Is the frequency of oscillation of a rotating disk affected if a mass hanging from (attached to the disk but pulled by gravity) the disk exerts a torque on the rotating disk?
In the following answer, I assume that your question is:
To simplify things I assume that 1.The disk doesn't rotate, but just oscillates. 2.The disk is flat out and infinitely thin (it's a circular disk with radius $r$) 3.There are two cases: when the disk is a. horizontal (all parts of the disk are same height from ground). b. vertical (perpendicular to the horizontal alignment.
In short, the answer is yes. Regardless of whether the disk is horizontal or vertically positioned, nonzero mass attached to any part of the disk other than its axis of oscillation will affect its frequency of oscillation.
In order to oscillate, there must be a force causing the disk to keep reversing its direction during the course of each oscillation. For example, there might be some spring system causing the disk to oscillate.
Here are some useful keywords to help you do further research on this question: Simple harmonic motion, torsion pendulum.