You speak of pistons coming out of cylinders and also returning to the cylinders, but in a real engine, the pistons always remain in the cylinders. The pistons move from one extreme position in the cylinder to a second extreme position (again, in the cylinder). What causes your piston to cycle...
There seems to be some very fuzzy thinking in the problem statement. Pistons do not expand; cylinder volumes do expand. Pistons do not return to the max volume condition by themselves; they are dragged there by the action of the crank and connecting rod. This looks like a case of confusion by...
The key is in your word "recently." That means the upward force was previously applied, but has now ended and when it ends, that when the downward acceleration begins.
To determine which forces are external requires the definition of a system boundary. Often it is useful to include all parts in the system within the boundary, but this is not always true. The choice of system boundary is a matter of experience, but a good choice makes life much simpler.
One...
Your first statement is
To solve is a verb that, in most cases, means to isolate one particular quantity on the left side of an equals sign. What are you trying to solve? I don't know, and I rather doubt that you do , so I suggest that you start again and consider where you want to go.
It is a long and treacherous road to the rank of Professor, and not all who start the trip make it to the end. Tenure is often awarded at the Associate rank, and many never rise beyond that. So while such position as the OP asked about certainly do exits, they are not plentiful.
I think the figure posted by @Lnewqban is actually intended to show the velocity gradient in a fluid flow. It just happens to fit this rolling situation, but that is purely coincidence.
This is very important, and for this reason, I have always required my mechanical engineering students to obtain the symbolic answer, provided it could be obtained without the use of numerical methods.
As per Berkeman's figure, just remember that the balls roll without slip on both the upper and lower plates. This should give you the necessary information easily.