An Analytical Approach to Pamphlet Folding

Most of us can say we’ve endured the frustration of folding letters or pamphlets, trying as much as possible to align the corners for a straight fold only to have them wiggle freely as the crease is made. And in an office like the one I work at, half-folding pamphlets is a fairly procedural task. Whenever this work is entrusted with me, I like to build a rig like the one shown below to minimize the chances of me pulling my hair out.

Fig. 1: Makeshift pamphlet folding rig

It's easy to arrive at a tool like this based on common-sense, but we can also take a more analytical approach as to why a simple 90° corner can make paper folding so much easier. We need to start with a surface-level premise about the universe we live in; as far as we can tell, our universe is comprised of three spatial dimensions, mathematically attributed as x, y and z and experienced as length, width and height, and one temporal dimension. The temporal dimension and the other however many must exist according to string theory or what have you aren’t as important in this task as the three spatial dimensions with which we interact on a daily basis. An unfolded pamphlet can move freely through these three spatial dimensions, as throwing them out the office window will demonstrate when they flutter down to the street below. But while they are sitting on my desk waiting to be folded, they don’t budge. This is because the desk on which they sit is applying an upward force on the pamphlet stack, counteracting gravity and preventing them from moving towards the center of the earth. Balancing forces like this is the foundation of mechanics and is the way that the following discussion manifests in the physical world we live in.

So what else is there to consider? Well, three free dimensions can also be referred to as three “degrees of freedom,” a fancy term that just says a variable can change as necessary. The more degrees of freedom that exist, the harder it is to pinpoint the position of an object that moves freely, like the corner of a pamphlet. What’s also cool about the three spatial dimensions is that even though by default we orient the basis of the defined dimensions to align one direction with gravity because this simplifies calculations, linear algebra tells us that the basis vectors x, y and z are easily translatable to different positions. Now let’s take my desk, oriented by default in the x-y plane, and translate it so that it now exists in the y-z plane. This is kind of similar to if I were floating with my body parallel to the desk, which would make more sense if I were in outer space since there is not as strong a default frame of reference when gravity is not apparent. What we see now is that my desk has somehow become like a wall! Amazing! This tells us that floors, walls and ceilings are not very dissimilar from each other, the common theme being that if I beamed a ball in a closed room, it would bounce off of all surfaces because its free motion in the three spatial directions has been truncated.

Fig. 2: Designation of three spatial dimensions adjusted to rig

Let’s return to the pamphlet folding rig. The rig is comprised simply of three walls: the desk and two others constructed from a box and a paper tray. Additionally, while folding gravity acts on the pamphlets to further constrict movement in the common z dimension, or up, while keeping the dimension still partially accessible so as to get the right side of the pamphlet over the left. My hands as well work to apply force to keep the pamphlet to the left against the box and away from me towards the paper tray while creasing, limiting the x and y dimensions. If we look at the two corners of the pamphlet tucked into the rig while half-folding, all three degrees of freedom have been collapsed, the pamphlets sandwiched between one surface and my hands along all three axes. Not only are these corners restrained, but the geometry of the pamphlet then guarantees that all other corners of the pamphlet should be aligned as well. So while in practice errors on my part and in my shabby rig’s construction limit the effectiveness of this folding method, in theory restricting the three degrees of spatial freedom of the pamphlet should produce a perfect fold every time without the frustration of any dancing corners.

Fig. 3: The final folded pamphlets

While some people may not appreciate such a lengthy analysis of as simple a contraption as a corner for folding pamphlets, I think really delving into why something so common-sense works from a physics standpoint demonstrates how approaching the world from an analytical perspective provides insights that can help everyday people live better lives. I hope you enjoyed this article, and if you have any feedback or questions, leave me a comment below. I love hearing from you guys. Thanks!

Iodine Salt to Treat Radiation?

Fig. 1: Next, movie cover (derricklferguson)

Over winter break I watched the movie Next starring Nicolas Cage with my dad. In the movie, the FBI was able to link a dead woman knifed in her room to a Russian nuclear attack plan because of a few potassium iodide (KI) pills found at the crime scene. The star FBI policewoman (who is also President Coin from the Hunger Games movies!) was quick in realizing that the only reason someone would take potassium iodide pills was to combat radiation poisoning. Before this movie, I had never heard of KI being used for this purpose, and expectedly I was skeptical. Before you judge me, imagine if someone had told you that a sugar pill could prevent you from dying of stomach cancer. That is the same magnitude of ridiculousness that I felt the whole KI pill thing had to be.

But, I was wrong of course. KI supplements are an established treatment for preventing thyroid cancer, one of the biggest health impacts observed after the Chernobyl meltdown [1]. So how is it that something so simple as a salt pill, because that’s what it is, can prevent one of the most odious conditions of modern times caused by a technology that took humans thousands of years to create? Turns out it’s by inhibition [2]. KI pills for thyroid cancer prevention aren’t made up of just any iodine; they are made of the ¹²⁷I isotope, which is iodine’s only stable form [3]. Ingested iodine is taken up by the thyroid gland, and if the iodine is of a radioactive isotope the subsequently produced radiation can cause thyroid cancer. KI pills work by dumping stable ¹²⁷I into the person’s blood stream, flooding the thyroid and reducing uptake of other radioactive iodine isotopes. KI pills only work in preventing thyroid cancer caused by radioactive iodine exposure, however, not other conditions caused by general radiation exposure.

Fig. 2: ²³⁵U fission product properties (Hochel, R. C.)

So, a few other questions obviously arise from this talk of iodide pills, one of which being where do the radioactive iodine isotopes come from in the context of nuclear fission? Fission of heavy atoms results in atoms of lighter weight and free neutrons that propagate the nuclear fission reaction. Some of the fission products of ²³⁵U are various iodine isotopes, including ¹³⁵I (6.33% yield), ¹³¹I (2.83% yield) and ¹²⁹I (0.9% yield) [4]. These are clearly not the main fission products of ²³⁵U, but they can still accumulate in contaminated environments, especially where large-scale nuclear fission reactions were involved such as nuclear meltdowns and atomic bomb testing sites. Another question to answer is how do the radioactive isotopes end up being ingested by people in contaminated regions? Scientists at Dartmouth, New Hampshire were able to measure increases in ¹³⁵I concentration, an indicator also of the presence of undetectable ¹²⁹I, on land but especially in local streams a year after the 2011 Fukushima meltdown in Japan [5]. They cited the increase as due to nuclear fallout from the Fukushima incident that blew across the continent and deposited itself in groundwater sources. This implies that the radioactive iodine isotopes can be both airborne and waterborne. If everything is coated in radioactive iodine, ingestion is believably imminent. To bring us full circle, there was also a run on KI pills in 2011 on the American West Coast due to fears of radioactive iodine finding its way into homes and food supplies there as well [6]. It’s funny how analyzing a simple movie premise can take us all the way to a not-so-late nuclear disaster.

Yesterday was the first day of classes, and soon enough school will be back in full swing. I've based my schedule this semester off of a google calendar with the idea that better organization will make hectic school life just a bit easier, so we'll see how that goes. My course load is two materials science classes, one materials science lab, orgo 2 and an anthropology class on modern culture. I'm hopeful that this semester will go better than last, and I'll keep you guys updated on what goes on. If you like this article or have ideas for another, be sure to leave me a comment below. Thanks for reading!

Can the Earth's Magnetic Field Support a Space Railgun?

Fig. 1: Railgun diagram (HowStuffWorks)

In physics class, you’ve probably learned about railguns and how they use magnetic fields and current to generate a propulsive force. And if you’re anything like me, you’ve probably wondered if we can launch a person or spaceship with one, because why not? What would be even cooler is if we could do it with the earth’s ambient magnetic field. Well, let’s test this idea a little bit.

We know that the force generated by a magnetic field on a charged particle is

1.       F = d/dt(L)q x B = qv x B (L is displacement, q is charge, B is magnetic field, v is velocity)

An equivalent statement more applicable to railguns is 

2.       F = d/dt(q)L x B = IL x B          (I is current)

This change of derivative position is acceptable because it represents a change in reference frame, the displacement derivative in the frame of the charged particle and the charge derivative from a fixed external frame of reference, watching the charges flow by. In the setup of a railgun, B and I are orthogonal and the resultant vector of the cross product is in the direction of propulsion. Solving the cross product yields

3.       F = ILB 

We also know the induced current through the railgun circuit to be related to the generated emf, 

4.       Ɛ = - d/dt(Φ_B) = - d/dt(BA) = - Bd/dt(A)          (Φ_B is magnetic flux, A is area)

By geometric analysis, we know the change in area to be the fixed base length/bar length times the change in height, or

5.       d/dt(A) = L d/dt(h) = Lv          (h is height)

The full expression is

6.       Ɛ = IR = - BLv

7.       I = - BLv/R

Appending our force equation to account for induced current that works against the initial current, we get

8.       F = (I - BLv/R)LB = LBI – (L²B²/R)v

This equation is useful because by setting the magnetic force to zero we can solve for a terminal velocity,

9.       v = IR/LB

This is no good! If we’re to launch someone high into the sky, we need a constant force to produce a constant acceleration to exceed that of gravity. So what if we keep a constant effective current to counteract the current induced by propulsion? Looking at equation 8, we see that this is possible if we establish current as a time-dependent function given by

      10.   I(t) = I₀ + BLv/R                                                 (R is resistance)

      11.   I(t) = I₀ + BL(at)/R = I₀ + BL(Ft/m)/R               (from kinematic equations v=at and F=ma)

The F term here should be substituted with the constant force we predict to be generated by the current adjustment

12.   F = I₀LB

Substitution and regrouping produces

13.   I(t) = I₀(1 + (B²L²/Rm)t)          (m is mass)

This equation can be verified by substitution into equation 8. Now that our force is constant, let’s calculate what current would accelerate the average person [1] on a weightless conductive beam unaffected by air drag 1m in length enough to counteract gravity (a pretty low bar for testing viability of concept). Since the magnetic field generated by the circuit would probably diminish too close to the walls of a 1m launcher, an external field is necessary. Let’s try using the upper end of the earth’s magnetic field strength range [2].

                     14.   F = mg = I₀LB

                     15.   I₀ = mg/LB = (62kg)(9.8m/s²)/((1m)(65x10^-6 N/Am)) = 9.3x10⁶ A

That’s a lot of current, but perhaps possible? Adjusting for a spaceship weighing 1,000 times the average person 10m wide with an acceleration of 1m/s² [3], we get a figure on the scale of 1x10⁷ A. In 2014, the CERN Superconductors team was able to pass 20x10³ A through an MgB₂ superconducting wire, a world record at the time [4]. This record value is clearly magnitudes less than even the current necessary to lift a single person, let alone a spaceship. Looks like the earth's ambient field is a no-go. But! If the person were in a magnetic field of an achievable 1T even, then with currently achievable currents it should be possible to throw them at least into the sky if not into space. Hooray for space railguns!

If you liked this post or have any ideas for another, let me know by leaving a comment. Thanks for reading!