Math 212
Name: __________________________________
Homework 8
1. The figure below shows a unit circle rolling along the B-axis. While rolling, the circle moves to the
right with a speed of " unitÎsec, and simultaneously rotates clockwise at a rate of " radianÎsec.
y
t = А2
t=Π
t = 3А2
t = 2Π
P
x
A point T is marked on the boundary of the circle. At time > œ !, the point T has coordinates a!ß !b.
As the circle rolls, the point T traces out the shown curve.
(a) Find a formula for the coordinates of the center of the circle at time >. (The center starts on the
C-axis at > œ !.)
(b) Find the coordinates of the point T at times > œ 1Î#, > œ 1, > œ $1Î#, and > œ #1.
(c) Let va>b be the vector from the center of the circle to the point T . Find a formula for va>b in
terms of >. Make sure to simplify your answer.
(d) Find parametric equations for the position of the point T at time >.
(e) Compute the velocity vector for the point T at times > œ 1Î#, > œ 1, > œ $1Î#, and > œ #1.
(f) Find the total distance traveled by the point T from > œ ! to > œ #1. You will need to use the
formula "  cos ) œ # sin# a)Î#b to evaluate the integral.
2. The file Homework8Data.xls (or Homework8Data.txt)
contains 101 data points for a curve G in the BC-plane. Use a
spreadsheet such as Microsoft Excel to answer the following
questions. After you are done, please staple a printout of
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C
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your spreadsheet to this homework assignment.
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(a) Compute the distance .= œ È.B#  .C# between each
consecutive pair of points. Add these together to obtain an
estimate for the total length of the curve G .
estimate for the total length here:
Write your
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-1
-2
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(b) Compute B# .= for each portion of the curve, using the B-value at the midpoint of each line
segment. Add these together to obtain an estimate for the value of ( B# .=. Write your
G
estimate for this integral here:
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