SUMMARY OF CURRENT LINEAR ALGEBRA MODULES

1. Mechanics, Linear Algebra and the Bicycle

The module consists of two main paths: "Training Wheels" and "Mountain Bike." "Training Wheels" starts with the easiest material and now has three subsections: (1) position vectors and force vectors; (2) free body diagrams and vectors; (3) two-dimensional rigid body equilibrium. A Java applet called the "geometry generator" allows students to interact with the vector dot product for finding the angles between the tubes of a bike frame. Using this they can design the geometry of their own bike frame. "Mountain Bike" is more advanced and introduces the topic of simultaneous equations and solution methods in the context of a truss model of the bike frame. Three subsections take students through the following steps: (1) formulating a truss model of the bike frame, including the relevant system of equations for the truss; (2) solving the system of equations using matrix methods; and (3) comparing the results with some real data that can be gathered from actual strain gage readings of the loading in each tube of the bike frame. Students can solve for the loadings in their own custom-designed bike using the techniques here. Approximately 50% of the intended content is up and running at http://www.rpi.edu/~portej/BIKE/bike.html

2. Moments, seesaws and the "swing weight" of golf clubs

Under construction, this new module will be a mechanics application that links to the calculus page dealing with centroids and moments.

The module will show how the ideas of moments, centroids, and seesaws are relevant to the design of a "matched" set of golf clubs, i.e., a set in which all of the clubs have the same value of the so-called swing weight.

The module will start with golf club manufacturers' definition of a club's swing weight in terms of a balancing procedure. Students will see that the procedure actually is quite relevant to the seesaw examples in the calculus module. For example, pp. 32-33 of The Physics of Ball Games by C.B. Daish (1972) explains that as one goes from the lower-numbered irons (e.g., a 2-iron) to the higher-numbered irons (e.g., a 9-iron), the club gets heavier and the shaft gets shorter, "...the two varying in such a way as to keep constant the so-called swing weight of the set." The book goes on to present a balancing procedure and a rather non-obvious formula (at least at first glance) for computing a number - e.g., 258 ounce-inches, or D9 -- that represents the club's swing weight.

However, the module will lead a student into the realization about the physical meaning of the swing weight formula; it will become clear that what is really being kept constant from club to club is not really a "weight" (force) per se, but rather the moment supplied by the player's hands on the grip of the club. That is, the hands exert the same moment from club to club so that all clubs "feel" the same. The logic for this approach will be demonstrated using static equilibrium of rigid bodies.

Also, the module will allow the student to do his or her own "virtual balancing" experiments on golf clubs -similar to what's done in the calculus seesaw module -- and to review actual experimental data from a torque cell that measures the net moment at the grip of each club in a set of clubs. Students will be able to see from experimental data that a set of clubs is matched if the measured moment (torque) to support each club is the same at the grip.

3. Material Properties and Matrices

Under construction, this module will demonstrate that some material properties of engineering materials are described mathematically by matrices rather than, for instance, simple scalar quantities.

We will work with the relationship j = s E, in which j is current density (a vector), E is electric field (a vector) and s is electrical conductivity (a 3 x 3 matrix representing a 2nd rank tensor). For an isotropic medium (in which s has a simple diagonal form), this equation indicates that the current flow is in the same direction as the electric field. However, for an anisotropic material, the 3 x 3 conductivity matrix can have off-diagonal terms, which means that a current j does not necessarily flow in the same direction as the applied electric field vector E. The module will "walk the student through" the basics of: (1) matrix multiplication; (2) coordinate system dependence of the matrix representing the conductivity (; and (3) the difference between isotropy and anisotropy of material properties.

Then, using a Jell-O-type material in Professor Newell's lab, we will do experiments on isotropic and anisotropic conductive media. These data will be built into the web module to allow a student to discover that the above mathematical description can be successfully used to describe real behavior.