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Use work energy theorem

\[1/2mv^2+1/2I_{sphere}\omega_{sphere}^2+1/2I_{pulley}\omega_{pulley}^2=mgH\]

w= v/r

you will get a equation in one variable and will be able to find solution also

Isphere=2/3MR²

@RaphaelFilgueiras you are welcome. Isphere is 2/5MR^2

You are right: I=2/3MR^2 for the sphere