Homework #6, due May 11, 2000 --- Note: the eres bulletin board is proving to be a
very valuable resource for identifying problems and resolving ambiguities
associated with the homework problems and answers, which are sometimes incorrect.
excercises/problems;
20-30
20-31
20-50
20-51
20-52
[NOTE: These last three numbers do not refer to problems in the book.
They refer to the problems, shown below. 20, because they relate to
chapter 20 (they are somewhat related to problem 20-35.)]
20-50. Consider a string of density 0.1 kg/meter, stretched at a tension
T=100 Newtons between two fixed posts a distance 100 meters apart.
a) what is the wave speed for this string?
b) what is the wavelength of the three lowest frequency modes
c,d...) Suppose that at t=0 the string is at rest and it is displaced
in the shape y(x,0)=A [sin(3.14x/100)]**3 :
c) plot y(x,0)
d) what is y(x,t) ? [Use everything you know here. This takes deep thought.
Don't calculate until you've thought it out.]
20-51 (same system, same initial conditions, etc.)
e) plot y(x,t) at t=0 (again) and t=1/(12f_1), where f_1 is the frequency of
the fundamental (the lowest frequency) normal mode (standing wave).
f) plot y(x,t) at t=2/(12f_1).
show how you did this (f).
[extra credit will be given for finding flaws associated with these problems.]
20-52 (same system) Consider the pulse y(x,t)=B exp{-((x-vt)/d)**2} ,
where d=3 meters.
a) plot y(x,t) at t=0.2 seconds and at t=0.4 seconds.
b) how long does it take this pulse to get from one end to the other?
express this time in terms of f_1 .
c) how long does it take the mixed mode that is a equal combination of
the fundamental (f_1) and 2nd harmonic (f_2) to go from a configuration of
constructive interference on the left side to
constructive interference on the right side.
d) to help with (c), plot y=A[sin(3.14*x/100)*cos(2*3.14*f_1*t) + sin(2*3.14*x/100)*cos(2*3.14*f_2*t)]
at t=0 and t=1/(2*f_1)
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Please continue to post comments on errors, ambiguities; and any others
thoughts, questions, insights, etc. on the eres bulletin board.