With my metaphorical “mathematics of sets” hat on, this is what physics looks like, to me. The further you go in the field, the more challenging the mathematics gets; also, the better (and more expensive) the toys become.
I’ve been using Zometools, available at http://www.zometool.com, to build interesting geometrical shapes since long before I started this blog. I recently found this: a 2011 photograph of myself, holding a twisting Zome torus. While I don’t remember who was holding the camera, I do remember that the torus is made of adjacent parallelopipeds.
After building this torus, I imagined it as an accretion disk surrounding a neutron star — and now I am imagining it as a neutron star on the verge of gaining enough mass, from the accretion disk, to become a black hole. Such an object would emit intense jets of high-energy radiation in opposite directions, along the rotational axis of this neutron star. These jets of radiation are perpendicular to the plane in which the rotation takes place, and these two opposite directions are made visible in this manner, below, as two dodecahedra pointing out, on opposite sides of the torus — at least if my model is held at just the right angle, relative to the direction the camera is pointing, as shown below, to create an illusion of perpendicularity. The two photographs were taken on the same day.
In reality, of course, these jets of radiation would be much narrower than this photograph suggests, and the accretion disk would be flatter and wider. When one of the radiation jets from such neutron stars just happens to periodically point at us, often at thousands of times per second, we call such rapidly-rotating objects pulsars. Fortunately for us, there are no pulsars near Earth.
It would take an extremely long time for a black hole to form, from a neutron star, in this manner. This is because most of the incoming mass and energy (mostly mass, from the accretion disk) leaves this thermodynamic system as outgoing mass and energy (mostly energy, in the radiation jets), mass and energy being equivalent via the most famous formula in all of science: E = mc².
An interesting phenomenon in physics, and physics education, is the simplicity of symmetric situations, compared to the complexity of similar situations which are, instead, asymmetrical. Students generally learn the symmetrical versions first, such as this static equilibrium problem, with the hanging masses on both left and right equal.
The problem is to find the measures of the three angles shown above, with values given for all three masses. Here is the setup, using physical objects, rather than a diagram.
The masses on the left and right are each 100 g, or 0.100 kg, while the central masses total 170 g, or 0.170 kg. Since all hanging masses are in static equilibrium, the forces pulling at the central point (at the common vertex of angles λ, θ, and ρ) must be balanced. Specifically, downward tension in the strings must be balanced by upward tension, and the same is true of tension forces to the left and to the right. In the diagram below (deliberately asymmetrical, since that’s coming soon), these forces are shown, along with the vertical and horizontal components of the tension forces held in the diagonal strings.
Because the horizontal forces are in balance, Tlx = Trx, so Mlgcosλ = Mrgcosρ — which is not useful now, but it will become important later. In the symmetrical situation, all that is really needed to solve the problem is the fact that the vertical forces are in balance. For this reason, Tc = Tly + Try, so Mcg = Mlgsinλ + Mrgsinρ. Since, due to symmetry, Ml = Mr and λ = ρ, Mr may be substituted for Ml, and ρ may be substituted for λ, in the previous equation Mcg = Mlgsinλ + Mrgsinρ, yielding Mcg = Mrgsinρ + Mrgsinρ, which simplifies to Mcg =2Mrgsinρ. Cancelling “g” from each side, and substituting in the actual masses used, this becomes 0.170 kg = 2(0.100 kg)sinρ, which simplifies to 0.170 kg = (0.200 kg)sinρ, then 0.170/0.200 = sinρ. Therefore, angle ρ = sin-1(0.170/0.200) = 58°, which, by symmetry, must also equal λ. Since all three angles add up to 180º, the central angle θ = 180° – 58° – 58° = 64°. These answers can then be checked against the physical apparatus.
When actually checked with a protractor, the angles on left and right are each about 53° — which is off from the predicted value of 58° by about 9%. The central angle, of course, is larger, at [180 – (2)53]° = 74°, to make up the difference in the two smaller angles. The error here could be caused by several factors, such as the mass of the string itself (neglected in the calculations above), friction in the pulleys, or possibly the fact that the pulleys did not hang straight down from the hooks which held them, but hung instead at a slight diagonal, as can be seen in the second image in this post. This is testable, of course, by using thinner, less massive string, as well as rigidly-fixed, lower-friction pulleys. However, reducing the error in a lab experiment is not my objective here — it is, rather, use of a simple change to turn a relatively easy problem into one which is much more challenging to solve.
In this case, the simple change I am choosing is to add 50 grams to the 100 g already on the right side, while leaving the central and left sides unchanged. This causes the angles where the strings meet to change, until the situation is once more in static equilibrium, with both horizontal and vertical forces balanced. With the mass on the left remaining at 0.100 kg, the central mass at 0.170 kg, and the mass on the right now 0.150 kg, what was an easy static equilibrium problem (finding the same three angles) becomes a formidable challenge.
For the same reasons as before (balancing forces), it remains true that Mlgcosλ = Mrgcosρ (force left = force right), and, this time, that equation will be needed. It also remains true that Mcg = Mlgsinλ + Mrgsinρ (downward force = sum of the two upward forces). The increased difficulty is caused by the newly-introduced asymmetry, for now Ml ≠ Mr, and λ ≠ ρ as well. It remains true, of course, that λ + θ + ρ° = 180.
In both the vertical and horizontal equations, “g,” the acceleration due to gravity, cancels, so Mlgcosλ = Mrgcosρ becomes Mlcosλ = Mrcosρ, and Mcg = Mlgsinλ + Mrgsinρ becomes Mc = Mlsinλ + Mrsinρ. The simplified horizontal equation, Mlcosλ = Mrcosρ, becomes Ml²cos²λ = Mr²cos²ρ when both sides are squared, in order to set up a substitution based on the trigonometric identity, which works for any angle φ, which states that sin²φ + cos²φ = 1. Rearranged to solve it for cos²φ, this identity states that cos²φ = 1 – sin²φ. Using this rearranged identity to make substitutions on both sides of the previous equation Ml²cos²λ = Mr²cos²ρ yields the new equation Ml²(1 – sin²λ) = Mr²(1 – sin²ρ). Applying the distributive property yields the equation Ml² – Ml²sin²λ = Mr² – Mr²sin²ρ. By addition, this then becomes -Ml²sin²λ = Mr² – Ml² – Mr²sin²ρ. Solving this for sin²λ turns it into sin²λ = (Mr² – Ml² – Mr²sin²ρ)/(-Ml²).
Next, Mc = Mlsinλ + Mrsinρ (the simplied version of the vertical-force-balance equation, from above), when solved for sinλ, becomes sinλ = (Mrsinρ – Mc)/(- Ml). Squaring both sides of this equation turns it into sin²λ = (Mr²sin²ρ – 2MrMcsinρ + Mc²)/(- Ml)².
There are now two equations solved for sin²λ, each shown in bold at the end of one of the previous two paragraphs. Setting the two expressions shown equal to sin²λ equal to each other yields the new equation (Mr² – Ml² – Mr²sin²ρ)/(-Ml²) = (Mr²sin²ρ – 2MrMcsinρ + Mc²)/(- Ml)², which then becomes (Mr² – Ml² – Mr²sin²ρ)/(-Ml²) = (Mr²sin²ρ – 2MrMcsinρ + Mc²)/(Ml)², and then, by multiplying both sides by -Ml², this simplifies to Mr² – Ml² – Mr²sin²ρ = – (Mr²sin²ρ – 2MrMcsinρ + Mc²), and then Mr² – Ml² – Mr²sin²ρ = – Mr²sin²ρ + 2MrMcsinρ – Mc². Since this equation has the term – Mr²sin²ρ on both sides, cancelling it simplifies this to Mr² – Ml² = 2MrMcsinρ – Mc², which then becomes Mr² – Ml² + Mc² = 2MrMcsinρ, and then sinρ = (Mr² – Ml² + Mc²)/2MrMc = [(0.150 kg)² – (0.100 kg)² + (0.170 kg)²]/[2(0.150 kg)(0.170 kg)] = (0.0225 – 0.0100 + 0.0289)/0.0510 = 0.0414/0.510 = 0.812. The inverse sine of this value gives us ρ = 54°.
Having one angle’s measure, of course, makes it far easier to find the others. Two paragraphs up, an equation in italics stated that sinλ = (Mrsinρ – Mc)/(- Ml). It follows that λ = sin-1[(Mrsinρ – Mc)/(- Ml)] = sin-1[(0.150kg)sinρ – 0.170kg)/(-0.100kg)] = 29°. These two angles sum to 83°, leaving 180° – 83° = 97° as the value of θ.
As can be seen above, these derived values are close to demonstrated experimental values. The first angle found, ρ, measures ~58°, which differs from the theoretical value of 54° by approximately 7%. The next, λ, measures ~31°, also differing from the theoretical value, 29°, by about 7%.The experimental value for θ is (180 – 58 – 31)° = 91°, which is off from the theoretical value of 97° by ~6%. All of these errors are smaller than the 9% error found for both λ and ρ in the easier, symmetrical version of this problem, and the causes of this error should be the same as before.
Because there’s nothing wrong with mixing a little Rolling Stones with your physics, that’s why.
In the Summer of 2014, with many other science teachers, I took a four-day-long A.P. Physics training session, which was definitely a valuable experience, for me, as a teacher. On the last day of this training, though, in the late afternoon, as the trainer and trainees were winding things up, some of us, including me, started getting a little silly. Physics teachers, of course, have their own version of silly behavior. Here’s what happened.
The trainer: “Let’s see how well you understand the different forces which can serve as centripetal forces, in different situations. When I twirl a ball, on a string, in a horizontal circle, what is the centripetal force?”
The class of trainees, in unison: “Tension!”
Trainer: “In the Bohr model of a hydrogen atom, the force keeping the electron traveling in a circle around the proton is the . . . ?”
Class: “Electromagnetic force!”
Trainer: “What force serves as the centripetal force keeping the Earth in orbit around the Sun?”
Me, loudly, before any of my classmates could answer: “God’s will!”
I was, remember, surrounded by physics teachers. It took the trainer several minutes to restore order, after that.
The TV series House MD first aired in 2004. I quickly developed a fascination with both the title character, and the show, and watched the entire run of the show, until it ended in the Spring of 2012. The following Fall, I began to seriously suspect — at age 44 — something that had occurred to me only as a possibility for a few years before that: I’ve been an “Aspie” my whole life, a hypothesis which I could tell had great explanatory power to explain my numerous peculiarities, as you can tell in this early blog-post on the subject, not long after I came to WordPress. By the time 2013 arrived, shortly before I turned 45, I had begun the necessary tests of this idea, which included, of course, discussions of the possibility with a psychiatrist, as documented in another blog-post, here.
This was a breakthrough for me, for it helped with a long-term project I begun in my teens: development of the ability to reprogram my brain’s own software, which I became aware was happening at night. I first wrote about this sleep-reprogramming here, just over a year ago, almost two months after getting married — to another math teacher. Thanks to my wife, the critical error in that sleep-related post (that I was reprogramming in the deep parts of non-REM sleep) was later discovered, for she noticed that I frequently stopping breathing while asleep. Sometimes she would have to shake me awake, urging me to resume breathing; other times she would be awakened, herself, by my own gasping for air, which it turns out I had been doing for years. At her urging, I discussed the possibility with my primary care physician, who immediately referred me to a specialist for a sleep study. This resulted in a definitive diagnosis of sleep apnea, revealing that I was only getting stage 1 and 2 sleep, but no significant quantities of stage 3 sleep, nor stage 4 sleep, nor REM sleep. This ruled out my “deep-sleep reprogramming” hypothesis.
I’ve been a teacher for over twenty years, and have more experience teaching science than any other subject. I started learning science very early in life, simply by hanging around in the science building of a university, as a little kid, where I could avoid interacting with children my own age (whom I did not understand), and communicate exclusively with professors and their college students, as described, along with a 5th-grade “sequel” involving “show and tell,” here. For these reasons, I don’t have to consciously employ the scientific method when empirical evidence shows a hypothesis to be flawed — it is automatically what I jump to. My deep-sleep hypothesis did not have to be thrown out completely, but only modified. The evidence prior to the sleep study did indicate this auto-reprogramming was happening in my sleep, but the sleep study proved that it could not be happening during deep non-REM sleep. What was not ruled out, though, was the only type of sleep I had been getting before I got the CPAP machine I use now: Stage 1 and 2 sleep. It is my current position, subject to further testing, that I sleep-reprogram in the shallow parts of non-REM sleep, most likely Stage 1, or in the “twilight” regions between wakefulness and sleep, hypnagogia (the “going to sleep” transition period), and/or hypnopompia, the “waking up” transition period. With my CPAP machine, I have now had the better part of a year to recover from the negative effects of chronic sleep deprivation. I now get the REM, stage 3, and stage 4 sleep needed for good physical and mental health. Of course, I still get Stage 1 and Stage 2 sleep, so I can still sleep-reprogram — and, for the last few months, I have been using sleep-reprogramming to seek ways to reduce the social-interaction difficulties involved with Asperger’s, while retaining the advantages.
One night, and I suspect it was last night, I used sleep-reprogramming to prepare myself to learn to do something I have not been able to do before: detect lies, and employ certain specific, related social skills, using alternative methods than those used by most non-Aspies. The program required watching House MD to activate it, however, as I discovered while watching it today, after a couple of episodes, while resting last night, before going to sleep — the first time I have watched the show in months. The changes that happened while I watched a single episode (season one, episode four, “Maternity”) were both rapid and dramatic; it was quite clear to me that an already-prepared unconscious subroutine was being activated.
