In the first part of this ongoing scientific, slice-of-life article, I introduced myself as a recently graduated astronomy and English literature major who settled on working as a bank teller while I figure things out in my life. While the job is…fine, it does not give me much satisfaction. And whenever it isn’t incredibly high-stakes, leaving me panicking about an upset customer holding me legally accountable for a spelling-error costing them hundreds of thousands of dollars, I’m bored. So, I set out to entertain myself at work and shake the dust off my physics textbooks before grad school application season by studying how the pneumatic tube system in the bank drive-through works. From there, I came up with a problem to solve: What is the minimum difference in pressure it takes to move the canisters through the tubes?
Any keen-eyed readers who knew me personally before reading the first article may have noticed that my name was spelled wrong — I suspect the so-called “independent” campus newspaper The Phoenix is secretly in the pocket of Big Finance and are trying to intimidate and silence me for my dangerous and shocking exposé: being a teller kind of sucks.
The Problem: There is a very simple way to find the answer to how much pressure is in the pneumatic tubes, too simple. I actually want more frustration here, but the kind of frustration that I like. Crack open your introductory physics notebook, draw a force diagram, set up a simple equation where all the forces add up to zero, and solve for the pressure difference.
What the above diagram tells us is that the pressure pushing upwards that comes from below the canister has to be balanced by the pressure pushing downwards that comes from above the canister and the force of gravity on the canister, which is just its weight.
That gives you this equation: 
Do some algebra and you get: 
So, by finding the weight of the canister and the area of its top and bottom, you can know the pressure difference required to suspend it in the air. But what I want is something that requires even more effort, something that makes me solve more equations to stave off the boredom. I tried taking notes of interesting things that happened to me during the day, but not that many interesting things happen at work unless something goes catastrophically wrong. Plus, I’m technically not supposed to write anything down, and I’ve already lost a few of the sticky notes I was using for this purpose. Instead, I will check my results against this simple mechanics problem, but I am going to use my astronomy degree for something!
Methods: Hydrostatic equilibrium is true for any system that is not moving or expanding. What happens is that if you have a big clump of stuff, the forces acting on any small subsection of that stuff look the same. You have gravity compressing the whole thing because it always points towards the center of mass, i.e., the middle of that big clump, and at the same time, you have the stuff above and below that little sub-section pressing down and up on it, respectively. When gravity and the downward pressure win, the clump compresses; when the upward pressure wins, the clump expands.
From this diagram, you end up writing a first-order, separable differential equation that looks like this: 
That all sounds kind of abstract and spacey, so how does it connect to the pneumatic tube system? The exact same physical process is happening there as prevents the Earth from collapsing in on itself. The canister starts at rest on a pedestal, and then a powerful fan creates a vacuum of low air pressure above it, eventually getting so low that the atmospheric pressure from below the tube pushes it upwards. In just the same way, any small piece of the Earth’s interior has pressure above and below it from the surrounding material which is in balance with the force of gravity on that small piece. It just felt particularly elegant as a demonstration of the concept to me because of how close it looks to the textbook version of the same concept, which always happens to represent the situation with cylinders of random materials.
Golston Product Solutions, 4.5” Clear Straight Body with Black Taper Bumper (41-5PT-7), 2024
Bradley W. Carroll & Dale A. Ostlie, Figure 10.1 Hydrostatic Equilibrium, 2017
It stuck with me, and I have spent weeks thinking about how I might write this out as a physics problem in preparation for some future time when I might want to teach the concept to someone. What I settled on was the idea that if you knew a handful of key numbers about the specifications of those little canisters, you could estimate the difference in air pressure (also known as the pressure gradient) that it would take to keep it suspended in the tube.
From the math angle, all you really need to know about the canister is how long it is, dr or Δr depending on the notation you are using, and how dense it is, ρ(r). The first one is easy: when you’re on your fifteen-minute break, just use the measuring tape app on your phone and measure it. Ideally, you will have done this before the bank implements a new policy banning phones in secure areas, but I’m not a cop. If you’re also working at this bank, the only problem with this step is that you have to dodge your fellow teller Danny who is so chatty your manager had to exile him to the drive-through just to keep the pace of the teller line steady. If he asks you about your thoughts on cryptocurrency, do not engage. Any response other than “I don’t know, man, I gotta go on break” will result in being sidetracked for at least two thirds of your break time.
The second one is a little more involved. Since we aren’t actually saying the density of the canister is a function of distance from the center of the Earth, r, we need to calculate the average density of the canister, which is its mass divided by its volume. To get those values, you need to measure the diameter of the canister so with that and its length you can work out its volume; then you have to weigh it or rather ask your manager the most out-of-nowhere question ever and get looked at weird, and then you have to covertly pull up the calculator tool on your desktop and pretend like you’re filling out your timesheet while you actually crack open the secrets of engineering and the universe.
This situation is quite difficult to navigate as your manager probably thinks you’re trying to pull off some kind of elaborate scam on the bank where the weight of the drive-through canisters is a crucial part of the plan, but if you’re awkward enough on the delivery of the question, she’ll hopefully just think you’re either desperately trying to make conversation or that you’re a physics nerd. Entering the data in the calculator is fine, the cameras mounted behind you will just see a diligent teller busy with more number tasks, but you have to balance it with engaging in conversation with Andrea, your retiree coworker who, when asked for the time, replies with “I’m thinking it’s gotta be Beer-thirty” concerningly early in the day.
