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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 32 "Exercise 4: Coupled Oscil
lations" }}{PARA 256 "" 0 "" {TEXT -1 9 "Solutions" }}{PARA 19 "" 0 "
" {TEXT -1 92 "Due at 5pm on Tuesday, September 28, in Matt's mailbox.
 (No separate problem set this week.)" }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 55 "Review of lecture material, Monday/Wednesday Sept 20/22" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 125 "For two masses 1 and 2, connected by three spr
ings a, b, and c, the differential equations of motion for the three m
asses are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "eq1:=m1*diff(x
1(t),t,t)=-ka*x1(t)-kb*(x1(t)-x2(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%$eq1G/*&%#m1G\"\"\"-%%diffG6$-%#x1G6#%\"tG-%\"$G6$F/\"\"#F(,&*&%#
kaGF(F,F(!\"\"*&%#kbGF(,&F,F(-%#x2GF.F7F(F7" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 51 "eq2:=m2*diff(x2(t),t,t)=-kc*x2(t)+kb*(x1(t)-x2(t))
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/*&%#m2G\"\"\"-%%diffG6$-%
#x2G6#%\"tG-%\"$G6$F/\"\"#F(,&*&%#kcGF(F,F(!\"\"*&%#kbGF(,&-%#x1GF.F(F
,F7F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "We simplifed this by \+
making both masses the same value as well as the three spring constant
s, and also by defining " }{XPPEDIT 18 0 "omega0^2 = k/m;" "6#/*$%'ome
ga0G\"\"#*&%\"kG\"\"\"%\"mG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 6 "m1:=m:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "m2
:=m:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "ka:=m*omega0^2:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "kb:=m*omega0^2:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "kc:=m*omega0^2:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 19 "eq1:=expand(eq1/m);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$eq1G/-%%diffG6$-%#x1G6#%\"tG-%\"$G6$F,\"\"#,&*&)%'om
ega0GF0\"\"\"F)\"\"\"!\"#*&F3F5-%#x2GF+F6F6" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 19 "eq2:=expand(eq2/m);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$eq2G/-%%diffG6$-%#x2G6#%\"tG-%\"$G6$F,\"\"#,&*&)%'omega0GF0\"
\"\"F)\"\"\"!\"#*&F3F5-%#x1GF+F6F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 27 "Into these, we substituted " }{XPPEDIT 18 0 "x1 = A1*exp(i*omeg
a*t);" "6#/%#x1G*&%#A1G\"\"\"-%$expG6#*(%\"iGF'%&omegaGF'%\"tGF'F'" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "x2 = A2*exp(i*omega*t);" "6#/%#x2G*
&%#A2G\"\"\"-%$expG6#*(%\"iGF'%&omegaGF'%\"tGF'F'" }{TEXT -1 57 ". We \+
then have the two algebraic equations for A1 and A2:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 44 "eq1a:=(omega^2-2*omega0^2)*A1+omega0^2*A2
=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eq1aG/,&*&,&*$)%&omegaG\"\"#
\"\"\"\"\"\"*$)%'omega0GF,F-!\"#F.%#A1GF.F.*&F0F-%#A2GF.F.\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "eq2a:=omega0^2*A1+(omega^2-2
*omega0^2)*A2=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eq2aG/,&*&)%'om
ega0G\"\"#\"\"\"%#A1G\"\"\"F-*&,&*$)%&omegaGF*F+F-*$F(F+!\"#F-%#A2GF-F
-\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 214 "These are two equation
s for A1 and A2 which have the only solution A1=A2=0, unless they are \+
actually the same equation. So, to avoid the trivial solution, we make
 them the same equation by multiplying the first by " }{XPPEDIT 18 0 "
omega0^2;" "6#*$%'omega0G\"\"#" }{TEXT -1 19 " and the second by " }
{XPPEDIT 18 0 "omega^2-2*omega0^2;" "6#,&*$%&omegaG\"\"#\"\"\"*&\"\"#F
'*$%'omega0G\"\"#F'!\"\"" }{TEXT -1 96 " and then forcing the coeffici
ents of A2 to be the same. This gives us an equation to solve for " }
{XPPEDIT 18 0 "omega^2;" "6#*$%&omegaG\"\"#" }{TEXT -1 85 ". The resul
t is the frequencies of the two \"normal modes\" of this coupled oscil
lator." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "sol:=solve((omega
^2-2*omega0^2)^2=omega0^4,omega^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%$solG6$*$)%'omega0G\"\"#\"\"\",$F&\"\"$" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 130 "Either of these two solutions gives us the same equation
 from the two we wrote down for A1 and A2. The first solution implies \+
that" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "simplify(subs(omega
^2=sol[1],eq1a));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&)%'omega0G\"
\"#\"\"\"%#A1G\"\"\"!\"\"*&F&F)%#A2GF+F+\"\"!" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 153 "This means that A1=A2. The two masses move together w
ith the same amplitude. The spring in between them never stretches. Th
e second solution implies that" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 36 "simplify(subs(omega^2=sol[2],eq1a));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,&*&)%'omega0G\"\"#\"\"\"%#A1G\"\"\"F+*&F&F)%#A2GF+F+
\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 299 "This means that A1=-A2.
