This question came up because of this diagram that I saw in my textbook of an angular simple harmonic oscillator. I've always struggled a bit with torque and rotational dynamics in general, and I thought that I understood it once I saw the relevant equations derived from Newton's laws and the conservation of energy. However, those derivations relied on the fact that there was a force $F$ being applied to the rotating object at distance $r$ from the axis of rotation and that $F \times r \neq 0$. In the case of a force applied at the axis of rotation, however, $r = 0$, and thus $F \times r = 0$ also. How can there be a torque in the case of an angular simple harmonic oscillator, and where is the force coming from that is causing this torque?
This is the webpage with the full material that is covered in my textbook.
Edit: another example where this seems to be happening in my textbook is during a section on the precession of a gyroscope. The book seemed to be saying that the gyroscope experiences torque after it has been set spinning, but that this torque is not applied at the edges of the gyroscope. Rather, it comes from the rod about which the gyroscope frame is rotating. Could the answer to my question in both of these cases be that, in fact, $r \neq 0$? Is $r$ then the radius of the string (in the case of the angular simple harmonic oscillator) and the radius of the central rod (in the case of the spinning gyroscope)? Or is the gyroscope example fundamentally different?