% BB 09 04 25
\documentclass[12pt]{article}
\usepackage{epsfig}
\textheight 9.5in \textwidth 6.0in
\topmargin -0.7in \headsep   0.4in  
\oddsidemargin 0.25in \evensidemargin 0.25in  

\def\pmbf#1{{\mbox{\boldmath${#1}$}}}
\def\undertext#1{$\underline{\smash{\hbox{#1}}}$}
\pagestyle{empty}

\begin{document}

\begin{center}
{\bf FINAL -- PHY 4936\ (December 15, 2011)}\\
\end{center}
\setcounter{equation}{0}
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%
\centerline{\bf PROBLEM 1 (25 points)}
\medskip \noindent
The Lagrangian of the 1D harmonic oscillator is
$$ L={1\over 2}\,m\,\dot{x}^2-{1\over 2}\,k\,x^2\ .  $$
\begin{enumerate}
\item Use the definition of the generalized momentum to find the momentum $p$.
\item Write down the Hamiltonian $H(p,x)$.
\item Write down Hamilton's equations.
\item Show that Hamilton's equations give Newton's force law.
\end{enumerate}
\bigskip
\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%
\centerline{\bf PROBLEM 2 (25 points)}
\medskip \noindent
Use the Lagrangian formalism to derive angular momentum conservation 
from the isotropy of space.
\bigskip
\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%
\centerline{\bf PROBLEM 3 (25 points)}
\medskip \noindent

Consider the effective potential of the Kepler problem
$$ U_{\rm eff}(r) = - \frac{\alpha}{r} + \frac{L^2}{2\,m\,r^2}\ . $$

\noindent (A) Calculate the value $r_0$ where $U_{\rm eff}(r)$ 
          has its minimum.

\noindent (B) Calculate $U_{\rm eff}^{\min}=U_{\rm eff}(r_0)$.

\noindent (C) Assume an energy $E<0$ and calculate the turning 
              points $r_{\min}$ and $r_{\max}$.

\noindent (D) For which energy range do we have two real solutions
              for $r_{\min}$ and $r_{\max}$?

\bigskip
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%
\centerline{\bf PROBLEM 4 (25 points)}
\medskip \noindent

Consider a thin disk composed of two homogeneous halves connected along 
a diameter of the disk. As shown in the figure (see backside), one half 
has density $\rho$ and the other has density $2\rho$.
\begin{enumerate}
\item Find the center of mass (CM) in the body frame. Do these values
      change when the disk rolls?
\item Let $R$ be the radius of the disk and $\theta$ be the angle 
      defined in the figure. Find the CM position in the lab frame, 
      $x_{CM},\,y_{CM}$, as function of $\theta$.
\item Use the parallel axis theorem to calculate the moment of inertia
      $I_3^{CM}$ for the axis perpendicular through the CM of the disk.
\item Write down the Lagrangian as function of the so far calculated
      quantities (assume gravity in the $y$ direction of the lab frame).
      It is {\bf not} not asked to evaluate the Lagrangian further (as
      it is done in a homework solution).
\end{enumerate}

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.5]{HW8-2b}
% \caption{} \label{fig:2b}
\end{center}
\end{figure}
      
\vfill
\end{document}



