I am struggling with balancing energy in a yo-yo. So, we have a yo-yo (massless string wrapped around solid cylinder). It is allowed to fall through a distance $h$ without rotation. The loss in potential energy $= mgh$ will be converted into kinetic energy (KE) and we can find the KE after it has fallen through a distance $h$.
Now, we allow the yo-yo to fall like a yo-yo so that it moves the same distance, but also rotates. So, again, we apply the conservation of energy. Now, the KE of the yo-yo will have 2 parts (i.e. due to the rotation around the center of mass and due to the translation of the center of mass). We find that the final KE in this case is 1.5 times than that of the previous case.
My question is, how come the 2nd case has more final energy when the initial energy in both the cases was $mgh$?