In the solution they are just using the conservation of energy principle, i.e.: PE1 + KE1 = PE2 + KE2 leaving out any work (smooth surface).

I don't understand how tension doesn't do any work. The rope is attached to the spool and the spool is rotating and sliding. 

Is it only because the displacement of tension is 0 because as it unravels, the points of contact change? That's the only logical explanation I can come up with...

[![enter image description here][1]][1]


**EDIT**:

"They" are the authors of the textbook. And here's the solution where they use T1 + V1 = T2 + V2, where T=KE and V=PE.

[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/t1DNP.jpg
  [2]: https://i.sstatic.net/yqEji.jpg