- I-Learn, Class Pictures, Learning Targets, Text Book Practice
- Prep Tasks: Unit 1 - Motion, Unit 2 - Derivatives, Unit 3 - Integration, Unit 4 - Vector Calculus
Day 11 - Prep
Task 11.1
We'll use Mathematica to create a program to compute all the quantities related to the TNB frame and curvature.
- Download the Mathematica file TNB-intro.nb.
- Read through the introductory examples, evaluating each block of code.
- Adapt the code to compute the TNB frame and curvature for a given curve.
Task 11.2
Recall that the curvature vector is $\kappa = \dfrac{d\vec T}{ds}$, with curvature being a the length of this vector. This vector tells us how much the direction of motion ($\vec T$) changes, as we increase the distance moved along the curve. As such, a tight corner will result in a large change of direction and hence a large curvature. Large corners will result in a small curvature.
The radius of curvature at a point, namely $1/\kappa$, provides the radius of a circle that approximates the shape of curve at that point. Large turns results in a large radius of curvature, while tight turns results in a small radius of curvature. This circle lies in the plane formed by $\vec T$ and $\vec N$ (so a normal vector to this plane is $\vec B$). We call this plane the osculating plane. The center of the circle can be found by following $\vec N$ from the point on the curve.
- For the curve $\vec r(t) = (3\cos t, 3\sin t, 4t)$, we have already computed $\vec T$, $\vec N$, $\vec B$, and $\kappa$. At $t=\pi/2$, evaluate these quantities.
- Give an equation of the the osculating plane at $t=\pi/2$. You'll need to identify a normal vector to the plane, and a point on the plane.
- Explain why the center of curvature is given by $\vec r + \frac{1}{\kappa}\vec N$.
- Give the location of the center of curvature for $\vec r(t) = (3\cos t, 3\sin t, 4t)$ at $t=\pi/2$.
Task 11.3
There are many ways to compute the TNB frame and curvature. In this problem, we'll develop a few others.
- Explain why $ \vec N = \dfrac{\vec r^{\prime\prime}_{\perp \vec r^{\prime}}}{ |\vec r^{\prime\prime}_{\perp \vec r^{\prime} }| } $.
- Explain why $ \vec B = \dfrac{\vec r^{\prime}\times \vec r^{\prime\prime}}{|\vec r^{\prime}\times \vec r^{\prime\prime}|}$.
- For a function of the form $\vec r(x) = (x, f(x))$, show that $\kappa = \dfrac{|f''(x)|}{(1+(f'(x))^2)^{3/2}}$.
Task 11.4
The last problem for prep each day will point to relevant problems from OpenStax. Spend 30 minutes working on problems from the sections below.
- section 3.3 exercises 113-151
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