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{\bf PHY 4938 HOME AND CLASS WORK -- SET 8}\\%
\vspace {3mm} (November 21, 2011) \\
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\begin{description}

\item{(32)} Assume $R = 1~{\rm meter}$ in the previous problem
and plot the velocity versus the angle $\theta$ over two periods
(for both cases A.\ and B.). Due November 28 before class (4 points).

\item{(33)} The angular velocity of a symmetric top can be written as
$$ \vec{\Omega}\ =\ \vec{\Omega}_{\rm pr} + \vec{\Omega}^*_3\
   =\ \Omega_{\rm pr}\,\hat{M} + \Omega^*_3\,\hat{x}_3 $$
where $\hat{M}$ is the unit vector in direction of the angular momentum 
$\vec{M}$ and $\hat{x}_3$ the unit vector of the $x_3$ axis, $\Omega_{
\rm pr}=M/I_1$. Compare figure~46, p.107 of Landau-Lifshitz. Express
$\Omega^*_3$ in terms of $M,\,I_1,\,I_3$ and $\theta$, all defined
in the book. Due in class (4 points).
% Add: The principal axes are (permanently) defined with respect
% to: (A)~The fixed frame. (B)~The moving frame. (C)~The body
% frame. (C)~The Lab frame.

\item{(34)} Inertia tensor of a homogeneous cube of density $\rho$,
mass $M$, and side length $b$.
\begin{enumerate}
\item Let one corner be at the origin and let the three adjacent
edges lie along the coordinate axes. Calculate the inertia tensor
with respect to these axes. Due November 30 before class (4 points).
\item Use the parallel axis theorem to find the inertia tensor
for axes through the center of mass, which are parallel to
the axes used in part~1. Due November 30 before class (4 points).
\item Is the cube a spherical top? Due November 30 before class 
(2 points).

\end{enumerate}

\item{(35)} Express the angular velocity components in Euler angles
with respect to the axis of the fixed (lab) frame, i.e., $\Omega_X,\
\Omega_Y,\ \Omega_Z$. Due December~2 before class (6 points).

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