The third formula you listed has already answered your question: If T 1 = T2-> dω/dt=ε=0,-> This is the case of conservation of momentum.

Because the pulley has mass, it has the moment of inertia j, which is the inertia measure of the rotating rigid body. The inertia of translational motion needs to be overcome by resultant force F, resulting in acceleration. ->; So as to change the motion state of the object; Similarly, to change the motion state of a rotating rigid body, there must be an angular acceleration ε->; A resultant force is needed to overcome the inertia of a rotating rigid body. -& gt； T 1≠T2。

Unless, regardless of the pulley mass (that is, m = 0), -> J=0 and the wheel has no inertia, -> T 1=T2.