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Exam 3 Analysis in Mechanical Engineering 11/16/93
Closed book Van Dommelen 2:45-4:00
Show all reasoning. One book of mathematical tables may be used.
- 1.
- The direction of the field lines of a two-dimensional
electrostatic field is given by:
![\begin{displaymath}
E_y {\rm d} x - E_x {\rm d} y = 0\end{displaymath}](img1.gif)
Find an algebraic expression for the field lines of the field:
![\begin{displaymath}
E_x = e^x \cos (y) + x - y\end{displaymath}](img2.gif)
![\begin{displaymath}
E_y = -e^x \sin (y) - x - y\end{displaymath}](img3.gif)
- 2.
- For the RCL circuit shown below, the voltage across the condensator
satisfies
![\begin{displaymath}
{{\rm d}^2 V\over {\rm d} t^2} + {R\over L} {{\rm d} V\over {\rm d} t}
+ {1\over LC} V = e(t)\end{displaymath}](img4.gif)
Assuming that the resistance R=3, capacitance C=0.5 and inductance
L=1, and the applied voltage is e(t)= - e-t/(1+et),
find the voltage across the condensator.
- 3.
- A projectile is fired through a constant area, but variable
density duct. Assuming the velocity variation is linear with distance,
the equation of motion for the projectile is
![\begin{displaymath}
\ddot x = a {(x - \dot x)^2 \over x}\end{displaymath}](img6.gif)
where a is a constant that cannot be scaled out.
Assuming that a=1, find an algebraic expression relating the projectile
velocity
to its position x.
Verify that the projectile never catches up with the stream:
projectile velocity does not become equal to position at large x.
'Author: Leon van Dommelen'