 The two masses move against each other, but again with the same ampli
tude. We can see what these normal modes actually mean in terms of the
 equations of motion by defining new quantities x(t) and y(t) and subs
tituting them into the differential equations for x1(t) and x2(t)." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "xyx1x2:=\{x(t)=(x1(t)+x2(t)
)/2,y(t)=x1(t)-x2(t)\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "
x1x2xy:=solve(xyx1x2,\{x1(t),x2(t)\}):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "neweqs:=subs(x1x2xy,\{eq1,eq2\}):" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 46 "solve(neweqs,\{diff(x(t),t,t),diff(y(t),t,t
)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/-%%diffG6$-%\"yG6#%\"tG-%
\"$G6$F+\"\"#,$*&)%'omega0GF/\"\"\"F(\"\"\"!\"$/-F&6$-%\"xGF*F,,$*&F2F
4F:F5!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 172 "These differential
 equations are much simpler, and show that the coordinate y(t), which \+
represents the distance between the two masses, oscillates with a freq
uency that is " }{XPPEDIT 18 0 "sqrt(3);" "6#-%%sqrtG6#\"\"$" }{TEXT 
-1 94 " times faster than the coordinate x(t), which represents the ce
nter of mass of the two masses." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
181 "The actual motion of the two masses can be solved directly in MAP
LE. Take the case where mass 1 starts from rest at x1=A, while mass tw
o starts from rest at equilibrium. Then we get" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 77 "paths:=dsolve(\{eq1,eq2,x1(0)=A,x2(0)=0,D(x1)(
0)=0,D(x2)(0)=0\},\{x1(t),x2(t)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%&pathsG<$/-%#x1G6#%\"tG,&*&-%$cosG6#*&%'omega0G\"\"\"F*F2F2%\"AGF2#
F2\"\"#*&-F.6#*(-%%sqrtG6#\"\"$\"\"\"F1F>F*F>F2F3F>F4/-%#x2GF),&F,F4F6
#!\"\"F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "A:=5: omega0:=1
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "plot([rhs(paths[1]),rh
s(paths[2])],t=0..4*2*Pi/omega0,color=[green,magenta]);" }}{PARA 13 "
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 278 "That is, the center of mass (blue
) and the difference between the positions (red) oscillate as sine wav
es all by themselves. It is also possible to calculate the kinetic ene
rgy of the two masses individually from the solution, and also the pot
ential energy in the three springs." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "assign(paths);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 24 "K1:=m*diff(x1(t),t)^2/2:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "K2:=m*diff(x2(t),t)^2/2:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 47 "U:=k*x1(t)^2/2+k*x2(t)^2/2+k*(x1(t)-x2(t))^2/2:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "U:=subs(k=m*omega0^2,U):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "m:=1:" }}}{EXCHG {PARA 0 ">
