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 = sE,
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.