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{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 9/19/01}\\
{\it Closed book \hfill Van Dommelen \hfill 10:35-11:25am}
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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.  Follow
the standard procedures; don't mess around at random.

\begin{enumerate}
  
\item A weight $W$ is hung from the ceiling by a net of ropes as
  shown.  The equilibrium equations governing the tension forces in
  the ropes are:
  \begin{displaymath}
    \begin{array}{c}
      T_2 - c T_1 = 0\\
      s T_1 -T_3 = 0 \\
      \alpha T_4 +T_3 - c T_6 = 0\\
      T_5 -\alpha T_4 + s T_6 = 0\\
      c T_6 = W/2
    \end{array}
  \end{displaymath}
  where $c\equiv\cos(30)$, $s\equiv\sin(30)$ and
  $\alpha\equiv\cos(45)$ Using the systematic procedures used in this
  class, solve this system and find the general solution for the
  tension forces in the ropes in as simple and clear a form as
  possible.  Keep the given order of the equations; do not use partial
  pivoting.  Using the result, identify a rope that you could leave
  out and still be able to suspend any weight.  Check that the tension
  forces in all the ropes are nonnegative: ropes cannot support
  compressive forces.

  \begin{displaymath}
    \epsffile{figures/1.ps}
  \end{displaymath}

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

\item Air above the sea is moving with a wind velocity $\vec v_a =
  3\hat i + 3\hat j$.  The sea water is moving with a velocity $\vec
  v_w = \hat i + 2 \hat j$.  To study the motion of small particles
  that fall into the sea, special coordinates can be useful: Instead
  of describing the position of these particles using the conventional
  Cartesian coordinates $x$ and $y$, it is easier to describe their
  position vector using the air and water velocities as basis vectors:
  \begin{displaymath}
    \vec r = x' \vec v_a + y' \vec v_w
  \end{displaymath}
  since changes in $x'$ and $y'$ are simply the amounts of time spend
  in air and water respectively.  Find the formulas that give the
  coordinates $x$ and $y$ if we know $x'$ and $y'$.  Also find the
  formulas that give the coordinates $x'$ and $y'$ if we know $x$ and
  $y$.

  \begin{displaymath}
    \epsffile{figures/2.ps}
  \end{displaymath}

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

\item The two coupled pendulae shown satisfy the following equations
  of motion:
  \begin{displaymath}
    \begin{array}{c}
      \displaystyle -\ddot \theta_1 = \frac{g}{\ell} \theta_1 + \frac{k}{m}
      \left(\theta_1 - \theta_2\right)\\
      \displaystyle -\ddot \theta_2 = \frac{g}{\ell} \theta_2 + \frac{k}{m}
      \left(\theta_2 - \theta_1\right)
    \end{array}
  \end{displaymath}
  Assume $g /\ell = k/m = 2$ Show that solutions of the form:
  \begin{displaymath}
    \theta_1 = A \sin(\omega t + \phi) \qquad 
    \theta_2 = B \sin(\omega t + \phi)
  \end{displaymath}
  lead to an standard eigenvalue problem.  Identify the matrix,
  eigenvector, and eigenvalue. Describe the motion(s) corresponding to
  the solution(s) of this eigenvalue problem.

  \begin{displaymath}
    \epsffile{figures/3.ps}
  \end{displaymath}

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

\end{enumerate}

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