Question

Would an object sitting on the floor have gravitational potential energy because its center of mass is a certain height above the floor?

Answer 1

Concept:- Newtonian physics cares only difference in potential energy, not it's actual magnitude so we are free to choose our reference for potential energy . So being at rest and possessing potential energy by an object is entirely two different concepts,they are totally unconnected to each other.

One more thing For extended object we always measure the height from center of mass to calculate its potential energy.

Therefore for given problem if we consider floor as a reference , object sitting on the floor have gravitational potential energy (because of its center of mass height) with respect to floor.