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$F\\^\\lFdd\\l7$$\"1`<[-TPWCFTFaiv7$F^cqFaiv7$Fd^\\lFaiv7$FccqFaiv7$F]
dqFaiv7$FgdqFaiv7$$\"1a$y4aww[#FTFb\\]l7$$\"1JJ5YIQ*[#FTFb\\]l7$$\"13z
A^&*3\"\\#FTFaiv7$F\\eqFb\\]l7$$\"1PAgm!4i\\#FTFb\\]l7$FaeqFb\\]l7$$\"
1ml(>eG8]#FTFaiv7$$\"1U85(3NI]#FTFb\\]l7$$\"1>hA#fTZ]#FTFb\\]l7$FfeqFb
\\]l7$$\"1rcZ-Y:3DFTFaiv7$$\"1Z/g26')4DFTFb\\]l7$$\"1C_s7wc6DFTFaiv7$F
[fqFbb\\l-F`fq6&FbfqF(F(F(-%+AXESLABELSG6$Q\"t6\"%!G-%%VIEWG6$;F($\"+B
TF8DFefq%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 
0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "The sum of the three energie
s is indeed a constant." }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 58 "Exerci
se: Initial conditions corresponding to normal modes" }}{PARA 4 "" 0 "
" {TEXT -1 58 "Use MAPLE to solve the coupled differential equations f
or " }{XPPEDIT 18 0 "x1(t);" "6#-%#x1G6#%\"tG" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "x2(t);" "6#-%#x2G6#%\"tG" }{TEXT -1 5 " for " }{TEXT 
256 4 "two " }{TEXT -1 221 "new sets of initial conditions. For one of
 these, start the oscillator off so that it excites only one of the no
rmal modes, and the other using the other set of initial conditions. I
n each case, show that the behavior of " }{XPPEDIT 18 0 "x(t);" "6#-%
\"xG6#%\"tG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y(t);" "6#-%\"yG6#%\"
tG" }{TEXT -1 20 " is what you expect." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 
"eq1:=diff(x1(t),t,t)=-2*omega0^2*x1(t)+omega0^2*x2(t):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "eq2:=diff(x2(t),t,t)=-2*omega0^2*x2
(t)+omega0^2*x1(t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "xyx1
x2:=\{x(t)=(x1(t)+x2(t))/2,y(t)=x1(t)-x2(t)\}:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 36 "x1x2xy:=solve(xyx1x2,\{x1(t),x2(t)\}):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "paths:=dsolve(\{eq1,eq2,x1(0
)=A,x2(0)=A,D(x1)(0)=0,D(x2)(0)=0\},\{x1(t),x2(t)\});" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%&pathsG<$/-%#x1G6#%\"tG*&-%$cosG6#*&%'omega0G\"
\"\"F*F1F1%\"AGF1/-%#x2GF)F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 16 "A:=5: omega0:=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "p
lot([rhs(paths[1]),rhs(paths[2])],t=0..4*2*Pi/omega0,color=[green,mage
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{EXCHG {PARA 5 "" 0 "" {TEXT -1 335 "That is, when the two masses are \+
started out with the same displacement, the oscillate with exactly the
 same paths and only the center of mass coordinate oscillates. Now do \+
it the other way, and start them out with equal displacements but on o
pposite sides of equilibrium. The opposite will happen instead, and th
e frequency is higher." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "p
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>&y%[ybH*F0$!1OIU'eW(R#*F07$Fin$!1$4(ei5&e0)F07$$\"1\"od;-F'e&*F0$!1iJ
