This is a problem-of-the-week format problem. It involves a format, used
in modern math courses, in which you solve a problem and present a written
description of your solution. An outline of your solution should follow something close
to the following outline (unless you are a very good writer and can come up with something better):
I. Problem Statement.
(after reading the problem (several times), restate it in your own words without
looking at the problem. Have a friend look at what you wrote and describe to you
what it says.)
2. Overview/summary of results.
(a clear concise summary of what you found, learned or showed. This is very important.
People tend not to read your work if this is not clear and focussed. Strive for clarity
coherence and emphasis.)
3. Solution.
organize steps. illustrate with graphs, pictures, as appropriate.
4. Discussion.
Discuss nuances, physical interpretation or analogies, justification, ...
5. Process.
Describe your process in approaching the problem, how did you start?
What didn't work? wrong turns? what was easy? what was hard?
6. Conculsions.
This can be similar to 2., but can be presented differently now
that all details have been revealed...
POW#1:
Consider a lightly damped oscillator, characterized by sqrt(k/m)=1 sec-1 , and a
damping, beta=0.1 sec-1 , or more generally, beta/w_0 =0.1 .
Suppose it is driven by a "square wave" force such that,
F(t)=F_0 for -d < t < d
and that F(t) is periodic in time with a period T, and,
F(t)=0 at all other times.
(Assume d < < < T, i.e., use d=T/100 in your calculations.)
a) Plot F(t).
b) What value(s) of T will give a particularly large response for x(t).
For that value of T, obtain an expression for x(t) which includes relevant terms
and discards ones that are too small to matter. (Describe your process in deciding
what to discard.) Plot x(t) and F(t) together (on the same time axis). Discuss.
c) Describe physically what is going on. explore analogies...
(optional extensions):
d) redo c) and d) for another frequency that gives big
response.
e) let the amplitude of the peaks of F(t) alternate between two different values.
(Making the period 2T.) Solve and discuss...