Racking my Brains

This week I got another question from my brother Nate. He's looking for a way to hang laundry to dry, and he found designs like this one appealing:

George & Willy

He figured it was simple enough that he could build one himself. He wondered though about the height of the triangles. He's hoping to hang the rack high enough that they can walk under it without getting pants in their faces, so the triangles should be as short as possible. Unfortunately, that makes it easier for the rack to tip if the clothes aren't perfectly balanced, which will bring some of the clothes lower. His question then was this: How do I choose the height so I have the most headroom?

As I started to diagram this out, I realized it was a much more complicated system than I first imagined. Looking at it from the side,

Here we have one side of the rack with more weight than the other, which makes the whole thing tilt by an angle ɸ. When we're dealing with rotation, we want torques in place of forces:

The X here is a cross product, which depends on the angle between r and F. What's really interesting is that since the weight always points down, this angle changes as the rack tips. For the right side of the rack, the torque is

After a bunch of simplification this becomes

and similarly for the left side,

We're interested in where the rack settles, so we can add these torques together and set the result to zero.

Now we have the angle, but what we really want is how low the rack gets. This is given by

Once again applying a bunch of trig identities gives

Trying to combine this with the equation for ɸ makes things even uglier than they already are, so let's plug in some values instead. If we imagine hanging 1 pound on one side of the rack, and 0.8 pounds on the other, we can try several heights and widths to find the shape of the depth function:

The minima seemed linear, so I threw the messy equations at Mathematica (thanks, UF site license!) and got something reasonably simple:

If you weigh some wet clothes, you can get some values for F1 and F2, plug them in here, and get the relationship you need for the width and height.

When I was talking to Nate about this question, he mentioned describing my blog to a friend. He noted that some of my posts involve long lists of complicated equations that end with me saying "Huh, that didn't work." Well, to Nate (and that friend if they're reading) I say not all my posts! Some involve long lists of complicated equations that end with a general equation for designing a laundry rack! Well, one.

Ezra & Phineas & Phineas & Ferb

This week, my nephew Ezra had a question about a movie he watched with his brother Phineas, Phineas and Ferb the Movie: Candace Against the Universe. In the movie, the main characters' sister is taken to an alien planet, which is surrounded by an "ion barrier".

There are two scenes that involve the ion barrier, and Ezra wanted to know whether either made sense, physically speaking. In the first scene, two groups of characters try to build portals to the other planet, but find,

There is an ion barrier around the planet. Our transporters were both deflected, which made them connect to each other.

Before we get into whether that makes sense, let's talk about what the word "ion" means. Atoms are made up of a nucleus of positively charged protons and neutrally charged neutrons, surrounded by a cloud of negatively charged electrons. Usually, there are an equal number of protons and electrons, which makes the atom neutral (no charge). However, sometimes it can gain or lose one or more elections, "ionizing" it. My guess is that the ion barrier is a layer of charged atoms surrounding the planet.

So, how could such a setup deflect a transporter? Well, assuming the portals used electromagnetic waves to transmit people, this might actually make sense. Electromagnetic waves include oscillating (moving back-and-forth) electric fields, which cause charged particles to oscillate as well. Those oscillating charges then create their own electric field, which can generate new electromagnetic waves. This is actually how mirrors work: The metal surface of the mirror is full of electrons that can move around freely – That's what it means to be a metal. When light hits those electrons, they reemit that light in the opposite direction, reflecting the image.

Supposing one transporter sent an electromagnetic wave, it could be reflected by this ion barrier, and be received by the other one. Unfortunately, the universe is a big place, and even going at light speed it would take 8 years to make this round-trip to the nearest star. They don't say anywhere in the movie how far away the planet is, but in any case their sister would be waiting an awfully long time!

In place of the transporter, the characters use a rocket ship to get to the planet, but once again encounter problems with the ion barrier:

So, if we go through the ion barrier without a shield, it could fry all of the electronics on the ship, rendering our navigation useless and stranding us in space! [...] In episode 206B of Space Adventure [the show's Star Trek stand-in], they were able to go through an ion barrier without a shield by spinning the USS Minotaur and scattering the ions as they went.

The first part of this makes sense: Charged particles, like the electrons in wires and the ions in the barrier, attract and repel each other. That means that while moving into the cloud of ions, the electronics in the spaceship could experience more current than they were designed for and fail (though likely not as spectacularly as the exploding control panels of Star Trek, and presumably Space Adventure). However, as far as I can imagine, spinning would make no difference, nor would "scattering" the ions, since the electric field they produce would still be present to disrupt the electronics.

As is often the case, storytelling is favored over scientific accuracy. While I nitpick the physics that appears in pop culture, I'm generally happy to suspend disbelief for the sake of a good story, and I'd say this movie certainly qualifies. Thanks for introducing me to it, Ezra!