Prediction $3\mathrm{.}1$: Before the collision, suppose the disk is rotating with an angular velocity vec$({)}_{\text{b}\text{e}\text{f}\text{o}\text{r}\text{e}\text{.}}$ In terms of $I_{\text{d}\text{i}\text{s}\text{k}},I_{\text{r}\text{i}\text{n}\text{g}},$ and vec$({)}_{\text{b}\text{e}\text{f}\text{o}\text{r}\text{e}},$ find a mathematical expression for the angular velocity of the disk and ring after the collision vec$({)}_{\text{a}\text{f}\text{t}\text{e}\text{r}}.$

Provide a scientific argument for your expression that relies on relevant physics concepts and principles.