From answers to the following question
[Why are tensions in the pulley different when the pulley has a mass or moment of inertia?](https://physics.stackexchange.com/questions/197048/why-are-tensions-in-the-pulley-different-when-the-pulley-has-a-mass-or-moment-of), I understand that the reason for that is because the difference in tension is the driving force behind the angular acceleration of the pulley.

However, I don't find this explanation completely satisfying - why does the pulley *has to* be set into rotation in the process in the first place?

Thus, I am wondering if there is an explanation for why tensions in the pulley are different when the pulley has a mass or moment of inertia **considering the interactions between the particles in the string**?

I think this may be a better way to address the causality of the process the pulley system undergoes as it proceeds in the following direction:

1. due to [answer to the above question], there has to be a difference in the tensions
2. due to the difference in the tensions, the pulley gains an angular acceleration during the process

A picture showing the pulley system that I was talking about:
[![enter image description here](https://i.stack.imgur.com/Ykbxz.png)](https://i.stack.imgur.com/Ykbxz.png)