Before this lie-detecting subroutine was activated, these had been long-term statements which accurately described me:
There is an important subgroup of the general population who form an exception to one of the common-among-Aspies characteristics above: not reading facial expressions or body language well. This group is the blind, and, to a lesser degree, the visually impaired. I wear glasses at work, and while driving, for mild-to-moderate myopia, but do not need them to function indoors, especially in familiar settings, such as home. Lately, I have simply stopped wearing my glasses while home. However, I certainly use my vision a lot, and think visually; the sheer volume of geometry (with apologies for the pun) on this blog is evidence enough of that. I have more familiarity with blindness than most sighted people, though, for three reasons: (1) I have read comic books since I was a kid (sometimes while hanging out in some science lab, although I was more likely to be found playing with a some dangerous chemical when very young), and my favorite title for most of my life has been Daredevil, the source of the lower-left part of the image above, (2) I have had close friends who were blind, and (3) I learned to read and write Braille as a college undergraduate, so that I could communicate with these friends by mail, and have not forgotten this skill, although it has lost speed — but I made the Braille lettering in the image above, which simply reads “everybody lies,” a line made famous by House MD.
There is also a fourth reason, unrelated to blindness, which is a common characteristic of those with Aspergers: periods of time when my senses become painfully over-sensitive, so that the sun, indoor lighting, etc., appear to be turned up extra-bright, everything is incredibly loud, etc. — except for my sense of smell, weakened by allergies, and exposure to chemical fumes. This used to happen during migraine headaches; now, I no longer get this state with headaches, but the amplified sensory perceptions are now so intense, at times, that this state is actually worse than a migraine headache, for at least those had pain to distract me, at least somewhat, from overwhelming sensory overload. When this happens, I blindfold myself to eliminate the visual overload, but I cannot silence the world, and you can read a description of what it sounds like, when I am in “Matt Murdock mode,” as I call it, in this blog-post. My perceptions don’t become so acute that I can hear the heartbeats of those around me (as the fictional Murdock can), and I’m fine with that; I don’t need, nor want, to have the volume turned up any further.
When House MD was airing new episodes, as I remember the show, Dr. Gregory House would be an incredible jerk to people at least six to ten times per episode, sometimes more, and I found it hilarious. Only tonight, while watching the episode mentioned above, “Maternity,” did I realize (after nearly everyone else on the planet who has seen even one episode of this show), that House is incredibly rude at a rate far closer to six to ten times per minute. I had no idea!
This made me quite surprised. I then suddenly noticed something else: Dr. Allison Cameron — lying. Next, Chase lied. Later, Foreman lied. A patient lied, but of course they always do — “everybody lies” is a recurring theme of the show, and the consequences of lying are the show’s most-used plot-device. While thinking about this, I caught House lying. And I knew about each and every one of these lies before House revealed them, as lies, later in the episode. How was I noticing these lies I had never heard, as lies, before?
Of course, I have seen every episode of House MD, so memory is definitely a factor, but had not watched season one in years. I began to focus on this puzzle. About ten detected-lies later, I figured it out: I was hearing not an increased volume, but tiny changes in pitch, no more than a half-step on the musical scale. Lying is a risky activity, and liars know this, so it causes increased anxiety when people lie — and, due to this anxiety, I reasoned, their airways constrict, at least slightly, at the end of a lie, causing an increase in the pitch and frequency of the sound of their voices at the end of a lie. House was the ideal tool for teaching this, for it allowed me to “calibrate” my internal lie-detector, by focusing on the voices intently, while using long-stored memories of Dr. House revealing lies to help me catch what I had never caught before.
Of course, House is a TV show. This needed to be tested — with something which did not involve actors, nor writers. And so, I devised a way to test it. I asked a scientifically-minded person I know to help me test a hypothesis, by choosing a random order for a true statement and a lie, and then simply telling both to me, and then see if I could accurately identify the true statement, as well as the lie.
The pitch of this person’s voice unexpectedly went down, slightly, at the end of one of the statements, but did not change when speaking the other statement. What was going on, I wondered? I then figured it out: I had chosen a scientifically-minded person — who was trying to outsmart my experimental test! I then noticed that one statement of the two — the lie — concerned the number of items in a box, and was therefore testable. The other statement was not testable. The other person, who is quite intelligent, had taken a minute or two to formulate both the true statement, and the lie — and clearly intended me to fail the test, and then be able to prove it had failed, by simply opening the box. Using logical reasoning (an old skill) and my new, sonic lie-detector, together, I can now even detect a lie that a police polygraph might not have been able to detect!
