Over the last year there’s been a lot of posting and questioning about what happens to the bottom of a slinky when you drop the top. Here’s a good video from my friend Derek showing people’s expectations, along with some good description of how things work:

There’s also this awesome blog post from my friend Rhett, showing how you can model a slinky as a bunch of mass/springs in series.

My motivation for this post was thinking about how the same thing doesn’t happen with all springs. For example, consider dropping a typical car suspension spring, you would definitely see the bottom moving well before the top gets there, in fact, you’d likely see it moving almost immediately. What I wanted to do was see what you’d have to do to Rhett’s model to see those effects.

The model

The model consists of point masses placed vertically and connected via springs to their nearest neighbors. One way to think about that is that each loop of the slinky is a spring. As Rhett pointed out, this model does a good job showing how the slinky works. Here’s my version of that model:

slinky with zero equilibrium length for all 50 springs (represented by a coil of the slinky)

The key to that model is that the equilibrium length of each spring was set to zero (which was the quickest way to code up this solution, and hence the first one I did). It’s kind of hard to see, but the slinky basically turns inside out. Here’s a plot of the location of each coil as a function of time:

plot of the height of each coil as a function of time with zero equilibrium length.

There’s definitely some cool things happening after the top flys by the bottom. However, this isn’t what really happens with the slinky, right?

Here’s the result of a simulation with a non-zero equilibrium length of the springs. Note how the coils don’t pass each other. Note also that the bottom moves before the top gets there:

20 coils with a non-zero equilibrium separation between them.

Here’s the coil plot for that movie:

coil positions for the previous movie

If I raise the strength of the spring constant, I can effectively turn the spring into a solid, rigid object where the bottom falls at the same rate as the top. What’s cool about all of these is a plot of the center of mass. It always looks like a parabola, even if I pre-compress the spring.

So, I think the slinky drop is cool, and I love how it makes people think. However, we should careful when ascribing its features to all springs.

code

Here’s a screenshot of the Mathematica code to do this:

Mathematica code for the calcuation

And here’s the graphics code:

Mathematica code for the graphics generation