Here is a goal: To smooth down the outside of a homemade ring.

Which way makes more sense to accomplish this goal?
Wrap some masking tape around a large drill bit so that it is slightly larger than the inside of your ring. Twist your ring onto the bit, making sure it's held firmly in place. If it's still loose, add more tape then try again. Once the ring is on the bit, attach the bit to an electric drill and turn it on. Wrap a small piece of wood in sandpaper, and hold the spinning ring against it to smooth the outside of the ring. Use increasingly higher grits to get it nice and smooth. Use a rough cloth once you're done sanding for a finishing touch.