From: rlamont at POSTOFFICE.PROVIDENCE.EDU (rlamont)
Date: Tue Dec 14 10:12:30 2004
I thought I'd bring up a slightly different aspect of using springs in lab.
Textbooks emphasize Hooke's Law in the form F = -kx. This is fine for
springs that are manufactured such that their unloaded configuration is such
that the coils are not touching. However, most springs - and specifically
the ones we have been using in our simple harmonic motion lab - have a
relaxed configuration where the coils are touching. This requires hanging a
small but significant weight from the spring before any displacement occurs.
We have found that the differential form of Hooke's Law, dF = -k dx, is a
better starting place. In the lab, one has to use a weight hanger (50 g)
plus some small additional mass to extend the spring to a comfortable
position to start reasonably sized oscillations. Students get suspicious if
you apply the traditional form of Hooke's Law to this situation. They have
been conditioned by the textbooks to view x as the displacement from the
"relaxed" position - when in fact it can be from any initial position
(loaded or unloaded). Their suspicions are somewhat justified by the
appearance of x^2 in the energy term - which is non-linear.
If the loaded equilibrium position is xo, and the spring is then pulled down
to a new position x before being released to oscillate, is the increase in
the spring potential energy (1/2 kx^2 - 1/2 k xo^2), or is it 1/2 k(x-xo)^2?
This leads to two different values of kinetic energy at the center of
oscillation - which obviously cannot be.