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{\bf EML 5060\hfill Analysis in Mechanical Engineering Exam 1 \hfill 10/1/92}\\
{\it Exam 1 \hfill Van Dommelen \hfill 2:45-4:10pm}
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\begin{enumerate}

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\item
A mass spirals outward, the distance from the center being given by
$\rho = 4 \sqrt{\theta}$.  If the rate of change in angular position
is $\dot \theta = 3$, find the radial and azimuthal components of
velocity $v_\rho$ and $v_\theta$, and those of the acceleration
$a_\rho$ and $a_\theta$.

\item
Find the moment of inertia $I_x = \int y^2 {\rm d} A$ for the inside
of the polar curve
$$\rho = \sin \theta \qquad 0\le\theta\le\pi$$
Note: for even integer $p$
$$\int_0^{\pi/2} \sin^p x {\rm d} x
=\int_0^{\pi/2} \cos^p x {\rm d} x = {1\cdot 3 \cdot 5 \ldots (p-1)
\over 2 \cdot 4 \cdot 6 \ldots p} {\pi\over 2}$$

\item
For the truss sketched below, the tension forces $T_1$, $T_2$, $T_3$,
and $T_4$ in the four bars satisfy the five equations:
$$ T_1 + T_2 = 0 $$
$$ -T_1 + T_2 + 2 T_4 = 200$$
$$ T_2 - T_3 = 0 $$
$$ - T_3 = 2 P $$
$$ T_3 + 2 T_4 = 200 $$
Write the augmented matrix of the system and determine for which value(s)
of $P$ a solution exists, and whether it is unique.

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\item
A coupled set of pendula as shown above has modes of the form
$$\theta_1 = A \cos(\omega t +\phi)\qquad \theta_2 = B\cos(\omega t + \phi)$$
For certain values of the spring constant, masses, and pendulum lengths,
the amplitudes satisfy
$$ - \omega^2 \pmatrix{A\cr B\cr}
= \pmatrix{-3&1\cr 1&-3\cr} \pmatrix{A\cr B\cr}$$
Find the natural frequencies $\omega$.
For each frequency, compare $\theta_1$ and $\theta_2$.

\item
The kinetic energy of a certain thin plate due to rotation around
its center of gravity has the form
$$T = 2 \omega_1^2 - 2 \omega_1 \omega_2 + 2 \omega_2^2
+ 4 \omega_3^2$$
In principal axes, there would be no $\omega_1 \omega_2$ term.
Find the direction of the principal axes of this body by diagonalizing
the quadratic form.

\end{enumerate}

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