In accordance with Phys-L tradition, let me pursue a
tangent. I realize this is not an answer to the question
that was asked, but even so, it may be of interest.
The wildest pendulum demo I've ever done is a
*parametric oscillator* which is almost the same as a
*parametric amplifier*.
Briefly, it works like this: Rather than driving the
pendulum with a horizontal /displacement/ or horizontal
/force/ at frequency ω, we modulate the *length* of the
pendulum at frequency 2ω. This is a dramatic, categorical
distinction.
Here's one way to proceed: Build a pendulum with a bob on
the end of a string. Near the top, the string passes through
a hole. The hole is stationary, as part of the structure.
The effective length of the pendulum is controlled by the
amount of string below the hole. We use a hole rather than a
fixed tiedown because we are going to *modulate* the length
by pulling a small amount of string through the hole.
In the simplest version, there is a segment of string above
the hole, tied off at the far end. You can modulate the
length by tugging sideways on the middle of this segment.
Shorten the string as the bob passes through the midpoint at
the bottom of each half-cycle; lengthen the string at the
turning-point at the end of each half-cycle.
To make this fancier, tie the string to a motorized crank,
or to a servo of some sort. Adjust the rate of modulation
to match the natural frequency of the pendulum, times two.
1) One option is to use a simple motor with a crank. A
universal "sewing machine" motor works fine, but you may
need a reduction gearbox between the motor and the crank.
2) An alternative is to use a servo, of the type used on RC
model aircraft. These things are widely available. They are
controlled by a 50Hz pulse-width-modulated signal, which you
can drive directly from the audio output of your computer.
Alternatively, for a few bucks you can get a USB dongle that
will control several servos.(*) https://www.pololu.com/product/1351
3) You can also get a stepper motor with a built-in
leadscrew and threaded follower block. Small ones are
cheaper than dirt. This may have advantages if your pendulum
has a rigid rod rather than a string, insofar as it makes it
easy to produce a purely vertical motion. USB stepper
controller/drivers are readily available.(*)
(*) The local robotics club will know all about such things.
===================
I realize that in the introductory HS course we usually
ignore damping. At a more advanced level, we add that
in; the result is a /damped/ harmonic oscillator.
The amazing thing about parametric drive is that can produce
*negative damping*. I kid the not. Doing the calculation to
obtain and explain this result is nontrivial, but the answer
makes a certain amount of sense, as follows:
For a zero-amplitude swing, modulating the length doesn't do
anything interesting. In contrast, when the bob is moving
through the midpoint, if you pull up on the string, you are
doing work against centrifugal force, adding energy to the
system. If you lengthen the string near the extremes of the
swing, there is no effect, because there is no centifugal
force. The magnitude of this energy input depends on how
much energy is already in the system, leading to exponential
growth away from equilibrium, as expected for negative
damping.
This is very different from ordinary force or displacement
drive at frequency ω, which converges exponentially to the
steady state. The convergence is a decaying exponential,
not a growing exponential.
Returning to the parametric oscillator: Note that if you
were to shift the phase, so that you were /lengthening/ the
string at the middle of each half-cycle, you would be doing
negative work against centrifugal force, sucking energy out
of the system. This would create additional positive
damping.
Because it amplifies one phase and deamplifies the conjugate
phase, it conserves area in phase space. This makes it
simple to analyze in terms of fundamental physics. It is
about the only type of amplifier that is simple to analyze.
Anything that purported to amplify both phases would have to
couple to other modes of the system; otherwise it would
violate Liouville's theorem, violate the second law of
thermodynamics, violate the Heisenberg uncertainty, etc.
(all of which are essentially the same thing). Coupling to
other modes brings in thermal noise. In contrast, the
degenerate parametric amplifier can be made almost
noiseless; its noise can be orders of magnitude less than
what you would expect based on a naïve application of the
uncertainty principle. I once had a talk of mine assigned to
the "crackpot session" at a conference, alongside talks on
exceeding the speed of light, on the grounds that I was
claiming to violate the uncertainty principle (which I was
not) and claiming something that was too good to be true. It
turned out that all the cool kids showed up my talk anyway.
There were some cancellations, so the moderator let my 10
minute talk go on for 20 minutes, plus 28 minutes of
questions. That was followed by standing in the corridor for
3 hours fielding more questions. It was epic.
Like I said, this is too fancy for HS students. However, for
undergrads, grad students, and faculty, their eyes bug out
the first time they see this.
As a homework assignment: Students (and others) can do this
on a playground swingset. Beware that it is very powerful,
meaning you can quickly build up a huge amplitude, so be
prepared for that.
The electrical analog of this is even easier to build than
the mechanical version.