Here is a link to page in a book which contains an example problem:
In the book, they work out the natural frequency of a hydraulic cylinder and come out with an answer in rad/sec. This number is then inverted to get a time constant, and the resultant unit is seconds.
I understand that a radian is dimensionless, and 1 rad/sec really equals 1/sec. So, it makes sense that you invert it and get seconds. However, you would also get seconds if you first convert the frequency from rad/sec to cycle/sec, and then invert.
My question is: how do you know which to use? When do you want to use sec/cycle, vs. sec/rad? It seems ambiguous, and the numbers would come out very differently.
I know the result of this equation is in radians. What if you experimentally measured the natural frequency in cycles/sec, and then inverted to get the time constant in seconds? You would get a different answer, but I don't think anything was done wrong.
Can anyone shed some light on this? I think I am missing something.