I then realized I was detecting more subtle emotions . . . catching more jokes . . . and generally doing, by rapidly analyzing sound, what other people do by somehow “reading” body language and facial expressions.
It is a myth that Aspies can’t do such things as understanding emotions. It is not a myth that Aspies think in different ways, however. Sometimes, individual Aspies simply have to figure out their own methods for doing something which comes naturally to non-Aspies — and that’s exactly what I did.
I could, presumably, use this new frequency-sensitivity to train myself to lie convincingly, by deliberately avoiding the “tell” of a slight pitch-increase at the end of a lie. However, I choose not to do that. The first week I went without telling a single lie, in my teens, was difficult, but the second week was much easier, and the third was easier still. Not having to try to keep track of previous lies began to give me a persistent, liberated feeling that can only be enjoyed by those who are honest — and that feeling would, of course, vanish if I chose to invest time, energy, and thought into becoming a skilled liar. To do that would be to move backwards in my life, rather than forwards, and I see no reason to do that. In other words, I’m still me . . . just the latest version of me, that’s all.
Definition of catbouncemax (shortened form of “maximum catbounce”): for any particular cat, its catbouncemax is equal to the takeoff kinetic energy of that cat if it suddenly and unexpectedly finds itself face-to-face with an adult copperhead snake.
I’ve actually seen this happen. Really. The cat reached a height I estimate as 1.4 meters.
Measured in joules, a cat’s catbouncemax can most easily be approximated by observing and estimating the maximum height of the cat under these conditions. For ethical and safety reasons, of course, one must simply be observant, and wait for this to happen. Deliberately introducing cats and copperheads (or other dangerous animals) to each other is specifically NOT recommended. Staying away from copperheads, on the other hand, IS recommended. Good science requires patience!
After the waiting is over (be prepared to wait for years), and the cat’s maximum height h, in meters, has been estimated, the cat’s catbouncemax can then be determined by energy conservation, since its takeoff kinetic energy (formerly stored as feline potential energy, until the moment the cat spots the copperhead) is equal to the gravitational potential energy (PE = mgh) of the cat at the top of the parabolic arc. In the catbounce I witnessed, the cat who encountered a copperhead (while walking through tall grass, which is why the cat didn’t see the snake coming) was a big cat, at an estimated mass of 6.0 kg. His catbouncemax was therefore, by energy conservation, equal to mgh = (6.0 kg)(1.4 m)(9.81 m/s²) = ~82 joules, which means this particular cat had 82 J of ophidiofeline potential energy stored, specifically for use in the event of an encounter with a large, adult copperhead, or other animal (there aren’t many) with the ability to scare this cat equally as much as such a copperhead. (I’m using a copperhead in this account for one reason: that’s the type of animal which initiated the highest catbounce I have ever witnessed, and I seriously doubt that this particular cat could jump any higher than 1.4 m, under any circumstances.)
It should be noted that the horizontal distance covered by a catbounce is not needed to calculate a cat’s catbouncemax. However, this horizontal distance will not be zero, as is apparent in the diagram above. Why? Simple: cats don’t jump straight up in reaction to copperheads, for they are smart enough not to want to fall right back down on top of such a snake.
It is more common, of course, for cats to jump away from scary things which are less scary than adult copperheads. For example, there certainly exist centipedes which are large enough to scare a cat, causing it to catbounce, but with that centipede-induced catbounce being less than its catbouncemax. The following fictional dialogue demonstrates how such lesser catbounces can be most easily described. (Side note: this dialogue is set in Arkansas, where we have cats and copperheads, and where I witnessed the copperhead-induced maximum catbounce described above.)
She: Did you see that cat jump?!?
He: Yep! Must be something scary, over there in that there flowerpatch, for Cinnamon to jump that high. At least I know it’s not a copperhead, though.
She: A copperhead? How do you know that?
He: Oh, that was quite a jump, dear, but a real copperhead would give that cat of yours an even higher catbounce than that! The catbounce we just saw was no more than 75% of Cinnamon’s catbouncemax, and that’s being generous.
She: Well, what IS in the flowerpatch? Something sure scared poor Cinnamon! Go check, please, would you?
He: [Walks over from the front porch, where the couple has been standing this whole time, toward the flowerpatch. Once he gets half-way there, he stops abruptly, and shouts.] Holy %$#@! That’s the biggest centipede I’ve ever seen!
She: KILL IT! KILL IT NOW!