This week we will continue our explorations of simple harmonic motion and the differential equations that describe them. The first video clip is just a quick description of drag forces and the distinction between the linear and quadratic terms. Throughout this lecture Walt made a number of mistakes which he corrects in the tape but it's very interesting to see and you need to be careful not to be confused by the small (and they are small mostly) errors.
This is valuable in part because it establishes language that is used throughout engineering and physics. Like Walt we will only be concerned with linear damping terms.
This next clip walks (quickly) how drag term enters Newton's 2nd Law and the redefined terms that Walt uses throughout these clips. You need to be able understand this process for some homework problems.
Differential Equation for a damped spring system:
Do you see why our simple harmonic solution (sines and cosines) can't work in the differential equation because of the drag term?
I have skipped the full explanation of how one finds the solution to this differential equation though you may find that you will want it at some point in MTH 256 so here is the whole thing (the full monty!) What I have linked here picks up at the end and follows the explanation of what the terms mean.
Make sure that you have tracked this discussion enough to understand the meaning of each of the different constants in the solution from the amplitude to the damping constant to the andgular frequency.
Here is the description of how the damped solution gets translated into a plot. Be sure you understand the relationship between the shape and the "gamma" constant.
Did you catch the statement about the behavior of the period of the damped oscillation over time? If you didn't go back and make sure you got it.
This discussion of the quality factor Q and it's relationship to the damping constant is very valuable, though complicated, and worth your attention.
Quality Factor and damped pendula:
What does Walt mean by the "1/e" point? What is the relationship between Q and the damping constant? We will explore this some in class because it's relevant here as well as later in the year in E Fundies.
Finally, if you have the time there are some interesting, though not critical discussion of overdamping that you might find interesting. We will not explore this in detail though I do expect you to understand what the term means. The example of the torsional pendulum is pretty to watch.
