Yes, but if you have perimeter = (2+π/2)r you can divide both sides by (2+π/2) to get r = perimeter/(2+π/2) (2+π/2) is about 3.57, so you can check your answer a third way if you want. :)
Guys. The student is not algebraic. Throttle way back. @Imoyram, you're not going to do this often enough to need to remember the formula for it. Guess-and-check is a reasonable strategy assuming you were provided with nice numbers. But, to feed your number sense and make future guessing easier: The circumference of a circle is 2 times pi times the radius. Pi is about 3, for mental math purposes, which means the whole circumference is about six times the radius. Half a circumference would be three times the radius. A quarter of the circumference would be half of that, 1 1/2 times the radius. So if you guess the right radius, you should end up with a curved-side that's one and a half times bigger. So a radius of 3 and a curved side of 4.7 means you had to be close - and indeed that turned out to be right. If you'd picked 2 for the radius, you'd have gotten a curved side of 6.7, when you expected it to be about 3 (one-and-a-half groups of two). So that would mean the radius was too short. If you'd picked 4 for the radius, you'd have gotten a curved side of 2.7, which is shorter than the radius. So a radius of 4 would be too long. Etc. etc. The key to good guess-and-check is being able to narrow down toward the right answer, instead of having to try every number in the universe. Even if the answer hadn't been exactly 3, if you'd known that 3 was, say, a little too small you could have tried 3.1, and so on until you had as many decimal places as you needed.
Man I am not a fan of this ixl thing. I can see how it'd be attractive to a teacher, but it doesn't leave the student much room to experiment/work things through/practise writing maths.
@TheSeer thanks. That does make sense, and was kinda like what my bro said, he said that the radius alone should be smaller than the curve, but two radius' together (the straight sides) should be bigger than the curve. Also, am I the only one who has learned to just do diameter x Pi for the circumference? I mean, I know that that is the same thing as radius x 2 x Pi, but we never learned it that way, we were told to use the diameter, just to simplify it I guess. @EulersBidentity yeaaaaahhh. I prefer paper math, but we are doing that too, so yknow. :/
It's all right @WithAnH . :) I'm just in grade 8, and while I could mostly understand what was happening theoretically, it just kinda hurt my brain trying to use it. But they did look super cool! :D (when I figure out how to use them I'm sure it'll be cool af)
Also it stops snarky kids from asking why they didn't just make pi twice as big. It does work either way, and I agree that pi x diameter is the version you should learn first. But since this was a problem where you could see the radius but couldn't see the diameter, it made more sense to use the radius version here.
Generalised coordinates. Erm. This concept is intangible to me. The textbook I'm reading says (missing a little context, but not tons): So far we have been thinking implicitly in terms of Cartesian coordinates. A system of N particles, free from constraints, has 3N independent coordinates or degrees of freedom. Why 3N? To account for 3 dimensional motion? What happens if you're modelling motion in more or fewer than 3 dimensions, or is the assumption that you won't be doing that? Edit: oh wait, if you're modelling in fewer than 3 dim that would count as a constraint, which is the whole point of degrees of freedom. So that answers that question. And I guess you don't go higher than 3 dim because this is classical mechanics.
Yup. In 3D space, you need three co-ordinates, whatever the hell those end up being. And each particle can move along these three co-ordinates, so there's 3N ways that can happen, 3N independently varying parameters.
Rad. Man, this textbook is way clearer than my printed course notes. I'm actually kind of enjoying myself.
Yeah, textbooks tend to be much better at this kind of thing, because they're not meant to go along with an expert with office hours (your lecturer) and worked examples in lectures.
I don't understand notation :| Defn of Work given as W12 = ∫21 F . ds What's the integral wrt? Time? Is the infinitesimal displacement a fn of t? Fake edit: hmmmmmmm. I do remember defining W as = Fs back at A-level when we only worked with constant F. It would make sense for F.ds to be the more general form of that defn. Gradually coming to terms with the concepts ._____. Real edit: a lot of this stuff would make more sense if I hadn't apparently FORGOTTEN ALL THE MECHANICS I EVER KNEW. velocity = displacement/time. So dr/dt = v = ds/dt => ds = vdt by chain rule => W12 = ∫21 F . ds = ∫21 F . v dt Right?
Position infinitesimals, like you said. I took a second and a glance at the Wikipedia page to refresh myself, and this is what I came up with.
In other news, writing that on a drawing program reminded me that Microsoft Journal exists, and I am now gonna use that if I have to write anything for this thread because WOW VERY NEAT WRITING.
Because mathematicians are mad that physics is applied mathematics and mathematics is supposed to be "pure" and not "tainted" by mundane things. :D
Convention is absolutely horrible, sometimes. There's an even worse gap between physics and maths conventions over at complex inner products.
Oh, and by grad school my physics teachers were using "log x" to mean the logarithm base e, not base 10.
Ah, I'm used to that. That's a thing in my maths course, too. It's the thing where the maths & physics defns are the converse of each other that gets me. Especially since some of my lecturers are mathmos and some are physicists. Edit: I don't think I've ever used a log base (not e)? At least not since learning what logarithms were & having to practise them to prove I understood.
