Angle of the Dangle

Since last year, Steve has been confined to a wheelchair. He gets in and out of it with help from a device called a Hoyer Lift:

Hillrom

The straps at the head and legs have several different notches to adjust the length from the hoist attachments. Along with those adjustments, Steve can also lie higher or lower in the sling, and adjust the angle between his legs and torso. Given these options then, Sally asks: How do we adjust things to get Steve to be more/less upright?

This turns out to be a surprisingly complicated geometry problem. I started off by diagramming it this way:

a and b are the lengths of the two straps, L is Steve's height, h is how far down the pad he's positioned, and θ is the leg-torso angle. We can use the Law of Cosines to first find the width of the two triangles, w, and then the angle between the lift straps:

What we need to figure out from here is the coordinates of that lowest point, where the sling bends. To do that, I rotated the setup so that a lay on the x-axis, and worked out the geometry from there. If the point we want is at (x,y) in these coordinates, then we can write

Solving these 4 equations together produces pages of messy equations, so instead I decided to do it numerically. Since my goal was to make an interactive tool my parents could use, I'm working in JavaScript, so I had to make my own equation solver. I decided to use the bisection method, in which we find the point where a function crosses zero by bracketing it more and more finely. For some combinations of parameters the algorithm fails to find a solution, and the angles don't always look right, but I think it can give a feel for how these different choices factor into the final angle that the lift rests. You can see the code here, or just play with it below!

Ozoning Regulations

Recently our local news in Michigan issued an air quality alert, which mentioned an increase in ozone levels. I was curious about this, since I know the ozone layer protects us from harmful UV light, and the hole over the South Pole is a major environmental concern. That hole was caused by certain pollutants, so I was surprised to hear increased ozone also being associated with poor air quality.

The key is where the ozone is in the layers of the atmosphere. The ozone layer that protects us is in the Stratosphere, while the one associated with poor air quality is in the Troposphere:

Wikipedia

The majority lies above 15-20 km and absorbs 97-99% of the UV light that comes from the Sun. The ground-level ozone results from byproducts of burning fossil fuels. Since ozone is made of 3 oxygen atoms, rather than the 2 that make up oxygen's breathable form, it's heavier, and so I wondered about the process that keeps it at those high altitudes. It turns out it's a continuous cycle, in which oxygen atoms bond and unbond in different configurations. The process is mediated by UV light, which is absorbed by ozone to break it into gaseous oxygen. Since it's absorbed up there, it doesn't make it to the troposphere to break up the ozone down here (or damage our DNA, so overall plus).

The benefits of ozone in the upper atmosphere, combined with its dangers in the lower atmosphere made me wonder if there were a way to transport it upward. Other people have wondered that, and unfortunately it doesn't work. Since ozone is heavier than oxygen, it won't naturally rise, and we'd need to carry it upward. That has its own problems though, since ozone corrodes many materials, so it would get expensive to keep making new containers. The other problem is that there just isn't a lot of ozone down here compared to what the upper atmosphere holds. It would be difficult to collect enough to make the process worthwhile. It seems our best bet is to stop making ground-level ozone in the first place, and to keep the upper-level ozone doing its job!

Everything But the Magnetic Sink

I was looking at some of the environmental data supplied by our PUC this week, and I had a thought on the magnetometer reading:

A magnetometer measures the direction and strength of the local magnetic field, similar to a compass. A significant difference though is that a compass only measures the horizontal direction of the field, while this reading shows all 3 dimensions. What's interesting about that is that we can see the field here has a significant z (vertical) component.

The Earth's magnetic field is approximately a dipole, with an offset of 11° from the rotational axis. We can plot the field lines over the surface:

The red line shows the rotational axis, and the magenta line shows the axis of the magnetic field. Notice that the lines come out of the South pole (a source), and go into the North pole (a sink). One of the strange quirks of how we define our magnetic poles is that because the North pole of a compass points toward the North geographic pole, this must be the South pole of the Earth's magnetic field. Using this plot, we can plot the relationship between the vertical angle of the magnetic field relative to the surface and your latitude on Earth:

The two curves represent the two halves of the Earth in the plot above: the right side, closer to the North pole, and the left side opposite it. The PUC gives us the horizontal dotted line, and we can look up our latitude here in Michigan for the vertical line (or "look up" to the North Star if you want to be fancy). In theory, we could measure our longitude by interpolating between the blue and green lines to the point where the red lines cross. Trying that out gives a position of -37.84°, while our true longitude is -83.86°. Maybe I'll stick to using GPS for now!

An Anod(yn)e Dock

This week my in-laws, Scott and Athena, bought a new dock for their lake. They decided on one made from anodized aluminum, and we were talking about its advantages over other materials. They told me that it doesn't heat up as much as other materials, and this made sense to me from baking with aluminum pans: They tend to heat up quickly and evenly, but there's also very little risk of burning yourself on them. I thought this was due to the heat capacity, the relationship between heat and temperature. Colloquially, we tend to equate these, but there's an important difference: Heat measures the internal energy of a substance, while temperature tells how easily it will give up that energy. Heat flows from high temperature to low temperature. Heat capacity measures how quickly temperature changes as heat flows in or out – Water has a relatively high heat capacity, which is why even a small amount of hot water can burn you. Previously, I had thought aluminum's low heat capacity meant that if you touched a high-temperature pan, your finger would cool it much faster than it would heat your finger. It turns out there's a bit more to it than that.

Scott and Athena mentioned that it was important their dock was made from anodized aluminum, rather than natural aluminum. Anodizing is a process that adds a layer of oxide to the surface of a metal, protecting it from corrosion. In the dock's case though, they said that this also made it feel much cooler than the natural aluminum, which would get uncomfortably hot in the sun. This didn't work with my explanation, since anodizing is a surface effect, which wouldn't significantly change the heat capacity of the bulk material. I decided to compare both the heat capacity and the thermal conductivity, which measures how quickly heat flows through substances:

Material	Heat Cap. [J/kg K]	Thermal Cond. [W/m K]
Stainless Steel	502	14.4
Natural Aluminum	921	236
Anodized Aluminum	921	1.07

Contrary to my previous understanding, aluminum actually has a higher heat capacity than steel! In reality the key difference is in the thermal conductivity, for which the three metals have vastly different values.

Kibble Quibble

We recently got a new bag of food for our dog Eros, and we try to taper him off the old food, since he has a sensitive stomach. As I was serving it up this morning, I got to thinking about how the mixture of foods changes depending on how long we spread the transition. If every day we feed him a total amount d, and we spread T days of old food across b days, we can write

where r is the rate we swap the foods. This is the series for the triangle numbers, so we can replace the sum and solve for the rate:

Using this, we can look at how the rate changes depending on the total amount, and the number of days we spread across:

Note that we require b > T, since otherwise we'd be giving more than a day's food in a day. Since b and T are both measured in days, I wondered if the ratio, representing how much we spread out the old food, had a clear relation to the rate of change in the mixture.

I expected that the points would all fall on a single curve, but there seems to be some variation depending on the specific values for b and T. 

Whenever I hear Eros's stomach making terrible noises, I'm reminded of a bit from Terry Pratchett's Guards! Guards! – "No wonder dragons were always ill. They relied on permanent stomach trouble for supplies of fuel. Most of their brain power was taken up with controlling the complexities of their digestion, which could distill flame-producing fuels from the most unlikely ingredients. They could even rearrange their internal plumbing overnight to deal with difficult processes. They lived on a chemical knife-edge the whole time. One misplaced hiccup and they were geography."