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.