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{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 10}\\%
\vspace {3mm} (March 2, 2009) \\
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\begin{description}

\item{(33)} One of the Apollo missions left a mirror on the moon. The 
McDonald Observatory in Texas flashes a laser wave at this mirror, which 
is received back after an elapsed time $\triangle t$. At what distance 
was the mirror, when it reflected the signal? Due March~5 in class 
(1~point).

\item{(34)} Let (in arbitrary units)
\begin{eqnarray} \nonumber
  &(a)&  ct=5,\ x^1=1,\ x^2=2,\ x^3=3\,, \\ \nonumber
  &(b)& ct=5,\ x_1=1,\ x^2=2,\ x^3=3\,, \\ \nonumber
  &(c)& ct=5,\ x^1=1,\ x^2=2,\ x^3=-3\,, \\ \nonumber
  &(d)& ct=5,\ x^1=0,\ x^2=3,\ x^3=4\,, \\ \nonumber
  &(e)& ct=5,\ x_1=0,\ x_2=3,\ x_3=4\,, \\ \nonumber
  &(f)& ct=5,\ x^1=2,\ x^2=3,\ x^3=4\,, \\ \nonumber
  &(g)& ct=5,\ x^1=0,\ x^2=3,\ x^3=-4\,.  
\end{eqnarray}
Write down the row and column vector for each case and calculate
$x^{\alpha}\, x_{\alpha}$. Due March~5 in class (7~points).

\item{(35)} A particle of mass $m$ and electric charge $q$ moves under 
the influence of a constant magnetic field of the form
${\bf B}({\bf r})\!=\!B_0\hat{\bf z}$. Obtain the most general 
solution for the velocity ${\bf v}(t)$ using Newton's second
law of motion in combination with the Lorentz force
$$ {\bf F}=m{\dot{\bf v}}=\frac{q}{c}{\bf v}\times{\bf B}\;.  $$
Homework, due March~15 before class (10 points). \bigskip
% \begin{enumerate} \item \end{enumerate}

\item{(36)} Consider the 2D rotation
$$ \pmatrix{x'^{\,1}\cr x'^{\,4}\cr} = \pmatrix{\cos(\phi)& \sin(\phi)
   \cr - \sin(\phi)&\cos(\phi)\cr}\, \pmatrix{x^1\cr x^4\cr} $$
and substitute $\phi = i\,\zeta$, $x^4=i\,x^0$, $x'^{\,4}=i\,x'^{\,0}$. 
Write out the equations for $x'^{\,1}$ and~$x'^{\,0}$. Due March~24
in class (8~points).

\end{description}

Read the {\bf  Notes on Relativity} up to section 1.1.3 (due
March~15).

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