First, neglect losses and assume perfect bounce. In other words all the GPE lost on the way down is restored on the way back up.ME wrote:the yo-yo's Centre of Gravity is offset from the vertical
Now let it go up at twice the offset, say.
There are only three possibilities.
It rises to the same height.
It doesn't reach the same height.
It rises above the starting point of the original yoyo.
If it rises above the starting point then we have gained energy. I'm sure you will agree with that statement.
So if you don't believe it has gained energy you must accept one of the other two outcomes.
You seem to be an intelligent chap, ME. I'm sure if you think about it long enough you will come to the right conclusion. :-)