Physics 410/609---Computational Physics
Assignment #3---Due Tuesday, March 20, 2012
NOTE: At the end of each problem, please indicate how long it took
to solve the problem.
1) a) Compute the period of the anharmonic oscillator for initial amplitude
1 for various strengths of the potential and determine the relation
between the period and k/m, where the potential is k*x**4/4 .
Use the fourth order Runge-Kutta method for this problem. This
method is explained in Numerical Recipes (see below) and the appendix
of CSM Chapter 3. Please note that in class I pointed out errors in
Eqs. (3.61g) and (3.61h).
To get an accurate value for the period, it is useful to run
several oscillation periods and then divide the time by the number
of periods.
The best way to see the relationship between the period and k/m is
to use a log-log plot.
b) Read sections 17.0-17.4 of Numerical Recipes (NR) in the chapter on
Integration of Ordinary Differential Equations. If you are looking at
an edition of NR other than the 3rd, it could be a different chapter
number, but the chapter title should be the same.
2) We looked in detail in class at the bias that is introduced when we try
to find 1/(average value) of a distribution. Consider instead what
happens when we look at log(average value). Again, the random
measurements are uniformly distributed between zero and one. You may
consider any of the methods we used in class. At least one of your
results should be a graph similar to ~sg/jackknife/jackknife.ax, that
is, it should compare results with and without jackknife bias reduction.
3) Using Mathematica, find the value of the integral of the function sin(x)/x
over the interval [0,5].
4) The local newspaper carried a story about jury selection for a trial.
The paper stated:
At a hearing Thursday in Owen County, Nardi denied a motion filed by
defense attorneys David Colman and Elizabeth Cure to disqualify the
jury panel. Colman and Cure observed that out of 80 potential jurors,
only one was younger than 30.
"Monroe County has a population of 120,000 people, of which between
30,000 and 40,000 are students at Indiana University," the motion stated.
"A truly random jury-selection process for Monroe County should result
in no fewer than 20 students from Indiana University."
This might be a good time for a lawyer joke, but I will resist that temptation
and ask you to analyze this situtation.
a) Assuming that there are 120,000 people in Monroe county and 30,000 of
those people are students at IU, what is the probability that a group of
80 randomly selected people would include fewer than 20 students?
Note, if you had been among the people called for jury duty, you probably would
have been excused from jury duty if you had pointed out the lawyer how stupid
his statement was. (I got out of jury duty some years ago when I told
the laywer that he had just contradicted himself. More recently, however,
I got asked by the defense attorney if I would be against a person just
because they were engaged in an alternate "life style." I responded,
"What are we talking about here, cannibalism?" The defense attorney could
not keep a straight face after that and could not ask any more questions,
but I did not get excused from that case.)
b) What is the probability if 40,000 of the 120,000 people, that there would be
fewer than 20 students in a randomly selected group of 80 people?
Note: I will be pleased if you can use Mathematica to solve this problem, but
if you prefer another approach, that is also fine. If you have taken a
course in statistical mechanics or a math course that covered combinatorics
you should have the theoretical background for solving this problem. If not,
come see me we can talk about drawers containing red socks and blue socks and
how many ways we can pick socks from the drawer.