This lab is about the vector valued function r(t) = (a...

This lab is about the vector valued function r(t) = (a cos(t), 2 sin(2t), 4bt), where a, b ? 0. Calculate the torsion at t = ?, as a function of a and b. Suggestion: First calculate v, a, da/dt at t = ?, and then use those to find ?. Q2 (99 points) Refer to Question 1, and assume a = 4 and b = 0, with result that the z-component of r(t) is always zero ie r(t) is a curve in the x, y-plane. A. I have calculated that a global minimum of the radius of curvature occurs at t = 0.6708. Calculate the radius of curvature and the center of the best fitting circle to the curve, at that value of t. B. Using a computer, plot the curve in the x, y-plane, from t = 0 to t = 2?, and on the same graph, plot the best fitting circle that you calculated in (A). Suggestion: The radius of curvature is R = \frac{|v|^3}{|v \times a|} and the center is at r + RN. To avoid complicated expressions, first substitute t = 0.6708 into r, v, and a, and then calculate ? and N from those.

Transcript

00:01 We're given this curve here, which i'm not sure what you even say about it.

00:07 It's at t equals zero, and it is, let's see, what it is zero.

00:16 Oh, zero, yes, i was looking at the wrong.

00:20 It goes through here, right? and then when, well, it technically comes down here.

00:31 Mathematic and most computer algebra systems, when it takes a fifth root of something, it will give you a complex number.

00:38 But you can tell it to give you the real.

00:43 Because when you ever take the fifth root or something, there's two roots.

00:46 There's a complex one and there's a real one.

00:48 And you can tell why it gives you, probably computer algebra systems give you the complex one has to do with basically looking at complex values of t.

01:00 But you can tell it to give you the real one, which i just, i guess i didn't do here.

01:03 So this curve should actually, i think, basically do this.

01:08 Anyway, we can, we're not really interested in that because we're at this point here.

01:17 T equals one -half.

01:19 So at t -equals one -half, that point is x equals one -half and y equals one -over -seven -five, or two -to -the -thifths.

01:31 So i don't know what that is about .35 looks like.

01:36 So here's a bunch of functions i define in mathematica to calculate.

01:41 Given a curve parameterized by t, i can just pass it in and get acceleration velocity, tangent, normal, binaural, portion, curvature, normal and tangential acceleration, and arc length factor.

01:56 So what i did is that, you know, i just took this and i said, okay, through this and it gave me a really ugly looking expression for this as a function of time or as a function of p.

02:09 I'm not sure if it's time not.

02:10 But then i plugged in t equals one -half.

02:13 I got, well, not a particular nice -looking thing, but again, i probably should have just read this numerically.

02:21 But this is one over nine plus 25 times two to the four -fifths square root of all that and whatever that is now oh um let's see here i forgot to actually let me let me get the numeric values for you here i forgot to actually write it down because it was so ugly um let's see here it was so ugly that i didn't write the actually even with the symbolic form was so ugly um let's see here here okay yeah um is that the curve that i'm looking for let me find the problem i did i forgot to yeah that's the curve oh okay up here yeah the curvature is just um the radius of curvature involves two to the the tenth root of two and other stuff so let me just get the value for that and i'm let me get them just so we can say what that looks like actually and so oops do that image all right so i move this down here so let's say what is n hat not in america i not do that oh wrong oh okay hold on a second i forgot to plug in t.

04:25 So that should give us a number...

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