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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 38 "Computer Intensive Physic
s\nPhysics 211" }}}{EXCHG {PARA 19 "" 0 "" {TEXT -1 163 "Steven R. Dun
bar\nDepartment of Mathematics and Statistics\nUniversity of Nebraska-
Lincoln\nLincoln, NE, 68588\n\nsdunbar@math.unl.edu\nhttp://www.math.u
nl.edu/~sdunbar\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "Topic:  Har
monic Motion, friction, damping\n\nTitle:  Damped Harmonic Motion\n\nK
eywords:  harmonic motion, friction, damping coefficient, " }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 165 "Audience:  Stude
nts in Computer Lab Setting\n\nFuture: Find a way to have Maple extrac
t solution directly in exp(-b*t/2m)*cos( omega *t) directly\n\nTime:  \+
30-45 minutes" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Derivation of th
e differential equation form Newton's Second Law." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}
}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 34 "Simple Harmonic Motion, No Dampi
ng" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 48 "Setting Up and Solving the \+
Differential Equation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "de0
 := m*diff(x(t), t,t) + k*x(t) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "ic := x(0) = A, D(x)(0) = 0;" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 22 "assume( m > 0, k > 0);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 34 "soln := dsolve( \{de0, ic\}, x(t)); " }}}}{SECT 
0 {PARA 4 "" 0 "" {TEXT -1 36 "Extracting and Plotting the Solution" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "motion := rhs( soln);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 62 "exptmotion := subs( \{ m = 0.840, k = 5.476, A = .15
\}, motion);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot(exptmo
tion, t = 0..10); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "What is amp
litude of the motion?  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 49 "How does that relate to the initial conditions?  " }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "What is \+
the ratio of the sucessive maxima of the motion?" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Does that make sense in v
iew of the physical assumptions for this system?" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 4 "" 
0 "" {TEXT -1 42 "Finding the Period Graphically-Numerically" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 159 "N
ote to take care to set your numerical solution range narrow enough to
 catch only the extreme point you desire, but not so narrow to exclude
 an extreme point." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 52 "critpt1 := fsolve( diff(exptmotion, t) = 0,
 t=2..3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "critpt2 := fso
lve( diff(exptmotion, t) = 0, t = 4..5);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "period := critpt2 - critpt1;" }}}}{SECT 0 {PARA 4 "" 
0 "" {TEXT -1 31 "Finding the Period Symbolically" }}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 129 "How would you find the symbolic equation for the \+
length of one period of oscillation from the motion?  (Set up on scrat
ch paper!)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 67 "Solve this equation symbolically for yourself!  (on scratch pap
er.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "period := sqrt(m/k)*2*Pi;
 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "subs( \{m=0.840, k = 5
.476\}, period);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(\"
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 57 "Damped H
armonic Motion, Friction Proportional to Velocity" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
48 "Setting Up and Solving hte Differential Equation" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 57 "de1 := m*diff( x(t), t,t) + b*diff(x(t), \+
t) + k*x(t) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ic := x
(0) = A, D(x)(0) = 0; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "s
oln := A*exp(-b*t/(2*m))*cos(sqrt(4*m*k - b^2)/(2*m) * t);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "subs( x(t) = soln, de1);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Is soln a
 solution of the differential equation?" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "dsolve( de1, x(t));" }
}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 40 "Investigating Period, Amplitud
e, Damping" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "Substitu
te in the experimental values of A, k, m from above and a trial value \+
for b and plot the solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 97 "Make sure that you plot over a long enough inte
rval that you can see four complete oscillations. " }}{PARA 0 "" 0 "" 
{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Wh
at is the numerical value of the period?" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "What are the magni
tudes of four succesive peak amplitudes, computed numerically?" }}
{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 59 "What is the ratio of the successive amplitudes?  (3 ratio
s)" }}{PARA 0 "" 0 "" {TEXT -1 52 "Are the ratios constant, increasing
 , or decreasing?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 92 "Are the ratios of amplitudes from your ex
perimental data constant, increasing or decreasing?" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 162 "Do you believe this is an adequate mathe
matical model of the experimental situation, espeically with regard to
 modelling the friction as proprotional to velocity?" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 20 
"Extra Credit Problem" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 202 "Extra Cr
edit (10 points):  By changing the assumptions on the way friction is \+
built into the mathematical model, can you construct a new, improved m
athematical model which more closely matches the data?" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "You have to consult
 with me to find out how to solve more difficult mathematical equation
s with Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}}}{MARK "7 4 1 3 0
" 0 }{VIEWOPTS 1 1 0 1 1 1803 }