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{SECT 0 {PARA 256 "" 0 "" {TEXT 259 44 "Calculations  for Ziggy's mass
-spring system" }}{PARA 257 "" 0 "" {TEXT -1 13 "Math 2250-3, " }}
{PARA 258 "" 0 "" {TEXT -1 12 "Nov 19, 2003" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 35 "The two mass, three spring sys
tem. " }{TEXT -1 304 " \nData:  Each ball mass is 23 grams.  Each spri
ng mass is 9 grams.  (Remember, and this is a defect, our model assume
s massless springs.)  The springs are identical, and a mass of 50 gram
s stretches the spring 15.0 centimeters.  (I checked this estimate bef
ore class -but we can do it again if you like.)" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Estimate k (from the data
 above):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "k:=solve(k*.15=.05*9.81,k);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"kG$\"&+F$!\"%" }}}{PARA 0 "" 0 "" {TEXT -1 54 "Time
 the two natural periods (which we discuss below):" }}{PARA 0 "" 0 "" 
{TEXT -1 154 "(For the fast one, in my office, I got 100 cycles in abo
ut 33 sseconds.  For the slow one I got 100 cycles in about 61 seconds
.  What do we get in class?)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Here's the mode
l:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "A:=matrix(2,2,[-2*k/m, k/m,
k/m,-2*k/m]);\n  #this should be the \"A\" matrix you get for\n  #our \+
two-mass, three-spring system.  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"AG-%'matrixG6#7$7$,$*&%\"kG\"\"\"%\"mG!\"\"!\"#F+7$F+F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvects(A);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$7%,$*&%\"kG\"\"\"%\"mG!\"\"F)F'<#-%'vectorG6#7$F'F'7%,$
F%!\"$F'<#-F,6#7$F)F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 47 "Predict the two natural periods from th
e model:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 257 7 "ANSWER:" }{TEXT -1 210 "  If you do the model correctly y
ou will come up with natural periods of  .304 and .527 seconds.  I pre
dict that these periods are too small compared to what actually happen
ed in our experiment.  What happened?" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 256 12 "EXPLANATION:" }{TEXT -1 733 "  The s
prings actually have mass, equal to 9 grams each.  This is on the same
 order of magnitude as the ball masses, and causes the actual experime
nt to run more slowly than our model predicts.  In order to be more ac
curate the total energy of our model  must account for the kinetic ene
rgy of the springs.  You actually have the tools to model this more-co
mplicated situation, using the ideas of total energy discussed in sect
ion 5.6, and a \"little\" Calculus.  You can carry out this analysis, \+
assuming that the spring velocity at a point on the  spring linearly i
nterpolates the velocity of the wall and mass (or mass and mass) which
 bounds it.  It turns out that this gives the same eigenvectors, but d
ifferent eigenvalues, namely" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6
#/&%'lambdaG6#\"\"\",$*&%\"kGF',&%\"mGF'*&#\"\"&\"\"'F'&F,6#%\"sGF'F'!
\"\"F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'lambdaG6#\"\"#,$*&%\"kG
\"\"\",&%\"mGF+*&#F+F'F+&F-6#%\"sGF+F+!\"\"!\"$" }}}{PARA 0 "" 0 "" 
{TEXT -1 108 "If you use these values, then you get period predictions
 of  .61 and .33 seconds per cycle.  Is that closer?" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 258 389 " I of
fer a prestigeous Math Department T-shirt (left over from our summer p
rogram 2 years ago) to the first student or group of students to submi
t a correct derivation of this \"corrected\" model , using total energ
y ideas.  (One T-shirt total).  I also offer a 10 midterm points (tota
l) to the successful student or group.  I am happy to have further dis
cussion with interested individuals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 2 "  " }}}{MARK "28 1" 83 }{VIEWOPTS 1 1 0 
1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }