PROBLEM SET 2 COMMENTS - SPRING 2003

Not much this week - these are pretty straightforward, essentially
redoing examples done in lecture or the text for different
applications.  The plots are crucial;  I'll be willing to put sample
MAPLE worksheets online and world-readable if desired.

      2.1:  Of course, the symbol "phi" for the angular displacement
        of the top of the rod is NOT a phase angle (too many angles,
        too few symbols).

      2.2:  Due partly to the storm, driven electrical circuits were
        not covered in lecture.  However, the simple series R-L-C
        circuit with a sinusoidal voltage can be done with your
        knowledge of 8.02 or the equivalent, and in fact might well
        have been done in that subject.  If you rely on another text,
        make sure that the notation agrees, specifically the sign of
        any phase angles  (and remember that we're using phase
        differences between 0 and Pi, not from -Pi/2 to Pi/2, which
        some other texts do).
	For part(b), that "Hint" is more than a hint;  without it, you
        won't be able to make the anology between displacement x and
        the corresponding electric quantity.

      2.3:  This works out nicely if done carefully.  For starters, in
        part (b) note that the quantity "phi" is the phase difference
        between the "effective driving force" and the floor
        displacement y(t), NOT the phase difference between y(t) and
        the table displacement or between F(t) and the table
        displacement.  (You really should use the hint, and measure
        the table displacement in an inertial frame.)  The phase
        difference between the table displacement and y(t) or F(t) is
        not part of the problem, so it's not worth it to bring in a
        new variable, unless you wish.

	Part (c) asks for a sketch, incorporating the limits you find.
	If you're ambitious, you can show that indeed A(omega) peaks
	near omega=omega_0 (the algebra is a bit trickier, since F_0
	is not as simple a function of omega as in other cases).  I
	get omega_max ~ omega_0*sqrt(1-1/(4*Q^2)), and this is an
	approximation to the solution of a quadratic.  I would
	recommend using the same mathematical tool as used in Problem
	2.1 to plot typical response curves as well as making a sketch
	"by hand".  There are some interesting aspects, especially at
	low Q, that might not be clear from your sketch.

      2.4: Of course, the parentheses need to be around the "cos omega
        t + phi" term (either that or they didn't copy well into the
        PDF file).  In part (a), the use of omega_1 was introduced in
        lecture today (February 19), and is French's notation.  We now
        have no fewer than three frequencies (the natural undamped
        frequency omega_0, the undriven damped frequency omega_1 and
        the driving frequency omega).  In general, when considering
        transients in driven oscillators, we'll need all three of
        these, and we have to keep them distinct.  One drawback is
        that if we have two or more driving frequencies, we'll want to
        take back omega_1 for labeling purposes, but considering
        transients with anharmonic driving terms is cruel, and will
        not (I sure hope) be done in detail in 8.03.