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{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 9/22/95 }\\
{\it Closed book \hfill Van Dommelen \hfill 9:00-9:50 EST}
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Show all reasoning and intermediate results leading to your answer.
One book of mathematical tables, such as Schaum's Mathematical Handbook,
may be used.

\begin{enumerate}

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\item
(33 points).
Six points $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, and $P_6$
in a certain structure have to be grounded.  The person in charge
did not do a good job. The person connected point
$P_1$ to $P_2$ with an electrical wire, and the person also connected
point $P_2$ to $P_3$. The person connected $P_4$ to $P_5$,
$P_5$ to $P_6$, and $P_6$ to $P_4$.  But the person did not connect
any point to the ground!  In view of the points that the person connected,
we can say the following about the electric potentials
$V_1$, $V_2$, $V_3$, $V_4$, $V_5$, and $V_6$:
$$V_1 - V_2 = 0 \qquad \hbox{wire 1}$$
$$V_2 - V_3 = 0 \qquad \hbox{wire 2}$$
$$V_4 - V_5 = 0 \qquad \hbox{wire 3}$$
$$V_5 - V_6 = 0 \qquad \hbox{wire 4}$$
$$V_6 - V_4 = 0 \qquad \hbox{wire 5}$$
Show that the rank of this system of 5 equations in 6 unknowns
is one less than what you would expect.  What does this mean in terms of
the wires that the person has used?  Do the potentials have to be zero at all
six points?  Do they all have to be the same?
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  <a href="figures/1.gif">Solution</a>
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\item
(33 points).
You are designing a machine to automatically polish the surface
$$z = \sqrt{x^2 + y}$$
Of course, the plane of the polishing disk needs to be tangential to the
surface at all times.  So what is the equation of the plane of the
polishing disk to polish the point where $x=2$ and $y=5$?
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  <a href="figures/2.gif">Solution</a>
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\item
(34 points).
The kinetic energy of a rotating plate is given by
$$T= 3 \omega_x^2 + 4 \omega_x \omega_y + 6 \omega_y^2$$
where $\omega_x$ and $\omega_y$ are the components of the angular velocity
vector.  Suppose that it is known that $T=1$.  What is the maximum
possible magnitude of the angular velocity vector $|\vec\omega|$ in
that case? Hint: simplify the quadratic form.
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  <a href="figures/3.gif">Solution</a>
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