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So, in economics, we read this thing about someone who took all the mints from a restaurant cashier. He was subtle at first, but eventually he just shoved them all in his pocket and left. So that was pretty funny, I'd like to dare one of my friends to try it some time. So I just finished that, and then I remembered I had to do a blog post (whoa, bye fourth wall), and it got me thinking about something I learned not to long ago. It's about napkin rings - more technically, spherical rings. I thought about them because, well, mints are toruses, as are napkin rings. That's about it. A napkin ring is an object that's the result of taking a solid sphere, and cutting out a cylinder from the center of it, all the way through the sphere. They look like, well, napkin rings. Now, there's a pretty interesting property of napkin rings, that is kinda physics-y, but it's more just mathematical. Although I'm sure there's some interesting physics going along with these, maybe some cool rotational inertia properties. Anyway, the property I'm talking about has to do with the volume of the ring. You see, if you have two napkin rings that are the same height - that being measure one the same axis along which the cylindrical hole was cut - they will always have the exact same volume. Isn't that kinda cool? You could take an orange (well, a spherically perfect orange, in the shape of a perfect sphere), and the Earth (again, a spherically perfect Earth - ours is actually fairly eccentric) and you cut them into napkin rings of the same exact height, they will have the same exact volume. Here's a video Vsauce made on the topic (I'll admit, it's not a very exciting video, it's just him going through some basic algebra, and proving this equal-volume property): So yeah, there. Something kinda (probably not really for most people, but whatever, I think it's cool) cool about a physical object. See what I did there? It's totally physics related. Hey! The first legitimate post, on what's sure to become a pretty cringey blog. See you next week!
I'm a big fan of sound. Music for me is a nice blend of science and art, and I take strides to better my understanding of it occasionally. And occasionally, I enjoy listening to chiptune songs - 8-bit music, as you may call it. A typical sound wave is sinusoidal, meaning it looks like a sine/cosine curve. This is the natural state of a pressure fluctuation that is sound. However, sound waves are (obviously) not all sine waves. Because of the constructive/destructive interference of waves, waves with a new shape - or timbre - like with a square or saw wave, essentially keeping the dominant frequency (pitch) of the note while still changing how it sounds. What is really happening when this is going on is that, in some ways, the frequency is changing, but just not the dominant frequency. In music, an octave occurs when one note has double the frequency of another, and by changing the amount of sound energy contained in a certain frequency that is an integer multiple of the base frequency (be it an octave or a different multiple), you can change the timbre without distorting pitch. This is the fundemental basis lying behind the Fourier transform, a method for breaking down a period function into an (often infinite) sum of sine waves with different frequencies. With bar-based music visualizers, the same things is happening, with wave shapes being analyzed for the frequencies they contain. But this phenomenom is what makes music sound the way it does, and it demonstrates that wave interference can have some interesting and melodic effects.
55 downloadsFor each problem, students are asked to begin by highlighting what information they are given, and what they are asked to find. Goal of the exercise is to have students begin to recognize start and end points of a problem, then search for pathways from the start to the end.