How freely should the spindle in my cartridge bearing bottom bracket spin?
Very Smooth with minimal resistance. No side to side play in the spindle. bk
Should it be able to continue to rotate on its own after I spin it?
Last one to the top...
Just my opinion.
Not unless it has a crank on it... there isn't enough mass to keep it spinning without a crank.
If you have a crank on it, then yes, it should keep spinning. However, I don't recall ever having a BB that would continue spinning without a crank attached.
Slow Ride Cyclists of NEPA
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If you're comparing to a cup and cone BB, then it spins less freely. Will definitely not continue spinning on its own without crank. With crank, yes, although in my experience, not much.
The viscosity of the grease and friction from the balls in the race will prevent it from spinning on its own. My Phil Wood BB won't do it without a crank on it. Sure is smooth, though. bk