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]


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