How should I apply Hooke's law to spiral helical springs?

Question

Say you have a spiral helical spring, like one you might find in a wind-up toy or a hairspring in a watch. The center is attached to a rod that can be twisted to wind it, and the outer end of the spring is permanently fixed to something.

I am trying to calculate the restoring force exerted by the spring, and I believe force applied is the force required to turn the rod attached to the center of the spring. But how should I measure the displacement of the spring?

Another application of Hooke's law that I am unsure of on a spiral helical spring is calculating force applied without an initial measurement. To obtain the Force applied using Hooke's law, I would need the stiffness (k) of the said spring. And unfortunately to find k, I would need to know the force applied. In this sense, I find myself unable to measure the force applied on the said spring without having an initial measurement of force, say 5N. Thus, without somehow measuring exactly how many Newtons of rotational force I exert to turn the rod with my fingers, for example.

Answer 1

If you're asking how to do the measurement, for the first case you can measure the torque to turn the knob by attaching a rod (maybe a small socket wrench) of length $L$. Then, pull the rod using a fish scale or luggage scale which will measure the force $F$, making sure to pull at 90 degrees to the rod. This gives you the torque $T=F*L$. You can turn the knob by a set angle $\theta$ by making markings on it. I would keep it at 45 deg or less to stay in the linear regime (if it is linear). Then your torsional spring constant is:

$k=T/\theta$

Keep in mind, if you hear a buzzing sound like a windup toy when you twist the knob, there is likely some kind of ratchet or servo motor in there, and it's not going to be a simple spring relationship.