I've attached a question that we went through I class. Could someone please help me understand why the tension in between the mass and the rope is 2K. Its also 2k in other pulley systems that I've seen where a pulley is attached with a rope from its centre to a fixed celling and two masses hanging on each side with tension k on each rope. I must be missing something very obvious but please help or provide a resource where I can lean more about such systems.
This isn't always true. In your case, it probably is.
There are pieces of information that haven't been explicitly stated.
1) the mass of the rope and the pulley are negligible. (We assume the masses are zero).
2) the two ropes on either side of the pulley are parallel (vertical).
As a consequence of (1), the tension on the rope on either side of the pulley is equal, and in your problem, you call this tension $K$.
Also as a consequence of (1) the total force on the (mass-less) pulley is zero (recall $F=ma$, and if $m=0$, then $F=0$). Since the two ropes are both pulling vertically upward on the pulley, each with force $K$, this must be balanced by the tension of the rope fixed to the axle of the pulley. So the tension in this rope is pulling down on the pulley must be $2K$. I hope this answers your question.
If the two ropes were not directly vertical, but more in a V shape, then you would have to deal with the forces by breaking them into the vector components and then add. In the case of a V shaped configuration, the tension would be less than $2K$.