I am making a rock tumbler and am a little perplexed as to how fast the "barrell" should turn. I will be using barrels of 5 1/2 inch and 6 1/2 inch. Anyone have any ideas on about the best RPM for general purpose smoothing? I do not have a variable speed motor available.

Here is one suggestion.

Tumbling is an economical finishing process because large batches of parts can be run with little or no supervision by the operator. A full cycle can take anywhere from 6 to 24 hours with the barrel turning at 20 to 38 RPM.

Ref:
Tumble finishing - Wikipedia, the free encyclopedia

Another good source is :

GemWorld-TUMBLING INSTRUCTIONS