I made it in GIMP. A cardioid is the envelope formed by a set of circles whose centers lie on a circle and which pass through one common point in space. This image shows the circle on which the centers of the circles in the above image lie. A cardioid is also the path traced by a point on a circle which is rolling along the surface of another circle when both circles have the same radius (here is a cool animation of that).
What is the cardioid's significance? Well, it looks like a heart, which is kind of cool. It's also the (2D) pickup pattern of certain microphones (I have a cardioid microphone). If a sound is produced at a given point in space, the pickup pattern shows an equal intensity curve. So, if I place a microphone at the intersection point of all those circles, the outside boundary is where a speaker producing, say, a 440 Hz tone would have to be to be heard at a given intensity. So, the best place to put it would be on the side where the curve is most round (the bottom in this picture) without being too far away from the microphone.
Another interesting fact about the cardioid is that it is the reflection of a parabola through the unit circle (r = 1. Here's what I mean). In polar coordinates, the equation of the above cardioid is given by
where a is a scaling factor, and theta is the angle relative to the positive x-axis. The origin is at the intersection of the circles. The equation of a parabola opening upwards and whose focus is at the origin in polar coordinates is just
which is an inversion of the cardioid equation through r = 1, or the unit circle.
The word meter (or metre for those who live across the pond/in Canada) comes from the word for "measure" in Greek/Latin (e.g. speedometers measure speed, pedometers measure steps, &c.), but the meter I'm talking about is the International System (SI) unit of distance. The original definition of the meter was one ten-millionth of the distance from the Earth's equator to the North Pole at sea level (not through the Earth). The first person to measure the circumference of the Earth was the Greek mathematician/astronomer/geographer Eratosthenes (and he was accurate to within 2% of today's known value) circa 240 B.C., so this value was readily calculable in 1791 when this standard was accepted.
In 1668, an alternative standard for the meter was suggested. The meter was suggested to be the length a pendulum needed to be to have a half-period of one second; in other words, the time it took for the pendulum to sweep its full arc from one side to the other had to be one second. The full period of a pendulum is
So, when L = 1 m and T = 2 sec, we get what the acceleration due to gravity, g, should be in meters per second per second (according to this standard of the meter). It turns out that g = pi2 meters per second per second, which is about 9.8696 m/s2. This is very close to the current value, g = 9.80665 m/s2 which are both fairly close to 10. In fact, for quick approximations, physicists will use a g value of ten to get a close guess as to the order of magnitude of some situation.
So, you may be wondering, why is it different nowadays? Well, among a few other changes in the standard meter including using a platinum-iridium alloy bar, we have a new definition of the meter: the speed of light. Since the speed of light in a vacuum is a universal constant (meaning it is the same no matter where you are in the universe, unlike the acceleration due to gravity at a point in space), they decided to make the distance light travels in one second a set number of meters and adjust the meter accordingly. Since the speed of light is 299,792,458 meters per second exactly, this means that we have defined the meter as the distance light travels in 1/299,792,458th of a second.
This is all nice, but it's not a very intuitive number to work with. After all, we humans like multiples of ten (due to having ten fingers and ten toes), so why not make a length measurement of the distance light travels in one billionth (1/1,000,000,000th) of a second (a.k.a. nanosecond)? That seems a bit more intuitive, don't you think? It turns out that a light-nanosecond is about 11.8 inches, or about 1.6% off of the current definition of a foot. In fact, one physicist, David Mermin, suggests redefining the foot to the "phoot," or one light-nanosecond, since it's based off of a universal constant while the current foot is based off the meter by some odd, nonsensical ratio.
This is why graduate-level texts frustrate me. The authors always assume that half the stuff they're discussing in their textbook is obvious to the reader/student who has maybe seen the material once before in an undergraduate course. While some of this material should be expected to be known already, you can't just chuck stuff at your reader and say "it is now obvious that" or "the proof is trivial" when neither of these statements is actually true. If you use either of these statements in your textbook, you're not a good teacher. Period.
The title of this entry comes from the fact that I'm comparing two Electromagnetic Theory textbooks, one by D.J. Griffiths and the other by J.D. Jackson. Griffiths' Introduction to Electrodynamics is a witty, conversational, and informative text that helps undergraduates cope with the fact the E&M is really hard and that most of the concepts are foreign to someone who has only ever dealt with classical mechanics. Jackson's Classical Electrodynamics, on the other hand, is a text where the reader can tell that the author really knows his stuff when it comes to E&M, but has no sense of how to convey that knowledge to someone who is not an advanced student of the subject.
For instance, let's say I were teaching the concept of projectile motion to someone who has never delved into the subject. If I were Griffiths, I would say something like, "All objects in free fall on Earth experience a force due to gravity toward the ground. This force causes all objects to accelerate at the same rate, meaning that the rate at which something speeds up/slows down in Earth's gravity is the same for all objects regardless of how heavy they are. Because this acceleration is constant near the ground, objects tend to follow a parabolic trajectory (if we ignore air resistance). The equations that show this follow from Newton's second law, F = m a. If you don't believe this, let's try it, shall we?"
Now wasn't that nice? This explanation is certainly very clear about what projectile motion is and what causes it. Griffiths enjoys taking concepts that may be hard to comprehend and then following through with some equations/proofs to try and clarify the situation, usually speaking to the reader as though he were sitting down with them helping them through a problem.
What about Jackson? He would probably say something along the lines of, "The reason projectiles follow parabolic paths is simple: if you solve the Hamilton-Jacobi equation in a uniform gravitational field, you will find that the path that minimizes the action is that of a parabola. This can be seen by setting the variation of the Lagrangian equal to zero."
Well that was simple, wasn't it? While technically correct, you probably have no idea what the Hamilton-Jacobi equation or Lagrangian are, nor do you probably know what "action" means in physics. Now you may be thinking, "well, these things are part of undergraduate courses, right?" Well, no, actually. I had no idea what the Hamilton-Jacobi equation was until I took graduate level quantum mechanics, and I was expected to have known that from my graduate classical mechanics course (which I didn't take until my second semester of quantum mechanics). Suffice it to say, there was a lot I had to learn on the fly, but you can probably see what I'm getting at. The assumption that students know everything you expect them to know and have it ready to go the minute you throw that curve ball at them is a terrible way to go about teaching and, in my opinion, does not foster good education.
On an unrelated note, I have a problem set out of Jackson due tomorrow which I haven't finished yet. So, how was your day?
Sombra < Chrysalis < Discord / Nightmare Moon
That said, though, I enjoyed the episode quite a bit. The songs were okay, but I've only listened to them once, really. However, it was nice for Spike to get a role as Twilight's supporting vocals. Cathy Weseluck amazes me.
I did think Rainbow Dash was a bit over the top, but I enjoyed everyone else.
También, sombra = shadow en español, por tu información.
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