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{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 9/20/02}\\
{\it Closed book \hfill Van Dommelen \hfill 11:50-12:40 pm}
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Show all reasoning and intermediate results leading to your answer, or
credit will be lost. One book of mathematical tables, such as Schaum's
Mathematical Handbook, may be used, as well as a calculator.

\begin{enumerate}

\item
The scaled cross-sectional area $A$ of a rocket nozzle depends on the
scaled Mach number as:
\begin{displaymath}
    A = \frac{\left(1+M^2\right)^3}{M}
\end{displaymath}
Neatly draw and discuss the graph of $A$ versus $M$, including
horizontal, vertical, and oblique asymptotes, $M$- and $A$-extents,
critical points, behavior of $A$ at large $|M|$, $M$- and
$A$-intercepts, cusps, and corner points.  If some of these features
are not present, explain why not, otherwise, give their relevant
numerical values.

\begin{rawhtml}
<A HREF="figures/1.jpg">Solution.</A>
\end{rawhtml}

\item
The inverse elliptic integral $\phi(t)$ of the first kind relates
deflection angle of a pendulum versus time.  While this function is
not found on your calculator, its derivatives at the origin
are relatively simple to find:
\begin{displaymath}
    \phi(0) = 0 \quad \phi'(0) = 1 \quad \phi''(0) = 0 \quad \phi'''(0) = -k^2
\end{displaymath}
where $k$ is a constant that you may take to be 2.
Approximate this function for small values of $t$ and use this
approximation to estimate at which value(s) of $t$ the derivative
$\phi'$ becomes equal to 0.99

\begin{rawhtml}
<A HREF="figures/2.jpg">Solution.</A>
\end{rawhtml}

\item
The edge of a tapered plate is described by:
\begin{displaymath}
    x^2 + y^2 = 6 x
\end{displaymath}
and the thickness is $t=0.01 x$.
Find the moment of inertia $I_0 = 0.01 \int x \left(x^2+y^2\right){\rm d}A$
for rotation around the origin for this plate.

\begin{rawhtml}
<A HREF="figures/3.jpg">Solution.</A>
\end{rawhtml}

\end{enumerate}

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