Re: Forced damped pendulum

[Leigh Palmer <palmer@sfu.ca>]

> I have a rather ambitious student who,  for a Senior project,  is attempting
> to analyze the motion of a driven damped pendulum.  We have gone through the
> more simple case of a mass on a spring found in most texts.
>
> He has developed a 2nd order differential (using Newton's 2nd law) to
> describe
> the motion and is now attempting to solve it.  Does anyone have any ideas for
> arriving at a mathematical expression for the pendulum's angular displacement
> as a function of time?  (I have plenty of ideas,  but won't bore you with the
> details ... yet.)

The solution to this ODE is, simply, tedious. It would be my guess that
what your student wants a notion of is the difference between this
solution and that of the linear case he has already studied: How different
from a mass on a Hookish spring is the motion of a pendulum?

May I suggest that he will learn best by adopting another approach, that
of numerical simulation? Modeling this system using a spreadsheet on a
fast computer (a Power Macintosh or so) will give him a double precision
comparison with both systems running in parallel. He can examine his
results graphically, plot the difference between them, vary the parameters
with ease, etc. The necessary equipment and applications are common these
days, even in high school environments or even the homes of students.

I think this would be a more appropriate exercise for a high school
*physics* project than the one he proposes. Perhaps the sophistication of
high school *mathematics* has now progressed to the point where this is an
appropriate project, but I sure don't see the product of any such courses
in my student intake at university level.

Leigh