A viscous damping force, modeling for example the viscous damping of the oil
in the bearing at the pendulum hinge, would to a good approximation be
proportional to the angular velocity of the pendulum, with a coefficient
we'll call alpha. The damping force will tend to slow down the pendulum,
which determines the sign of the new term in the damped pendulum equation
of motion:

The amplitude for alpha = 0.1 should decay roughly by a factor of two
after two periods. **Find the value of the damping**
coefficient alpha needed to make the amplitude decay to precisely
half of its value after two periods, and add it to your writeup.
Try to measure this alpha to better than two significant figures.

(G2a.2)
Take your numerical solution for the first part of this question
(with your optimal value for alpha), and make a combined plot of the
energy, the potential energy, and the kinetic energy as a function of
time, labeling each curve. Save this plot and include it in your lab
writeup. In your writeup, **address the following questions**:

- Is the total energy the sum of the kinetic and potential energy?
- At the initial time, is the energy coming from the potential energy or the kinetic energy? What initial condition would make the energy all kinetic energy?
- After two periods (as measured in the last part), is the energy mostly potential or kinetic? The answer, in retrospect, should be obvious.

Notice that for large damping and for small damping you needed to run for a long time to see the decay, but for intermediate damping, the decay is quite a bit faster!

(G2b.2)
Roughly find out (to better than one significant figure) at what
value of alpha the oscillations stop. **Include your value in the
lab writeup.** This is called *critical damping*. Putting
the damping just large enough to kill the oscillations also removes
the energy as fast as possible: making the damping even larger is
counterproductive.

(G2b.3)
Imagine our pendulum is a screen door, with the ball at the end representing
the doorknob, and the pivot of the pendulum representing the hinge.
Screen doors are often attached with springs - dissipation is low, so
they are roughly described by the harmonic pendulum equation with
alpha=0. Spring-attached screens slam when they close: when the pendulum
reaches Theta=0 it hits the door frame hard. There are fancier gizmos
to pull screen doors shut, that we can model as a dashpot filled with oil
- well described by our damped equation of motion above, with alpha > 0.
**What value of alpha best balances** the need to shut the
door fast (avoiding mosquitos), without slamming? Are the two needs
incompatible?

Damping occurs in most physical systems. Sound is damped in air, light is absorbed as it passes through water, guitar strings don't vibrate forever. The wave equation we study in this course as a model for these systems ignores this damping. This is in many circumstances an excellent approximation, and furthermore ignoring this complication allows us to study the fascinating concepts of wave propogation, interference, and diffraction more simply and clearly - just like ignoring air friction allowed Newton to understand trajectory motion more clearly.

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).