Introduced the damped harmonic oscillator, with frictional force proportional to velocity. Solved Newton's law by methods we learned for differential equations with constant coefficients. Found three regimes of physical behavior:
1. Underdamped. Here 
from the quadratic equation we obtained is
imaginary, and solutions look like 
. That is, a sinusoid modulated by a decaying
exponential whose initial slope is 
. So an underdamped
oscillator displaced from equilibrium springs back, but overshoots,
and oscillates back and forth with ever declining amplitude.
2. Overdamped. ( real). Here solution was 
. That is, a linear
combination of decaying exponentials with initial slopes 
, 
. So an overdamped oscillator displaced
from equilibrium springs back, but never oscillates.
3. Critically damped. ( 
) Here our quadratic equation gave
only one solution, 
. We claimed a
second solution 
exists; we will
derive it from the limit as 
of the overdamped
case, next time.
-- KB