\nopagenumbers
\def\pt#1{\hbox{#1) }}
\input amssym.def
\input amssym.tex
\def\R{{\Bbb R}}
\font\srm=cmr8
\magnification=\magstep1

\def\u{{\bf u}}
\def\d{\partial}

\headline={ESP Math 408D - AP\hfil Fall 1996}
\footline={\ifnum\pageno>1 \hfil Worksheet 31\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}

\centerline
{\bf WORKSHEET 31} 
\bigskip

\item{1.}  In class Friday, you were given two unit vectors which are
useful when using polar coordinates.  At a point $(r_0,\theta_0)$, define
$$
\u_r=(\cos\theta_0,\sin\theta_0)\qquad\u_\theta=(-sin\theta_0,\cos\theta_0).
$$

\itemitem{a)}  Pick three points in the plane and draw both of these
vectors.  Why are these vectors named $\u_r$ and $\u_\theta$?

\medskip
\item{} Another way to come up with basis vectors at a point
$(r_0,\theta_0)$ is to define a vector ${\d\over\d r}$ by
parameterizing a path $\gamma_r$ starting at $(r_0,\theta_0)$ at time
$t=0$, which keeps $\theta$ constant and has $r(t)=t$.  Then set
${\d\over\d r}=\gamma_r'(0)$.  Define the vector ${\d\over\d\theta}$
in the corresponding way as $\gamma_\theta'(0)$ for an appropraite
path $\gamma_\theta$.

\medskip
\itemitem{b)}  Write down the paths $\gamma_r$ and $\gamma_\theta$ in 
polar coordinates.  Then convert these paths to Euclidean coordinates.
Compute ${\d\over\d r}$ and ${\d\over\d\theta}$ in Euclidean
coordinates.

\medskip
\itemitem{c)}  Write ${\d\over\d r}$ and ${\d\over\d\theta}$ in terms
of $\u_r$ and $\u_\theta$.

\medskip
\itemitem{d)}  Find the dot products
$$
{\d\over\d r}\cdot{\d\over\d r}\qquad
{\d\over\d r}\cdot{\d\over\d \theta}\qquad
{\d\over\d \theta}\cdot{\d\over\d \theta}.
$$

\medskip 
\item{2.}  Some believe that the positions of the planets at the time
of birth influence the newborn.  Others deride this and say that the
gravitational force exerted on a baby by the obstetrician is greater
than that exerted by the planets.  To check this claim, calculate and
compare the gravitational force exerted on a 6-kg baby
\medskip
\itemitem{a)} by a 70-kg obstetrician who is 1 m away,
\itemitem{b)} by the massive planet Jupiter ($m=2\times10^{27}$ kg) at
its closest approach to earth ($6\times10^{11}$ m), and
\itemitem{c)} by Jupiter at its greatest distance from earth
($9\times10^{11}$ m).
\medskip
\itemitem{d)}  Is the claim correct?

\medskip
\item{3.}  Let $r(t)$ and $\theta(t)$ be polar coordinates for a
particle moving in the plane.  Derive formulas for the velocity and
acceleration of the particle.
\item{}  Hint:  First find ${d\u_r\over dt}$ and ${d\u_\theta\over
dt}$ using the chain rule.  Then use the fact that the position is
given by $r\u_r$, and differentiate.

\medskip
\item{4.}  Below, $r$ and $\theta$ are the polar coordinates of a
particle moving in the plane.  For each motion, express the velocity
and acceleration in terms of $\u_r$ and $\u_\theta$.
$$
{\eqalign{r&=a(1-\cos\theta)\cr {d\theta\over dt}&=3\cr}}\qquad
{\eqalign{r&=a\sin2\theta\cr {d\theta\over dt}&=2t\cr}}\qquad
{\eqalign{r&=a(1+\sin t)\cr \theta&=1-e^{-t}\cr}}
$$

\medskip
\item{5.}  Without introducing coordinates, give a geometric argument
for the validity of the equation
$$
{dA\over dt}={1\over2}\left|{\bf R}\times{d{\bf R}\over
dt}\right|{\atop,}
$$
where {\bf R} is the position vector of a particle moving along a
plane curve and $dA/dt$ is the rate at which that vector sweeps out
area. 

\medskip
\item{6.}  A weather satellite is in geosynchronous orbit, hovering
over Nairobi, which lies very close to the equator.  If its orbit
radius is increased by 1 km, at what rate and in what direction would
its reference spot, which was formerly stationary, move across the
earth's surface?
\medskip
\item{}  Useful data:  Earth's mass = $5.975\times10^{24}$ kg,
Equatorial radius of Earth = 6378.533 km, Gravitational constant $G =
6.6720\times10^{-11}{\rm Nm}^2{\rm kg}^{-2}$ 



\bye