I want to talk about the constrain added by introducing rotation of a rigid body to a simple case:
An homogeneous ring at rest is dropped from height $H$ of an declined surface without any kind of friction. To find it's translational velocity once it hits the horizontal floor.
Conservation of energy gives: $2g(H-R) = v^2 + R^2\omega ^2$
I want to know if $\omega$ and $v$ are independent variables, usually correlated with the additional assumption of rotation without slipping, (and thus the problem is unsolvable without any additional assumptions ) or if there is no need for more information in order for the problem to be solved and rotation without slipping is a natural consequence of this configuration and therefore $\omega$ and $v$ are not independent in first place.