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

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.

• If you don't know the reference (no-load) configuration and cannot remove the load, then you cannot find the spring constant. You can only measure the instantaneous restoring force at the existing displacement. Commented Jan 5, 2022 at 2:26
• What do you mean by load? If I don't touch the spring it has no compression force Commented Jan 5, 2022 at 3:14
• I don't really know what you're asking. If you have access to the unloaded spring, then apply a torque, measure the angular displacement, and obtain the spring constant in N-m/radian. Commented Jan 5, 2022 at 4:41

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:
$$k=T/\theta$$