Chain Change

Recently I've been rereading the book Seveneves by Neal Stephenson, which opens with the Moon being destroyed by an unknown object, forcing humanity to flee the surface of the Earth before the debris comes through the atmosphere. One of the engineers that goes into space notices something interesting about the chain he wears:

[He] had got it spinning around his neck. It had opened up into a broad, undulating oval that didn't touch his neck or collar anywhere, so it was just orbiting around him in free space. [...] He had learned a few tricks for speeding it up and coaxing it into different shapes by blowing on it with a drinking straw or flicking it with a fingernail. [...He] poked an index finger up into the chain's path. It caught on his knuckle, hiccupped, and suddenly wrapped around his hand in a chaotic tangle.

I was really curious if I could simulate some of these effects, so I came up with a simple model for the chain: A circle of points that are connected to each other by simple springs. The force applied by the springs is proportional to how far each point is from its neighbors, so the circle will tend to collapse to a point, but if we set it spinning, then just like an orbit we can keep the points from falling inward. We can see what happens if we tap one of the nodes:

This shows some really nifty behavior: Part of the poke moves around the circle in the direction of rotation, but there's also a dent that stays fairly stable until the other part of the poke comes around again. You might also notice when the animation loops that the chain has moved off-center by a fair amount. That made me wonder what would happen if I poked it symmetrically on each side:

Now the chain stays centered, but becomes more ellipsoidal, with the long axis rotating with the nodes. So far, so good – Let's try sticking a finger in it!

Hmm, not quite what we were looking for. To model the "finger," I checked at every step whether each node was inside the finger's radius. If it was, I would move it to the nearest point outside. Unfortunately, this meant that if I moved the finger much further in than I do above, links would start popping to the outside, which sounds painful for our simulacrum! I do like the faux-3D helix the chain forms on the other side of the finger though. 

The Fault in Our Ground

Yesterday, we took the camper to refill propane and get it cleaned. When we came back to our site, and plugged in to the campground power, we found nothing was flowing! We tried a couple ideas to solve it, but we were exhausted and it was already dark, so we turned up the (newly filled) propane heat and went to sleep. This morning, after much more troubleshooting, we discovered the issue was the ground fault circuit interrupter (GFCI), which we had tried the previous night, but hadn't performed the necessary reset steps correctly. I thought I'd get some good out of the frustrating experience, and write a post on GFCIs.

Humans are essentially big bags of salty water, which makes us good at conducting electricity. Unfortunately, letting anything more than around 100 milliamps flow through us can be fatal. The 120 volts that the US uses for mains power can create this much current under the right conditions, so we need safety mechanisms to prevent it.

A "ground fault" refers to the amount of outgoing current not matching the amount of current returning to the power source, meaning that charge is finding another path to flow down. We can measure the current in each line using a loop of wire, which picks up the magnetic field of the moving charges and generates a voltage in the loop. If the two lines don't match, we break the circuit:

In this diagram, the green loops measure the current, and the red is the GFCI's circuitry to compare and break the connections. Initially, I had wondered how a single outlet could cut power to all outlets, but this diagram makes that clear: Typically houses are wired with a pair of lines going from the source to some endpoint, and outlets are wired across the two in parallel. However, if we can break these lines immediately after the source, none of the downstream plugs will be powered.

That brings us to our issue: Once the GFCI is tripped, what do you do next? Typically, GFCIs have two buttons on them, Test and Reset. The test button will artificially pop the breakers so you can ensure they work, but critically, this requires having power. Once we had discovered the power wasn't working, we disconnected from the line in case whatever unknown problem was a fire risk. The real issue though was that to reset the GFCI, you need to push those breakers back in. They're designed to stop power within a tenth of a second, so they use powerful springs to open the connection quickly. That means you need to push a tiny button deep inside the outlet with incredible force, something we were incapable of doing last night! Thankfully in the morning we worked it out, and our power is back on without any expensive fix.

Fraught Freight

Now that I'm actually going into an office every day, I'm once again a car commuter, and I'm filled with transportation-related questions! Our route mostly involves the highway, and I was curious about the rollover signs on off-ramps:

rxrsignals.com

This shows a truck tilting at a 30° angle, while going 25 mph. The tilt comes from the fact that the truck is going around a curve, which requires applying a centripetal force. Since this force is applied to the tires, there's a torque produced, which will act to tip the truck toward the outside of the curve:

The red dot marks the truck's center of mass, with m the total mass of the truck, and a the acceleration required to make the turn. This is given by

where R is the radius of the turn. To find this, we can use Google Maps' measuring tool on the ramp by our home:

Given this R, along with the speed and dimensions of the truck, we can find the height of the center of mass that would produce the given tilt – The diagram above is too high, and the truck would tip over, but to balance g and a, we would need

where θ is the tilt angle. Using our 30° case, and the turn shown above, we can check the heights for several different speeds:

In most cases, we should expect the truck to be loaded such that the center of mass is in the center of the truck, which means the posted limit for our turn of 25 mph is just about right!