COMMENTS ON THIRD PROBLEM SET - FALL 2002

	First off, read French's footnote on Page 119, specifically
"And it is not as mathematically formidable as it may appear at first
sight."  Prof. French speaks the truth.  It is not necessary to know
the details of linear algebra, and in fact both French and B&B use
what I call, in all honesty, "8.03 math" - linear algebra is of great
utility, but if you have other things to do while at MIT, maybe you'll
want to defer learning about determinants and eigenvalues (B&B do use
determinants, though).  You will learn this eventually, and I'd like
to suggest the sooner the better, but for now it's your decision.

	If you do want more math, and you want it now, I have lots
(maybe even too much) online, at "Coupled Linear Oscillators",
"Normality of Modes of Discrete Coupled Oscillators" and "Normal Modes
of Coupled Pendulums."  Help yourself.

	In B&B Figure 1.39, note that the middle spring is labeled
with a force constant "K" while the derivation in the text on Page 102
uses "kappa."  It happens.  Figure 1.40 has it right.

	Problem 3.1: I've always thought that the point of problems
similar to this is to show how complicated the analysis can become if
the masses are not the same; you've got some work to do here.  By way
of full disclosure, it took me about as long to do this using MAPLE as
it did to do by hand (of course I used MAPLE to do the plots).  The
algebra is not that hard.  You might want to check to see if you get
the known simpler results if m=M.  As another check, make sure that
your frequencies are both real, and (omega)^2 is positive for both
roots of your quadratic equation (there's really no way to avoid
solving a quadratic equation).

	Problem 3.2:  This is very similar to what's done in French.
By way of a suggestion, make sure in your work that you distinguish
between the lengths l and l_0.

	Problem 3.3: There's a very interesting and important subtlety
to this problem (read part (c) before beginning).  In fact, if you do
this problem "by hand", as opposed to using a program such as MATLAB
or MAPLE, you would find yourself solving a cubic equation.  However,
if you do part (c) first, you'll find a root of this equation,
reducing the cubic to a quadratic, which should hold no fear for you.

	Problem 3.4:  I don't have a kit, but the experiments don't
change much from year to year (note the date).  For this experiment,
you want to pendulums to swing perpendicular to the supporting
(horizontal) string, in directions perpendicular to the plane of the
diagram.  Knowing knots helps;  I go with a taut-line hitch at the
top, and you should indeed do as suggested, and fasten the string to
the weight by using the small piece of wood, allowing easy adjustment
of the length.