Consider an initially vertical thin rod of mass $m$ and length $L$ (i.e. with one end touching the ground) that is allowed to fall. Assume no friction between the rod and the ground and that gravity is uniform. Will one end of the rod always remain in contact with the ground?
I've borrowed the graphic above from this post which seems relevant enough to what my question is asking. I tried to condense the main features of my question into the more textbook-y sounding version above, but the gist of it is
Under what conditions can a sliding rod enter into case 3 (in which the normal force vanishes)?
Additionally, can a falling rod with no initial velocity (just outside of unstable equilibrium) ever reach that state?
What would the locus/equation of motion of some point initially $y$ off the ground look like?
I tried to approach this problem in a couple of ways, but I kept getting stuck at finding equations of motion for some non-CM point. I'm comfortable with use of formalisms for the more concrete questions (my initial method was finding the angle for which the constraint $\lambda(t)$ equals zero), but I think the bulk of it might be more intuitive with forces/torques on the center of mass.
If you'd like, I can add some of my non-constructive attempts at answering these questions, but really I'm just curious as to the intuition behind it, since some physical experimentation makes it seem like case 3 never happens.