I printed some SMD cartridges and I'm generally really happy with the design, but I noticed a few things I thought could be a bit better.

The first is that the path that the film exits through is quite wide. Normally that's probably not a problem, but I'm a bit clumsy and prone to knocking things over, so I'd prefer a design where a part couldn't slip out through that gap.

So I put together a quick design in Fusion360 with a narrower film gap and tried putting my roll of WS2812B tape into the magazine and it didn't quite fit, so I'll want to print a bigger version.

How much bigger? It would be good to get a number before getting started^{[1]}

Well, how much tape of a given thickness can be held by a given diameter of magazine?

The tape follows a path that resembles an Archimedean Spiral

In polar coordinates the spiral is described by

where a is a constant offset from the center and b controls how much the radius increases per radian traveled.

We're starting at the middle, so a is zero.

The radius increases one tape thickness for each full circle. Taking a given tape thickness t and substituting into the above equation gives:

The tape I'm using is between 2mm and 2.5mm thick. Substituting those values give b of 0.318 and 3.98

For a given radius and thickness theta can be found

Subbing in 30mm for the radius and our b values that gives thetas of 94.25 and 75.4. That's the end of where we need to integrate. The start should be some number greater than or equal to zero. A tape can't be wound down to a point, so I'll use 2pi as the lower bound of the range.

Taking those numbers and plugging them into into Wolfram Alpha gives

So, somewhere between 1.1 and 1.4 meters.

How close is that result to reality?

Only thing to do is to mark where the tape starts to go into the magazine

and measure

1.37 meters! That's in the expected range.

Using that to refine my model suggests that a 40mm radius magazine would hold 2.4m of tape and a 50mm radius would hold 3.8m of WS2812B tape.

[1] - It would've been a good idea, but I just doubled the diameter and started modeling a prototype in Fusion360. I'm going back and writing up notes because I got curious.

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