Up: Return
EML 5060Analysis in Mechanical Engineering Exam 1 10/1/92
Exam 1 Van Dommelen 2:45-4:10pm
- 1.
- A mass spirals outward, the distance from the center being given by
. If the rate of change in angular position
is
, find the radial and azimuthal components of
velocity
and
, and those of the acceleration
and
. - 2.
- Find the moment of inertia
for the inside
of the polar curve
![\begin{displaymath}
\rho = \sin \theta \qquad 0\le\theta\le\pi\end{displaymath}](img8.gif)
Note: for even integer p
![\begin{displaymath}
\int_0^{\pi/2} \sin^p x {\rm d} x
=\int_0^{\pi/2} \cos^p x {...
...t 5 \ldots (p-1)
\over 2 \cdot 4 \cdot 6 \ldots p} {\pi\over 2}\end{displaymath}](img9.gif)
- 3.
- For the truss sketched below, the tension forces T1, T2, T3,
and T4 in the four bars satisfy the five equations:
T1 + T2 = 0
-T1 + T2 + 2 T4 = 200
T2 - T3 = 0
- T3 = 2 P
T3 + 2 T4 = 200
Write the augmented matrix of the system and determine for which value(s)
of P a solution exists, and whether it is unique.
- 4.
- A coupled set of pendula as shown above has modes of the form
![\begin{displaymath}
\theta_1 = A \cos(\omega t +\phi)\qquad \theta_2 = B\cos(\omega t + \phi)\end{displaymath}](img11.gif)
For certain values of the spring constant, masses, and pendulum lengths,
the amplitudes satisfy
![\begin{displaymath}
- \omega^2 \pmatrix{A\cr B\cr}
= \pmatrix{-3&1\cr 1&-3\cr} \pmatrix{A\cr B\cr}\end{displaymath}](img12.gif)
Find the natural frequencies
.For each frequency, compare
and
. - 5.
- The kinetic energy of a certain thin plate due to rotation around
its center of gravity has the form
![\begin{displaymath}
T = 2 \omega_1^2 - 2 \omega_1 \omega_2 + 2 \omega_2^2
+ 4 \omega_3^2\end{displaymath}](img16.gif)
In principal axes, there would be no
term.
Find the direction of the principal axes of this body by diagonalizing
the quadratic form.
Up: Return
'Author: Leon van Dommelen'