much force to static. I now don’t think it can happen.. still need to sit down and work it out…