Okay guys, this is really important. I need you to explain to me what a phase portrait is. More to the point: what do these red lines mean?! from http://scienceworld.wolfram.com/physics/Pendulum.html
x is what? y is what? One of them is time, right? And what the fuck does that pair of equations mean with those dots on top of the x and stuff and how does it make that portrait? Also what does everything else on that entire page mean? Like OK something about the “moment of inertia” is just d squared theta over d t squared but… why squared? Cuz it has to do with acceleration? Oh that’s just what acceleration is, isn’t it? 2nd derivative? Ok……..
The last time I took physics was 10th grade, ok? =( I don’t know any of this.
I went to the Exploratorium a couple weekends ago with Fritz and his lab partner for their physics class, and I was completely enthralled with 10 bobs set up such that the biggest one swung 15 times in 30 seconds, the second one swung 16 times in 30 seconds, and so on; the littlest one swung exactly 24 times every 30 seconds. I stood there setting it off and watching it for maybe like 10 minutes straight. This kid ran by and screamed “IT’S DNA!!!” and ran off again, which is the problem with science for kids, I guess. It all starts running together after a while.
It’s so beautiful because when you set it off, letting all the bobs go at the same time, you get this period of patterns, right? At the 15-second mark, they HIT like BANG JAZZ HANDS and are EXACTLY alternating-opposite–the biggest one has swung 7.5 times so it’s at its highest point on the other side, while the next one is fully through with swinging 8 times so it’s at its highest point where it began, and so on, and it is glorious. And then they phase back and at 30 seconds they line up again because, well, 30 is the, um, first common multiple of 15/30, 16/30, 17/30, …. and 24/30 ? I mean, right?
Here is a picture of it: http://www.flickr.com/photos/exploratorium_exs/2361101581/in/photostream/
Look at it! Is it not the prettiest thing ever?!?!?!?!?!?!?!? Oh god I love it so much. All I want to know is like, what is the relationship between the string length and the period of the swing? Which one is the right formula for that? And what kind of curve is it or whatever, if it has a classification? Even if I don’t know what the classifications are? I just want it to have a name in my head and I will be happy. I was really struck by the curve of the wood from which all the bobs are hanging, making the bobs level with each other for the greatest effect. If you make one at home, it’s probably easier to hang them from a straight thing, so the “curve” appears along the bobs, which probably still looks pretty cool in action, but you can definitely see why they lined them up in the exhibit.
Edited to add: holy shit ok here it is, from http://www.exo.net/~pauld/activities/sound/pendulumpipes.htm
The frequency of a pendulum, f, depends on its length, L, and on the acceleration of gravity,g.
f = (1/2p) (g/L)0.5
If the length of the pendulum is cut to 1/4 of its original length the frequency of its swing will double.
This website suggests that you get 10 people and assign them to a pendulum and give them all a musical instrument, and get them to play it when their pendulum swings to them. I am very visual, so the .wavs don’t make as BIG of an impact on me, but still THEY MAKE ME OVERCOME WITH JOY. I DON’T THINK YOU UNDERSTAND WHAT KIND OF JOY THIS BRINGS OK. Kind of reminds me of Pipe Dream which is another thing that makes me OVERCOME WITH JOY. Man, I remember when Animusic first came out, my brother was so into it that he did the thing where you can download whatever you call the source of an animation and render it yourself on your own video card instead of just playing back a video. I love how each new instrument comes out and is ridiculous and maybe inefficient but hilarious…. ahhh. Really you guys it is so easy to make Judy happy.
So, acceleration of gravity. So if I took the pendulum snake, or like a really long pendulum snake, and took it to Mars………………………….. it would still make a snake……. but …. they’ll “line up” after a different amount of time… right? How different is gravity on Mars anyway? Fuck I don’t know ANYTHING. Thank Zeus for the internets. How would I find out how long it will take for the Earth Pendulum Snake to make a full rep of its pattern if I took it to Mars? Or how long to cut my strings so that they’d line back up at 30 seconds on Mars? I guess you just chug the numbers! Duh!
Edited again to add this picture I took of Pendulum with my camera phone when we saw them a few months ago in SF:
So the x and y are just position vectors: how far the pendulum is in each direction. It looks like the phase portrait has a few flaws, though, if only for inappropriate labelling.
The dots above the x and y are shorthand for the time-derivative of the vector. Each x-dot could also be written as dx/dt. X double dot would be acceleration.
More, perhaps, later.
How far the pendulum is in each direction, x being distance to center (what do you call place where a pendulum rests?), and y being height or something? That doesn’t make sense, cuz then y = 0 (I’m assuming if the height = 0, then the pendulum is at its “lowest point” in the center, ie aforementioned at rest place) at a bunch of x distances?
Unless they’re vectors?
Hmmm… if x is like, velocity going in the horizontal direction, and y is velocity going in the up and down direction… if x is 0 that means it has reached its highest “swing” point on either side and at that point the y will be the biggest it’ll get… and if y is 0 that means at that point it’s probably at the lowest and only going horizontally…
Time-derivative is velocity, right? So x dot is a shorthand for dx/dt, which is just velocity, and x double dot is d2x/dt2, which is just acceleration, then… never seen the dots before, so thank you =)
Also unfamiliar with calling velocity “time-derivative” but you call it that just because… that’s just what it is?
In physics is it useful to derive it by other things or something to find other stuff… I dunno… maybe some liquid is pouring into a container and it’s gaining mass over time or something, … I guess that would be a time-derivative too, though.
Yes, velocity is the time-derivative of position, and acceleration is the time-derivative of velocity. X is definitely horizontal position and Y could maybe be vertical position, but that’s often ignored in favor of just using theta to describe the angle. Neither one is a velocity without a dot.
Now that I’m thinking of it, the phase portrait is likely to be a visual relation between the varied pendulums, with Y being a function of the length of the pendulums. Think of all those ratios coming together at 30 seconds – that could be when they cross the y-axis! The word phase usually means the difference between two angles.
Sometimes it’s useful to see how something changes not with time but with distance. For instance, dy/dx is usually how height changes with length. In order to find the smallest length of thread needed to move from point A to point B, you derive by distance.
There are other uses, too. Usually a derivative is used to find the minimum or the maximum one can use of something.
Hollin posted https://www.facebook.com/video.php?v=691439280939587&pnref=story so I dug up this post. UNADULTERATED JOY!!!!!