Hello my class has been talking about centers of mass, centroids, and density the last couple of days and instructor told us to come up with an argument to the following statement, If you have a plate with density delta and finite area does it have a center of mass?

He told us to consider a ring where it would have a definite density and area but no center of mass. When I was thinking about this I was wondering if you could just spin it on its y axis and balance it upright. Anyways any ideas or suggestions that would help add to my input would be appreciated.

Thanks.