EML 5060 Analysis in Mechanical Engineering 9/20/02
Closed book Van Dommelen 11:50-12:40 pm

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.

1.

The scaled cross-sectional area A of a rocket nozzle depends on the scaled Mach number as:

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. Solution.

2.

The inverse elliptic integral 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:

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 becomes equal to 0.99

Solution.

3.

The edge of a tapered plate is described by:

x² + y² = 6 x

and the thickness is t=0.01 x. Find the moment of inertia for rotation around the origin for this plate.

Solution.