## Hanging Slinky Analysis 2: The Pre-Tension Wrinkle

Last time, we saw how the length of a hanging slinky is quadratic in the the number of links, namely,

$\Delta x = \int \mathrm{d}x = \int_0^M \sigma mg \, \mathrm{d}m = \frac {\sigma M^2 g}{2}$,

where M is the mass of the hanging part of the slinky, g is the acceleration of gravity, and $\sigma$ is the “stretchiness” of the material (related to the spring constant k—but see the previous post for details).

And this almost perfectly fit the data, except when we looked closely and found that the fit was better if we slid the parabola to the right a little bit. Here are the two graphs, with residual plots: