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:

$$E_y {\rm d} x - E_x {\rm d} y = 0$$

Find an algebraic expression for the field lines of the field:

$$E_x = e^x \cos (y) + x - y$$

$$E_y = -e^x \sin (y) - x - y$$

2.

For the RCL circuit shown below, the voltage across the condensator satisfies

$${{\rm d}^2 V\over {\rm d} t^2} + {R\over L} {{\rm d} V\over {\rm d} t} + {1\over LC} V = e(t)$$

Assuming that the resistance R=3, capacitance C=0.5 and inductance L=1, and the applied voltage is e(t)= - e^-t/(1+e^t), 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

$$\ddot x = a {(x - \dot x)^2 \over x}$$

where a is a constant that cannot be scaled out. Assuming that a=1, find an algebraic expression relating the projectile velocity $\dot x$ 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.