Please look at the problem attached as a screenshot.
Assuming frictionless, Ei = Ef, which means objects that are the same will end up in the same heights (so we can group A&C, B&D, and E&F).
For A&C and E&F, mgh = KE_rot + KE_trans
For B&D, it is mgh = KE_trans.
Also, v = rw (I know it's omega, but for convenience, I'll write it as w).
I _solid sphere = 2/5mr^2
I_hollow sphere = 2/3mr^2
The Attempt at a Solution
After all these equations set up, now it's pretty much plugging things in. For A&C ,
mgh = 1/2mv^2 + 1/2Iw^2
mgh = 1/2mv^2 + 1/5mr^2w^2
gh = 1/2v^2 + 1/5 r^2(v/r)^2
gh = 1/2v^2 + 1/5v^2 = 7/10v^2
so v^2 = 10gh/7
then KE at the end is then 1/2mv^2 = 5mgh/7, and that can be converted to the new GPE. so the height will be 5/7 the original height.
For E&F, I can use a similar process, only now I = 2/3mr^2. That gives me a KE of 3/5mgh, so the new (final) height will be 3/5 the original height.
For the blocks (B&D), there's only KE_trans, so Ei = Ef and mgh = 1/2mv^2. With no rot KE to lose trans KE to, the final height should be the same as the original height.
Thus, in order, it is B&D, A&C, E&F.
Could someone please check if I don't have any flaws in my reasoning?