EML 5060 Analysis in Mechanical Engineering 9/17/96
Closed book Van Dommelen 9:00-9:50am

Show all reasoning and intermediate results leading to your answer. One book of mathematical tables, such as Schaum's Mathematical Handbook, may be used.

1.

(33 points). For a cylindrical barrel, assume that the cost of the flat ends is f dollar per square feet, while the cylindrical surface costs 2f dollar per square feet. Find the ratio of diameter D to height h of the barrel that gives the lowest total cost for a given barrel volume V. (It seems that the people who invented barrels may not have known much about cost reduction?) Solution.

2.

(33 points). A vehicle moves along the hyperbolic curve

y² - x² = 1

with a constant, unit speed. Find the time to travel from x=0 and y=1 to x=0.5. Note: you will need to make an approximation. Include the first three nontrivial terms in this approximation. Solution.

3.

(34 points). A force field is described by

$$F_x = \sqrt{x+y+z} + y^2 + z^2$$

$$F_y = \sqrt{x+y+z} + 2 xy$$

$$F_z = \sqrt{x+y+z} + 2 xz$$

Find the work that this force does on a body travelling from the origin along the half circle

$$(x-{\textstyle\frac12})^2 + y^2 +z^2 = {\textstyle\frac14} \qquad y=z\ge 0$$

to the point (x,y,z)=(1,0,0) on the x-axis. Hint: a brute force approach may be a fun challenge, but more than you can do in the allotted time. If you can simplify this problem, explain why, and do it. Solution.