I'm learning mechanics and I don't understand the following (simple, I guess) concept. Here's a pulley-rope system:
The force of tension, as I understand it, is defined as the force exerted on an object connected to the rope/string. For example, $\vec{T_1}$ is the force of tension exerted on the object with mass $m_1$. But what are the forces $\vec{T_1}'$ and $\vec{T_2}'$? My textbook says that these are tension forces too, but I don't understand on which body they act - on the pulley? Well, there's no friction here. So the only reasonable explanation is that these are internal tension forces (which is very different from $\vec{T_1}$ for example). In this case:
Why do we even care about them? Don't they all cancel out along the rope (when it's massless)?
Why are we interested in the forces $\vec{T_1}'$ and $\vec{T_2}'$ which are acting on the rope tangentially to the pulley and not, for example, in the tension force acting on the pulley itself, that is:
Or maybe in the internal tension forces acting on the rope in some other places?