This is an exercise without a conclusion, intended to examine a phenomenon that, in the relativistic world, seems most difficult to comprehend, let alone explain – relativistic rotational angular momentum. Many have tackled the problem, first cited by Ehrenfest in 1909, and known since then as the “Ehrenfest Paradox,” primarily in terms of analyzing effects on the size and geometry of a relativistically rotating disk. Length contraction and time dilation interplay with aspects of general relativity and hyperbolic geometry according to the experts. Tackling this aspect is beyond me, so I try to examine a somewhat more predictable phenomenon, relativistic angular momentum as a result of relativistic mass increase in both a rotating rod and disk. No conclusions are drawn, but observations are offered as food for thought for the reader.