EML 5060 Analysis in Mechanical Engineering 9/19/01
Closed book Van Dommelen 10:35-11:25am

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.

When numerically solving partial differential equations on a computer, normally a ``mesh'' of points must be defined. Algebraic mesh generation requires that certain functions are chosen to define the mesh. In one of my programs, I am using, in part, the following function:

where C is a constant. The reason I am using this function is that it has a horizontal asymptote for large negative x and has a 45 degree asymptote for large positive x. Do the following:

(a)

Verify the horizontal asymptote at large negative x by finding . (The value of this limit is the height of the asymptote.)

(b)

Find the y-intercept of the 45 degree asymptote. This intercept is given as the limit for of the following function:

(Hint: you might want to find the limit of the function e^g first.) (The fact that the intercept exists verifies that f does indeed have a 45 degree asymptote.)

Solution.

2.

The rim angle of a solar energy collector made up out of plane mirrors satisfies the equation:

where C is a constant that depends on the desired concentration factor, mirror density, height, etcetera. Assuming that is small, (i.e. that C is close to a half), write an approximate cubic equation for the rim angle . (You do not have to solve the cubic equation unless you want to.)

Solution.

3.

Find the moment of inertia for the flattened axis and .

Solution.