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\def\pt#1{\hbox{#1) }}
\input amssym.def
\input amssym.tex
\def\R{{\Bbb R}}
\font\srm=cmr8
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\def\pmb#1{\setbox0=\hbox{#1}%
               \kern-.025em\copy0\kern-\wd0
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               \kern-.025em\raise.0433em\box0 }

\headline={ESP Math 408D - AP\hfil Fall 1996}
\footline={\ifnum\pageno>1 \hfil Worksheet 28\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}


\centerline
{\bf WORKSHEET 28} 
\bigskip

\item{1.}  Consider a path around the unit circle parametrized by
\pmb{$\sigma$} on the interval $[0,1]$ with
$$
\hbox{\pmb{$\sigma$}}(t)=\big(\sin(5\pi t-\pi),\cos(5\pi t-\pi)\big).
$$

\medskip
\itemitem{a)}  Where does this path begin?  Where does it end?  Does
it travel clockwise, counterclockwise, or sometimes one then the other?

\medskip
\itemitem{b)}  At time $t=0$, what is the velocity vector?  What is
the speed?  What is the acceleration vector?

\medskip
\itemitem{c)}  Is an object moving along this path accelerating at
time $t=0$?  Is the speed increasing or decreasing?

\medskip 
\item{2.}  Find the parametric equations
$$
\hbox{\pmb{$\gamma$}}(t)=\big(x(t),y(t)\big)
$$
and a parameter interval for the motion of a particle which travels
around the ellipse
$$
{(x-2)^2\over9}+{(y+1)^2\over16}=1
$$
once clockwise.

\medskip
\item{3.}  For the curves in Problems 1 and 2, find a formula for the
{\bf speed} of an object moving along the path. 

\medskip
\item{4.}  {\bf Curvature.}  The {\it curvature} of a path \pmb{$\sigma$}$(t)$
is defined to be the rate at which the unit tangent vector is changing
with respect to arc length.  In this problem, we develop this idea more
explicitly.

\medskip
\item{}  The unit tangent vector is defined to be
$$
{\bf T}={\hbox{\pmb{$\sigma$}}'(t)\over|\hbox{\pmb{$\sigma$}}'(t)|}
$$
where $|\hbox{\pmb{$\sigma$}}'(t)|$ denotes the length of 
$\hbox{\pmb{$\sigma$}}'(t)$ or the speed at $t$.

\medskip
\itemitem{a)}  Why is ${\bf T}$ called the unit tangent vector?

\medskip
\item{}  Now we define the {\it curvature vector} to be 
$$
\hbox{\pmb{$\kappa$}}={1\over|\hbox{\pmb{$\sigma$}}'(t)|}\dot{\bf T}.
$$
The curvature vector points in the direction in which ${\bf T}$ is
turning, perpendicular to ${\bf T}$.

\medskip
\itemitem{b)}  Does the curvature vector always point in the same
direction as the acceleration vector?

\medskip
\item{}  We define the {\it scalar curvature} $\kappa$ to be the
length of the curvature vector.  That is
$$
\kappa=|\hbox{\pmb{$\kappa$}}|.
$$
For a circe of radius $r$, the curvature vector \pmb{$\kappa$} points
toward the center of the circle and has length $\kappa=1/r$.  For a
general curve, the best approximating (or {\it osculating}) circle has
radius $1/\kappa$.


\medskip
\itemitem{c)}  Without doing any computations, answer the following
questions:
\itemitem{}  i)  What is the scalar curvature $\kappa$ at any point on
the path in problem 1?  
\itemitem{} ii) Is $\kappa$ constant for the curve in problem 2?  
\itemitem{} iii) If not, where is it the greatest?  the least?  

\medskip 
\itemitem{d)} Prove that the curvature vector is always perpendicular
to the tangent vector. 

\medskip
\itemitem{e)}  When will the curvature vector and the acceleration
vector point in the same direction?


\bye