*3/22/07*

Be familiar with the complex plane: _{}, definitions: _{}

To multiply complex numbers, multiply the mods, *add* the
phases. Know the all-important formula:

_{}

and be able to interpret it as a
point on the *unit circle*.

*Some useful small x approximations:*

_{}

and for small angles

_{}

Be able to solve the equation

_{}

_{ }

and write down the velocity and
kinetic energy at any time. Be able to
sketch a graph of the *potential energy*
as a function of position, both for a horizontal and a vertical spring. Be able to derive the angular frequency and
the period. Know how to find the
dependence of the period on *k*, *m* using dimensional arguments.

You should know the equation of motion

_{}

and that a solution is _{}_{ }Be able
to substitute this in the equation to find _{} and from that the
condition _{}for exponential decay.

Be able to use dimensional arguments to find that *m*/*b*
and *b*/*k*
*both* have dimensions of
time, and for *large* damping give a
physical interpretation of when these very different times are physically
relevant.

I would not ask you to solve the equation of motion in this
case, but you should know the following facts: the energy in the oscillator
decays in time as _{} where _{} The *Q* factor _{} measures how many
radians the oscillator goes through during the time the energy drops by a
factor 1/*e*. You should be able to sketch how the
amplitude decays in time, showing the exponential “envelope” of the
oscillations.

Know the condition for critical damping, and be able to sketch how the amplitude decays above, at and below critical damping.

For the damped simple harmonic oscillator reviewed above, if
*f*_{1}(*x*) and *f*_{2}(*x*) are
solutions, so is *A*_{1}*f*_{1}(*x*) +*A*_{2}*f*_{2}(*x*)
for any constants *A*_{1}, *A*_{2}. However, this is *not* the case for the driven oscillator reviewed below: in fact, if *f*_{1}(*x*)
is a solution for the driven oscillator, 2 *f*_{1}(*x*) would
be a solution if the driving force were doubled. But one *can*
add to a solution of the driven oscillator any solution of the same but undriven oscillator—and this is usually necessary to fit
initial conditions.

Be able to write down the equation of motion _{}and understand it as the real part of the equation _{} Know how the put in
the trial solution _{} and from that deduce
the amplitude _{}. Be familiar with the
form of this as a function of driving frequency, especially near resonance.

Know the equation of motion of a simple pendulum, how it relates to a simple harmonic oscillator, how to handle the equation if the pendulum is a rigid body rather than just a point mass on a light rod or string. Know how to find the potential energy as a function of angle.