Homework #5
Instructions. This is due on March 23 at the beginning of class.
It is worth 20 points. Late papers lose 3 points per day late. You must work by
yourself, but feel free to consult with me. Explain what you did and why you
did it: good explanations are essential! Your paper should have an
introduction, a main body and a conclusion. Start early, ask questions and do
well.
Introduction. In this assignment, you
will again use ODE Architect.
Problems. Start by running “Multimedia ODE Architect” and choosing “Second Order
ODEs” and then “Earthquakes.” The opening screen has
a demonstration with narration. I suggest listening to the explanation. When
it’s finished, click the right arrow in the lower right corner of the
screen.
On the new screen, read the description and click
the first three buttons to reveal the equations. Write out the details on
combining the first two equations to get the third equation. In particular,
show the formulas for the constants L and w0. Click the software’s formulas for these constants and make sure they
are the same. When you have this right, click the right arrow to go to
the next page.
Start on this new page by showing how to get the
“simplified ODE” from the equation you just finished deriving. Now, plot y(t) and determine if it is in phase with the
motion of the ground. (“In phase” means that the peaks and troughs occur at the
same time, even if they aren’t the same heights.) Then build the frequency
response curve for the default values of c = 1 and w0 = 3. (You only need to
use w values of 1, 1.5, 2, etc.) Then change w0 to 4 and build the
next frequency response curve. Discuss any similarities and differences that
you see. When you have these questions answered, click the right arrow to go to
the next page.
On this page, change w0 to w0 = 0.8 and solve with w
= 0.8, 0.85, 0.9, 0.95 and 1. In each case, briefly describe the motion of
the seismograph. Briefly describe what “resonance” looks like; which value of w produced resonance? Briefly
describe what “beats” looks like; which value of w produced beats? The text says
that “undamped Seismo hates
small | w - w0 |.” Why
is this true? By solving for w = 0.81,
w = 0.82 and so on, find how small “small” is in this case.
Now for three general
questions to answer. You may need to review the module to answer these. First, from a Seismo Grapher read-out, how
could you compute the magnitude of the earthquake? How is the magnitude
affected by the frequency of the harmonic motion of the ground? In physical
terms, why should the frequency play a role in measuring the severity of an
earthquake?