I am having some difficulty when dealing with complex number using mathematica function Abs. I am trying to use Abs[1+ I w] to get Abs[1+ I w]=Sqrt[1+ w^2]. But, it seems that I can't do it in a straight forward way. That is

  $Assumptions = \[Omega] \[Element] Reals
    Abs[1 + I \[Omega]]

Using the above lines, I don't get the right answer. After reading some previous post related with complex numbers, I tried to use ComplexExpand function.

 ComplexExpand[Abs[1 + I \[Omega]], TargetFunctions -> {Re, Im}]

Using the above function, I get the right answer.

My question is why I failed in the first attempt, or what is the logic that Abs[1 + I [Omega]] will not give me what I want. Is there a way to get the expected answer by using Abs.

Many thanks