This question actually seems somewhat subpar to me, although I may be making some terrible errors in judgment. I'm not qualified to comment on the leadin, other people mentioned the resonance problem with electrical vibrations, so I'll start after that.
Ike wrote: When it affects motion, this phenomenon is modeled by the second term of the differential equation (mass times y double dot, plus mu times mass times little g times the sign of y single dot plus k times y equals 0), where y stands for displacement.
This clue probably isn't generally true - this describes Coulomb damping at low velocities, sure, but isn't accurate, for example, in the case of viscous damping using a fluid or a dashpot, which I thought could be proportional to either velocity or velocity squred (incidentally, would you have prompted on Coulomb/viscous damping here?), and which definitely affects motion. In addition, a buzz with friction absolutely warrants a prompt here, since what you are describing is exactly the damping of a SHO due to Coulomb friction.
Ike wrote:One way to visualize this phenomenon is to take the maximum N, and divide it by the maximum number N+1, and taking the log of that. That quantity is the log-decrement.
As someone who recently covered the log-decrement while discussing underdamped harmonic motion, I do not believe this is an accurate description of it (and actually can't really figure out what the first sentence is talking about at all). Perhaps "Experimental observations of this phenomenon often calculate the natural log of the ratio of two distinct extrema, or the log-decrement," would be better, although that seems somewhat transparent. While in general I support the usage of standard symbols for certain quantities, this is one example where N means different things to different people and ultimately is utterly unhelpful. Furthermore, you seem to be treating N as a measure of the number of cycles elapsed at a certain point, in which case this is fully incorrect, since the log-decrement takes the ratio of the amplitudes. Nevertheless, the usage of the log-decrement in questions is (in my opinion) a good idea.
Ike wrote: When this phenomenon is present, the power dissipated is equal to the time derivative of energy.
If I ignored other clues, I think I would be totally justified here with a buzz on "friction," "electrical resistance," or literally any other physical process in which internal energy is radiated or dissipated in some form. This really is not a unique clue.
Ike wrote:Systems are said to be over this if it is unable to complete one cycle, or critically this if there is no oscillation at all. For 10 points, identify this physical effect which reduces the amplitude of a harmonic oscillator over time, usually caused by a non-conservative force.
ANSWER: damped harmonic motion (accept damping or word forms, accept underdamping prompt on simple harmonic motion )
It's worth noting that there really isn't any oscillation in overdampted oscillators either, since they don't complete any cycles - the distinction being that overdamped oscillators' motion is modeled in the form ae^{\lambda_1 t}+be^{\lambda_2 t}, while critically damped oscillators have motion modeled by (a+bt)e^{\lambda t}. In addition, the non-conservative force thing might actually work better pre-FTP, since it doesn't work that well as a giveaway and helps give more context to those who are uncertain.
Ike wrote: As for that differential equation, from my understanding its simply ma = mu * g * m* sgn(y) + ky, where I rearranged the terms to be my double dot + mu * m * g * sgn(y) + ky = 0.
I guess so, but it seemed to me like the physics/ODE classes that treat damped SHM go more in-depth with the viscous damping model (my'' + \gamma*y' + ky = 0), although this is based on my experience and a sample size of n=3.
Ike wrote: Of course, one can be a bit skeptical, but I don't think that the clue is useless or egregious. If there is a better way to expressing the damping term more basically, please let me know.
On the whole, the motivation behind this tossup was (I think) good, since it focused on things anyone who covered basic SHM with damping would need to know. (Perhaps you could've added something about the nonlinear damping term in the van der Pol oscillator, for example, if you wanted to go deeper.) However, I personally feel (looking at the tossup) that the execution was not as well done, since it was overly general in some parts and sort of inaccurate in others.
Ike wrote:I guess what I'm also asking for is whether or not these type of tossups are a good idea. This was basically written right out of lecture notes that I used in engineering physics, with the leadin coming from a book on nonlinear dynamics that I have.
I am emphatically not an authority, but I think questions that focus on the more mathematical aspects of important physical systems are in general much better than those that deal with named things and effects. At the same time, such tossups require more work to make sure that they use unique and useful clues.