Problem Set 6 — Arc Length and Curvature

Purpose

This exercise reinforces your understanding of arc length and curvature. It therefore contributes to the following learning outcomes for this course:

• Outcome 3.1. Analyze vector functions to find their derivatives and tangent lines
• Outcome 3.2. Analyze vector functions to find their arc length and curvature.
• Outcome 12. Use technological tools such as computer algebra systems or graphing calculators for visualization and calculation of multivariable calculus concepts.

Background

This exercise is mainly based on material in section 2.3 of our textbook. We covered that material in classes between March 1 and 3. The exercise also asks you to do calculations related to arc length and curvature in Mathematica. We talked about general Mathematica functions for vector calculations and derivatives and integrals in class on February 27th, and about calculating curvature on March 3.

Activity

Solve each of the following problems.

Problem 1

In problem set 5, you met an ant who was crawling along a coil of wire in such a manner that \(t\) seconds after the ant started crawling it was at position \(\langle 2t, \sin (\pi t), \cos (\pi t) \rangle\), in some coordinate system in which distance is measured in inches.

Part A

How far has the ant walked 5 seconds after it starts? Solve this problem by hand up to the point where you have a symbolic numeric answer (in other words, an answer that is technically a single number, but that might most naturally be given as an expression involving various irrational square roots, constants such as \(\pi\) or \(e\), etc.) You may use Mathematica to evaluate that symbolic answer to a decimal number if you wish.

Part B

What are the ant’s coordinates after it has walked 5 inches? Solve this problem by hand up to the point where you have a symbolic numeric answer, but you may then use Mathematica to evaluate that symbolic answer to a decimal number if you wish.

Problem 2

Find the length of one turn of the 4-dimensional helix \(\vec{r}(t) = \langle 2\sin t, \sqrt{5}\,t, 2\cos t, 4t \rangle\). Do not use Mathematica on this problem.

Problem 3

Find a unit vector that points in the direction the curve \(\vec{r}(t) = \langle \cos (e^t), \sin (e^t), 0 \rangle\) is turning when \(t = \ln \pi\). Also find the curvature of \(\vec{r}(t)\). Use Mathematica to carry out and organize the calculations.

Follow-Up

I will grade this exercise during an individual meeting with you. That meeting should happen on or before the “Grade By” date above, and should ordinarily last half an hour. During the meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Sign up for the meeting via Google calendar. Please have a written solution to the exercise ready to share with me during your meeting, as that will speed the process along.