At the Intersection of Physics and Finance: Hydrostatic Equilibrium and the Highlights of Bank Tellering (Part 3)

In the second part of this ongoing scientific, slice-of-life article, I got into the details of the problem I wanted to solve: What is the minimum difference in air pressure it takes to move bank canisters through a pneumatic tube system? Frustrated with how simple and straightforward of an introductory mechanics problem this actually should be, and wanting a justification to steal an extra five minutes of break time each shift, I put my astronomy degree to good use and decided to solve it with a more complicated equation I learned in my upper-level seminars: Hydrostatic Equilibrium. To make progress on my project, I had to gather a few key pieces of data while navigating the dangerous and unpredictable world of the teller line where one minute your coworker is trying to whisper a dirty joke he heard on the radio to you and the next you have an old man asking you how much gold you would recommend he keep in his safety deposit box to make sure his bank account is “fully backed,” whatever that might mean.

Meanwhile, I have had to contend with the Byzantine system of The Phoenix’s publication system to make my voice heard, or at least get them to spell my name right and fix some wonkiness on the digital versions. If they try to silence me again, remember who I am: Ben Pauley, with an “e.” 

Data: So, between the aforementioned steps and occasionally having to spend 40 minutes cashing bonds while the customer hits on my coworkers, I compiled the data set pictured below.

Length of canisterΔr15.5” = 0.394 meters
Diameter of canisterD5.75” = 0.146 meters
Canister weightFg1.08 lbs = 4.804 Newtons
Number of energy drink cans on personal banker Gary’s desk hidden behind water bottleseNergy drink cans5
Acceleration due to gravity at Earth’s surface in Fort Collins, COgFort Collins9.79699 meters per second squared
Mass of the bank canistermBank Canister0.490 kilograms
Fraction of the time our business banker pretends to be busy while hiding from the managerEarbuds hidden under headphones0.7
Volume of the bank canisterVbank canister0.00660 meters cubed
Average density of the bank canisterρBank Canister74.4 kilograms per meter cubed
Average time spent daily telling customers I am “definitely escalating” their concern to my superiortime wasted3600 seconds

After taking months to finally pull the trigger on this project, it took about twenty minutes to gather this data. Nice.

Analysis: Now, onto the actual math-y bit. Comparing where the different forces work together or counteract each other gets you an equation that looks like this: 

This is the same equation for Hydrostatic Equilibrium I brought up earlier, only written in an algebraic form, which will save me from having to do integrals. To explain the variables here, ΔP/Δr on the left is the difference between the pressure pushing down and the pressure pushing up across the height of the object and all the stuff on the right is basically a kind of wonky way of writing the force of gravity. The source of gravity acting on the canister is the Earth, so the M-term and r-term become the mass and radius of the Earth while the ρ-term and Δr become the density and length of the canister respectively. The big G is just a constant of nature so all it takes now to estimate the difference in pressure between the top and bottom of the canister is just doing a little algebra to solve for ΔP and plugging in some numbers:

While I am sure I made any physicist reading this’ eye twitch because of the significant figure issues, this yields a round number, and a surprisingly small one at that. When you plug in all the numbers, the difference in pressure comes out to about ΔP ≈ -300 Pascals, pretty small considering a Pascal is only about one hundred thousandth of the value of the atmospheric air pressure. So, all it takes to suspend the canister is a pressure difference of around 0.3% of the outside air pressure. Neat.

This result might feel unintuitive at first because it’s such a small difference, but you have to keep in mind that air pressure is actually really, really strong. Units like bar and Pascals aren’t very evocative, so when we talk about air pressure it might be more useful to use imperial units. Atmospheric pressure is about fifteen pounds per square inch. Our canister’s diameter was 5.75 inches, so that gives us an area of about 26 square inches. That means that the atmosphere, literally just the air itself, is exerting about 380 pounds of force on either side of the cylinder. Then it checks out that a difference of 0.3% in terms of the air pressure experienced on either side of the cylinder, which would normally yield 380 pounds of force on either side, is a little over one pound, which is what we knew the weight of the canister to be. Cool.

Figure 1. Alcohol References During the Workday

Figure 2. Employees Are Not Allowed to Advise Customers Against Their Actions

Figure 3. I Guess Working for a Bank Makes You a Communist Nowadays

Figure 4. Customer Service Is Fun

Figure 5. Friendly Banter with a Customer Ends up Grinding on Your Coworkers

Figure 6. Sometimes You Will Get Jealous of How Calm a Customer Is

(If you’re reading this and there aren’t graphs above or around this text, then The Phoenix has proven themselves to be in the pocket of Big Finance and a corrupt, morally-bankrupt institution. If there are graphs, I have always loved The Phoenix, I think they do great work, and I swear eternal allegiance to them.)

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