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{SECT 0 {PARA 256 "" 0 "" {TEXT 258 43 "Calculations  for a 2 mass- 3 \+
spring system" }}{PARA 257 "" 0 "" {TEXT -1 13 "Math 2250-1, " }}
{PARA 258 "" 0 "" {TEXT -1 17 "November 26, 2008" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 35 "The two mass, three spri
ng system. " }{TEXT -1 316 " \nData:  Each ball mass is 50 grams.  Eac
h spring mass is 6 grams.  (Remember, and this is a defect, our model \+
assumes massless springs.)  The springs are \"identical\", and an extr
a mass of 50 grams stretches the spring 18.0 centimeters from equilibr
ium.  (We can recheck this.).  Thus the spring constant is given by" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Digits:=5:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(k*.18=.05*9.8,k);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#$\"&As#!\"%" }}}{PARA 0 "" 0 "" {TEXT -1 60 
"Let's time the two natural periods (which we discuss below):" }}
{PARA 0 "" 0 "" {TEXT -1 183 "(For the fast one, in my office, I got 5
0 cycles in about 25.14 seconds.  (Hard to count this one!) For the sl
ow one I got 20 cycles in about 18.03 seconds.  What do we get in clas
s?)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Here's t
he model:" }}{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  #o
ur two-mass, three-spring system.  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%\"AG-%'matrixG6#7$7$,$*(\"\"#\"\"\"%\"kGF-%\"mG!\"\"F0*&F.F-F/F07$F
1F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvects(A);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$7%,$*&%\"kG\"\"\"%\"mG!\"\"F)F'<#-%'ve
ctorG6#7$F'F'7%,$*(\"\"$F'F&F'F(F)F)F'<#-F,6#7$F)F'" }}}{PARA 0 "" 0 "
" {TEXT -1 47 "Predict the two natural periods from the 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 -1 216 "  If you do the model correctly and our data a
grees with my office attempts, we will come up with natural periods of
  .49 and .85 seconds.  I predict that the real natural periods are a \+
little longer.  What happened?" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT 256 12 "EXPLANATION:" }{TEXT -1 866 "  The \+
springs actually have mass, equal to 6 grams each.  This is not quite \+
the same order of magnitude as the mass masses, and causes the actual \+
experiment to run more slowly than our model predicts.  In order to be
 more accurate the total energy of our model  must account for the kin
etic energy of the springs.  You actually have the tools to model this
 more-complicated situation, using the ideas of total energy discussed
 in section 5.6, and a \"little\" Calculus.  You can carry out this an
alysis, like I sketched for the single mass, single spring oscillator \+
(nov4.pdf), assuming that the spring velocity at a point on the  sprin
g linearly interpolates the velocity of the wall and mass (or mass and
 mass) which bounds it.  It turns out that this gives a non-diagonal M
 matrix, and ultimately an A-matrix the same eigenvectors, but differe
nt eigenvalues, namely" }}{PARA 262 "" 0 "" {XPPEDIT 18 0 "lambda[1] =
 -k/(m+5/6*m[s]);" "6#/&%'lambdaG6#\"\"\",$*&%\"kGF',&%\"mGF'*&#\"\"&
\"\"'F'&F,6#%\"sGF'F'!\"\"F4" }{TEXT -1 0 "" }}{PARA 263 "" 0 "" 
{XPPEDIT 18 0 "lambda[2] = -3*k/(m+1/2*m[s]);" "6#/&%'lambdaG6#\"\"#,$
*&%\"kG\"\"\",&%\"mGF+*&#F+F'F+&F-6#%\"sGF+F+!\"\"!\"$" }{TEXT -1 2 " \+
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 57 "If you use these values, then you get per
iod predictions " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "m:=.05:
\nms:=.006;\nk:=2.722;\nOmega1:=sqrt(k/(m+(5/6)*ms));\nOmega2:=sqrt(3*
k/(m+.5*ms));\nT1:=evalf(2*Pi/Omega1);\nT2:=evalf(2*Pi/Omega2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#msG$\"\"'!\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"kG$\"%AF!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'
Omega1G$\"&].(!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Omega2G$\"&7C
\"!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T1G$\"&;$*)!\"&" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#T2G$\"&@1&!\"&" }}}{PARA 0 "" 0 "" {TEXT 
-1 51 "of  .89 and .51 seconds per cycle.  Is that closer?" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 2 "  " }}}{MARK "14 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 
1 }{PAGENUMBERS 0 1 2 33 1 1 }