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F07$$\"1\\u?J\"Gk#=Fao$\"19a9'oB<w*F07$F]r$\"1UC`10nV%*F07$$\"1p*[LI%z
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$$\"1#*fk=\"*QB>Fao$!1]CbPdQ:KF07$$\"1+D%H`)**H>Fao$!1J)*Rv`'fF%F07$Fc
r$!1$46hMo0G&F07$$\"1p0Po<`\\>Fao$!1)p#Rgk\"R.(F07$$\"1I@]*ebC'>Fao$!1
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\"1A*oKDo]*>Fao$!1\"*=s-:)*****F07$$\"1\"fsZFL=+#Fao$!1fhu@/mL**F07$$
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$\"1cUz#[&[o?Fao$!10MKX3nlHF07$$\"1oTa-5+v?Fao$!1oQk<5Mr=F07$$\"1!3%HA
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Yfqu\\\"F07$$\"1dbz@1@,@Fao$\"1e?(4$4/EEF07$$\"15sz\"oty5#Fao$\"1)zYf1
u'>PF07$$\"1k))zTn`9@Fao$\"1^#*Qd,#Qw%F07$$\"1=0!=!)*>@@Fao$\"1\"37au(
eWdF07$$\"1DQ!=#f_M@Fao$\"1,q7[6\"[Y(F07$F_s$\"1MI`?W5*y)F07$$\"1*f3[
\"y1h@Fao$\"1m.h#Hy@k*F07$$\"1m+\"ye$Gu@Fao$\"1-V?OJB#***F07$$\"1L/1Jv
ex@Fao$\"1#=l\"fRT)***F07$$\"1+3Ju9*3=#Fao$\"1-2r(yg=(**F07$$\"1n6c<a>
%=#Fao$\"1'QlZbgE\"**F07$$\"1L:\"3O*\\(=#Fao$\"1G)yq225#)*F07$$\"1nAJZ
s5%>#Fao$\"1wC^=fkT&*F07$Fbs$\"1o$\\0uKu8*F07$$\"1hUmc!fG@#Fao$\"11FS]
Hu(3)F07$$\"1@b^zH+DAFao$\"1-ZYD)[:o'F07$$\"1#ymB!p9PAFao$\"1aQ0;[$3)
\\F07$Fes$\"1VK6'on01$F07$$\"16BBy*\\iD#Fao$\"15&p&Q&=O*=F07$$\"1zlCJ
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Fao$!1)fI,XCEq\"F07$$\"1#Q*G!f'3%G#Fao$!1/#[wdd^(GF07$$\"1\\OIVd/\"H#F
ao$!1an*fDof+%F07$$\"1<zJ'*[+)H#Fao$!1>kM$)oky]F07$Fhs$!1kv.LuixgF07$$
\"1Qa;[FT<BFao$!1<=\"yy?fj(F07$$\"1\"p)*pWh)HBFao$!17&=7b&eS))F07$$\"1
V>$e95BM#Fao$!1O?ZiA$ej*F07$F[t$!1*>$o)4K[)**F07$$\"1#Q<?2x!eBFao$!1rP
,w9(*****F07$$\"1o&p$*H&RhBFao$!1<w!39)3#)**F07$$\"1a<sENrkBFao$!1ip&)
[6CJ**F07$$\"1SR2a<.oBFao$!1$f+7T)fZ)*F07$$\"16$y(3#oYP#Fao$!1KJeMz8$e
*F07$$\"1$o#[jYI\"Q#Fao$!1$Q\")y6(>#>*F07$$\"1E9*GdxXR#Fao$!1@42oX6`!)
F07$F^t$!1Ti?zxG!\\'F07$$\"1#zWiT-UT#Fao$!1Gr3hDv:cF07$$\"1;%*=]Vb?CFa
o$!1))e-R@JtYF07$$\"1RS8%G1pU#Fao$!1*y=FUiVn$F07$$\"1i'y!=#eKV#Fao$!1B
#eui#)4j#F07$$\"1'GB?:5'RCFao$!1^]G8#*yb:F07$$\"14z'f3ifW#Fao$!1ZBByK$
yh%F,7$$\"1LD\"*>SJ_CFao$\"1L*[\\-k!yjF,7$Fat$\"1,'e=$QoH<F07$$\"1iiNu
>\\lCFao$\"1n[()\\)y$zGF07$$\"1n`&[*zJsCFao$\"1&**zqrr)))RF07$$\"1tWN:
S9zCFao$\"14Fg#frE/&F07$$\"1yN&e.qf[#Fao$\"1aY@E_1EgF07$$\"1*y^o2A'*\\
#Fao$\"1'o%[hv)ys(F07$Fdt$\"1#y[)zki***)F0-Fgt6&FitFjtF(F(-%+AXESLABEL
SG6$Q\"t6\"%!G-%%VIEWG6$;F($\"+BTF8DF\\u%(DEFAULTG" 1 2 0 1 0 2 9 1 4 
2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 
171 "This time, it is the difference coordinate that oscillates and th
e center of mass does not move, while the individual masses oscillate \+
against each other, all as expected." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 286 "Finally, we pointed out that there are standard techniques for
 finding the frequencies and \"normal coordinates\" (which we called x
(t) and y(t)) using matrix techniques. The procedure is called the \"e
igenvalue\" method, and is done withing MAPLE using the \"linalg\" pac
