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{\bf PHY 4936 HOME AND CLASS WORK -- SET 7}\\%
\vspace {3mm} (November 11, 2011) \\
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Read Landau-Lifshitz p.96 up to p.101 (\textsection 31 and
\textsection 32).

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\item{(29)} Consider a thin disk composed of two homogeneous halves 
connected along a diameter of the disk. If one half has density 
$\rho$ and the other has density $2\rho$, find the expression for 
the Lagrangian when the disk rolls without slipping along a horizontal 
surface as shown in the figure (the rotation takes place in the plane 
of the disk).  Due November 18 before class (10 points).
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\item{(30)} A. Calculate $\vec{a}\times(\vec{b}\times\vec{c})$ as
  superposition of two of its vectors using
  $$[\vec{a}\times(\vec{b}\times\vec{c})]_i =
    \epsilon_{ijk}\epsilon_{klm}a_jb_lc_m\ . $$
  Due in class (3 points). B. Does the expression $\vec{a}\times\vec{b}
  \times\vec{c}$ make sense? Yes or No with reason. Due in class (1 
  point).
\item{(31)} The total, conserved energy of a system is given by
  $$ E\ =\ \frac{1}{2}MR^2\dot\theta^2\left[\frac{3}{2} 
      - \frac{8}{9\pi}\cos\theta\right] 
      + MgR\left[1-\frac{4}{9\pi}\cos\theta\right]. $$
Consider the following energies:
$$ {\rm (A.)}\ E\ =\ MgR\,,\quad {\rm (B.)}\ E\ =\ MgR\,
   \left(1-\frac{4}{9\pi\sqrt{2}}\right)\ . $$
Are there turning points? If yes, give their $\theta$ values.
Due November 21 before class (4 points).
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Read Landau-Lifshitz p.105 up to p.111 (\textsection 33 and
\textsection 35).

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