EML 5060 Analysis in Mechanical Engineering 9/20/00
Closed book Van Dommelen 10:45-11:35am

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.

A house is to be build on a plot with the following boundaries:

The edge of the street along the plot is given by the x-axis and the front of the house needs to be parallel to the street. Find the rectangular house with the largest floor area that fits on the plot. Solution.

2.

For an automobile with a continuously varying transmission, the fastest acceleration is obtained by running the engine constantly at the rpm at which the engine produces peak power. However, if the car has only a finite number of gears, you cannot just drive at a single rpm. For any given gear, the rpm varies between the rpm that you shift into the gear and the rpm that you shift to the next gear. The average power you will get while in that gear is now given by

where is the average rpm in the gear and T is the torque produced by the engine; T depends on the engine rpm, , and is measured using a dynometer. Assume that in the range of rpm operated, the torque does not vary too much: where T_a is the average torque and is small compared to T_a. By expanding up to errors that are allowed to be of order ,show that

In the top you recognize the famous ``area under the torque curve." In any case, the power you get in a gear is average rpm times average torque in that gear. Make sure you clearly show all steps in your derivation. Solution.

3.

Find the volume of an axle

that has rounded ends described by

x² + y² + z² = l²

(With l>a). Solution.