kage. For this case," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rest
art;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "Omega:=matrix([[-2*
omega0^2,omega0^2],[omega0^2,-2*omega0^2]]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&OmegaG-%'matrixG6#7$7$,$*$)%'omega0G\"\"#\"\"\"!\"#F
+7$F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }
}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}
{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "eigenvalues(Omega);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*$)%'omega0G\"\"#\"\"\"!\"$,$F$!\"\"
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "eigenvectors(Omega);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%,$*$)%'omega0G\"\"#\"\"\"!\"$\"\"\"
<#-%'vectorG6#7$!\"\"F+7%,$F%F1F+<#-F.6#7$F+F+" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 41 "The eigenvalues are the negatives of the " }
{XPPEDIT 18 0 "omega^2;" "6#*$%&omegaG\"\"#" }{TEXT -1 547 " values fo
r the normal modes, and the eigenvectors corresponding to each of thes
e two eigenvalues (the two component vector in the last element of eac
h of the two lists returned from \"eigenvectors\") are the relative va
lues of the \"A\" coefficients, and tell us the way to constuct the no
rmal coordinates. As shown here, the first eigenvalue has the two mass
es oscillating with the same amplitude but opposite to each other, and
 the second has the two oscillating at the same amplitude and in phase
, just as we found with the detailed algebra above." }}}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 19 "Extension Exercises" }}{EXCHG {PARA 4 "" 
0 "" {TEXT -1 75 "1) Two masses are connected to each other by a sprin
g with spring constant " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 
-1 86 ", and each is connected to a fixed wall on either side by sprin
g with spring constant " }{XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT -1 148 "
. (This is the same as the case worked in class, but the spring in the
 middle no longer has the same spring constant as the the two on the o
utside.)" }}{PARA 15 "" 0 "" {TEXT -1 80 "Write the equations of motio
n and show that the two normal modes have frequency " }{XPPEDIT 18 0 "
omega;" "6#%&omegaG" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "omega^2 = o
mega0^2;" "6#/*$%&omegaG\"\"#*$%'omega0G\"\"#" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "omega^2 = omega0^2+beta^2;" "6#/*$%&omegaG\"\"#,&*$%'om
ega0G\"\"#\"\"\"*$%%betaG\"\"#F+" }{TEXT -1 7 " where " }{XPPEDIT 18 
0 "omega0^2 = k/m;" "6#/*$%'omega0G\"\"#*&%\"kG\"\"\"%\"mG!\"\"" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "beta^2 = 2*alpha/m;" "6#/*$%%betaG
\"\"#*(\"\"#\"\"\"%&alphaGF)%\"mG!\"\"" }{TEXT -1 1 "." }}{PARA 15 "" 
0 "" {TEXT -1 45 "Show that the equations decouple if we write " }
{XPPEDIT 18 0 "x = (x1+x2)/2;" "6#/%\"xG*&,&%#x1G\"\"\"%#x2GF(F(\"\"#!
