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