Computer Algebra Recipes

3.3. SPECIAL FUNCTION MODELS 137
3.3.2 The Vibrating Bungee Cord
Wisdom consists in being able to distinguish among dangers
and make a choice of the least harmful.
Niccolμo Machiavelli, from The Prince (1469{1527)
While on vacation in Rainbow County, Jennifer is nervously watching her sister
Heather getting ready to bungee jump from an abandoned railway bridge
spanning a deep gorge. At present, the uniform elastic bungee cord has no one
attached to its lower end, but is simply hanging vertically downward and displaying
small vibrations transverse to its length. This reminds Jennifer that the
famous mathematician Daniel Bernoulli ¯rst studied this vibrational problem
nearly 300 years ago, and was able to solve it. To put a modern spin on an old
problem, Jennifer recently showed her mathematical physics class a computer
algebra derivation of the solution.
With Jennifer's permission, we shall now reproduce her treatment. To aid
in understanding the physics of the problem, a free-body diagram is shown in
Figure 3.9, which shows the relevant forces on the bungee cord and introduces
the notation that will be used.
y
y+dy
T () y θ
ψ ( y,t) ψ(
y+dy,t )
dy
ds
dψ
T(
y+dy)
Figure 3.9: Free-body diagram for a segment of vibrating bungee cord.
Jennifer begins her recipe by loading the plots and plottools library packages,
which will be needed for the animation of the transverse vibrations of the
vertical cord.