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**** Actually f = {\omega_1 \over 2\pi} and T = 1/f, but you knew that.=)
I wrote this explanation for the student (please correct me
if I have it wrong):
\theta (t) = A \cos (\omega_1 t + \phi)
{d\theta \over dt} =
\omega_2 (t) = -\omega_1 A \sin (\omega_1 t + \phi)
{d^2\theta \over dt^2} = {d\omega \over dt} =
\alpha (t) = - \omega_1^2 A \cos(\omega_1 t + \phi)
Now \omega_1 is a constant angular velocity whose
sinusoidal PROJECTION determines the period or
frequency of oscillation. For instance,
T = {\omega_1 \over 2\pi}. \omega_1 in SHM is not a
function of t.
\omega_2 (t) is the time varying angular velocity of an
angular oscillator (physical or classic pendulum). It
describes the actual physical angular velocity of the
pendulum as a projection of the constant angular velocity
\omega_1.
So \omega_1 \neq \omega_2 (t) in general.
Since all of the introductory calc texts discuss the equations
for SHM and physical pendula periods and SHM as a
projection of circular motion, you might expect this
question to arise more frequently IF students are actually
interpreting the problems they are solving. The four
freshman mech w/calc texts I have here (I've been
reviewing for dept adoption) do not distinguish between the
two omegas.
However, I have only ever been asked this question once in
lecture (I teach 500 students who I feel are amongst the
brightest freshman engineers extant). Similarly, I have
only ever been asked a few times why the units of torque isn't
joules. I suggest that for all of the talk about problem-
solving and conceptual understanding, traditional physics
texts and instruction (for too many of us the text IS the
curriculum) doesn't contain nearly enough in-depth
illustration and exemplar variety.
What use is knowing and being able to solve equations like:
f = \omega \over 2\pi
if students don't know which omega to use, and the texts
ignore the question entirely in the race to get the topic over
with in one chapter? Are we REALLY teaching problem-
solving skills or are we teaching plug-and-chug in a variety of
contexts with unexplained notation?
Dan M
Dan MacIsaac, Visiting Assistant Professor of Physics,
danmac@physics.purdue.edu
http://physics.purdue.edu/~danmac/homepage.html (yes, white socks)