Can anyone please help explain why the arrows in this diagram are drawn like they are? I'm having a tough time understanding why they are like this. I really want to understand this before I tackle the equations. Also looking at the diagram would the tension (T1 and T2) just be spring force? Can someone please help, I'm pretty confused over this.
The tension in a string has no inherent direction. It's just the force with which the lengths of string to the right and left of a point are trying to pull the string apart. When we draw arrows to indicate tension it should be done in such a way that what force they are representing is clear. It is not clear in this figure.
The opposite arrows for $T_1$ on the top line represent the forces the string exerts on the pulleys at the ends. The bottom line is more problematic. If the spring $K_1$ is massles the tension will be the same to right and left of the spring, but the middle $T_2$ should point the other way so as to represent the force on the block. The resulting equation of motion should be $M\ddot y= T_1-T_2$