A uniform disk of mass $$M = 8.0 \;\mathrm{kg}$$ can rotate without friction on a fixed axis. A string is wrapped around its circumference and is attached to a $$m = 6.0 \;\mathrm{kg}$$ mass. The string does not slip. What is the tension in the cord while the mass is falling? In this problem, I understand that there is a tension force within the rope that causes the disk to turn and pulls the mass hanging up. However, I am unable to judge the magnitude of force that the ropes tension puts on the disk. If we imagine the rope as a bunch of massless pieces, then we can see that each piece gets pulled by a force of $$T$$ ($$T$$= Tension Force). Thus, I am wondering if the tension force actually accumulates to a greater amount on the disk. Any ideas?

I am unable to judge the magnitude of force that the ropes tension puts on the disk.

For a "light" rope, we assume the tension is uniform throughout, and equal to the magnitude of the force pulling on each end of the rope.

If we imagine the rope as a bunch of massless pieces, then we can see that each piece gets pulled by a force of T (T= Tension Force).

It's unclear exactly what you mean by "pulled". What direction does this pulling occur? Tension in a rope doesn't have a direction.

Because each "piece" of the rope can be considered massless, there can be no net force on it. Otherwise from $$F=ma$$ or rearranged, $$a = \frac Fm$$, the acceleration would be undefined. In practice, the light rope is able to accelerate rapidly enough that it moves to eliminate any difference in forces from each side.

In this case the rope connects the two other objects (rim of disc and block) and constrains them to move with the same velocity and acceleration. The tension in the rope is equal in magnitude to the force the block pulls on the rope and the force the rope pulls on the pulley.

Without the rope joining them, we can see the block will accelerate downward at $$1g$$, while the pulley will remain stationary.

If we place a small force on each (say $$1\text{N}$$), we can slow the acceleration of the block, while beginning the acceleration of the pulley. At some larger value, the slowing of the block and the increasing speed of the pulley intersect and they move at the same rate and same acceleration. That value will be equal to the tension in the rope.