\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = x1-x2;" "6#/%\"yG,&%#x1G
\"\"\"%#x2G!\"\"" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 19 "Find
 the solutions " }{XPPEDIT 18 0 "x1;" "6#%#x1G" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "x2;" "6#%#x2G" }{TEXT -1 43 " where mass 1 starts from \+
rest at position " }{XPPEDIT 18 0 "x0;" "6#%#x0G" }{TEXT -1 44 " and m
ass 2 starts from rest at equilibrium." }}{PARA 15 "" 0 "" {TEXT -1 
37 "Make a plot of the two solutions for " }{XPPEDIT 18 0 "t = 0;" "6#
/%\"tG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "200*pi/omega0;" "6#*(
\"$+#\"\"\"%#piGF%%'omega0G!\"\"" }{TEXT -1 7 " using " }{XPPEDIT 18 
0 "beta/omega0 = .2;" "6#/*&%%betaG\"\"\"%'omega0G!\"\"$\"\"#!\"\"" }
{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 118 "Plot the kinetic energ
ies of the two masses for the same conditions.  Briefly describe what \+
is going on here in words." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
8 "restart;" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 90 "We'll start from t
he same place as the beginning of class, and make the new substitution
s." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "eq1:=m1*diff(x1(t),t,
t)=-ka*x1(t)-kb*(x1(t)-x2(t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 51 "eq2:=m2*diff(x2(t),t,t)=-kc*x2(t)+kb*(x1(t)-x2(t)):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "m1:=m: m2:=m: ka:=k: kc:=k: \+
kb:=alpha:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "k:=m*omega0^2
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "alpha:=m*beta^2/2:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "eq1:=simplify(eq1/m); eq2:=
simplify(eq2/m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/-%%diffG6$
-%#x1G6#%\"tG-%\"$G6$F,\"\"#,(*&)%'omega0GF0\"\"\"F)\"\"\"!\"\"*&)%%be
taGF0F5F)F5#F7F0*&F9F5-%#x2GF+F6#F6F0" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%$eq2G/-%%diffG6$-%#x2G6#%\"tG-%\"$G6$F,\"\"#,(*&)%'omega0GF0\"\"
\"F)\"\"\"!\"\"*&)%%betaGF0F5-%#x1GF+F6#F6F0*&F9F5F)F5#F7F0" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "Omega:=matrix([[-omega0^2-be
ta^2/2,beta^2/2],[beta^2/2,-omega0^2-beta^2/2]]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&OmegaG-%'matrixG6#7$7$,&*$)%'omega0G\"\"#\"\"\"!\"\"
*$)%%betaGF.F/#F0F.,$F1#\"\"\"F.7$F5F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning
, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new
 definition for trace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ei
genvalues(Omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*$)%'omega0G\"
\"#\"\"\"!\"\",&F$F)*$)%%betaGF'F(F)" }}}{EXCHG {PARA 5 "" 0 "" {TEXT 
-1 101 "These are the (negative of the squares) of the normal mode fre
quencies, as indicated in the question." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 48 "xyx1x2:=\{x(t)=(x1(t)+x2(t))/2,y(t)=x1(t)-x2(t)\}:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "x1x2xy:=solve(xyx1x2,\{x1(
t),x2(t)\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "neweqs:=sub
s(x1x2xy,\{eq1,eq2\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "f
actor(solve(neweqs,\{diff(x(t),t,t),diff(y(t),t,t)\}));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#<$/-%%diffG6$-%\"yG6#%\"tG-%\"$G6$F+\"\"#,$*&F(
\"\"\",&*$)%'omega0GF/\"\"\"F2*$)%%betaGF/F7F2F2!\"\"/-F&6$-%\"xGF*F,,
$*&F5F7F?F2F;" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 88 "The equations in
deed decouple, with the frequencies we got from the eigenvalue solutio
n." }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 104 "MAPLE will do a better job
 with the solution if we give it a new variable for the second eigenfr
equency." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "beta:=sqrt(omeg
a1^2-omega0^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG*$-%%sqrtG6
#,&*$)%'omega1G\"\"#\"\"\"\"\"\"*$)%'omega0GF-F.!\"\"F." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "paths:=dsolve(\{eq1,eq2,x1(0)=A,D(x
1)(0)=0,x2(0)=0,D(x2)(0)=0\},\{x1(t),x2(t)\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&pathsG<$/-%#x1G6#%\"tG,&*&-%$cosG6#*&%'omega0G\"\"\"
F*F2F2%\"AGF2#F2\"\"#*&-F.6#*&%'omega1GF2F*\"\"\"F2F3F;F4/-%#x2GF),&F,
F4F6#!\"\"F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "beta:='beta
':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "paths:=subs(omega1=sq
rt(beta^2+omega0^2),paths):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
34 "A:=5: omega0:=1: beta:=0.2*omega0:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 91 "plot([rhs(paths[1]),rhs(paths[2])],t=0..200*Pi/omega0
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