A great deal of effort went into making spring energy sources* for clocks
that generate force as nearly constant as practicable throughout their
unwinding (and winding as well, as those of you old enough to remember
winding clocks will know). The trick most commonly employed involved
coiling a leaf spring tightly so that it only straightens a little bit,
and over a varying length, as it unwinds. It is never allowed to unwind
completely unless (as readers old enough to have disassembled clocks in
their youths will know) it is allowed outside its surrounding capsule.
The trick with the watch spring is that only a part of it is relaxing at
any time due to the stops at the core and the capsule, and the part that
is relaxing only unbends part way, since its fully relaxed condition is
one which is much larger in radius. It's very difficult to get the spring
back in its capsule if it expands suddenly, incidentally!
The fellow in the next office volunteered a scheme for making a non-
Hookeish clockspring that he devised many years ago. It involved making
the spring width vary over its length. Another example of a non-Hookeish
spring is the recurved bow. If it never occurred to you to wonder how
this clever piece of technology works, you should figure it out. It is
elegant! A little less elegant, but clever still, is the compound bow
which has cams to vary its spring constant in a most non-Hookeish way.
The restoring force actually decreases with extension near full extension.
When I teach my students Hooke's law I always inform them of its status
as a first order engineering approximation, citing the examples above and
others as counterexamples, all of which are drawn from engineering, so my
classification will not appear to be derogatory.
Finally, let me mention that it is somewhat difficult to make a helical
spring that is suitably Hookeish for use in an undergraduate laboratory
designed to demonstrate that phenomenon. Making an oscillator isochronous
with amplitude is difficult (this is a sensitive test). I made some Hooke
springs once by winding brass wire on a conical mandrel when I was in
grad school. I can't recall why that works, but the extension of a
helical spring involves torsion in the wire of which it is made, and that
is at the heart of the explanation as I recall.
Leigh
*Weight driven clocks do not have this problem, of course.