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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 17 "Foucault Pendulum" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "We progra
m equations (6.80) from Knudsen and Hjorth (" }{TEXT 264 31 "Elements \+
of Newtonian Mechanics" }{TEXT -1 84 ", Springer 2001, 3rd ed.) in ord
er to display the motion of the pendulum (at Paris)." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 282 "The equations were der
ived under certain simplifying assumptions. It would be interesting to
 append a solution of the full equations in order to demonstrate that \+
the approximations were justified, and also in order to have equations
 which are also valid for higher rotations speeds." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The latitude at Paris cor
responds to sin(phi)=0.75" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 
"restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "sf:=3/4;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#sfG#\"\"$\"\"%" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 49 "DE1:=diff(x(t),t$2)+g/L*x(t)=2*w*sf*diff(y(t
),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$DE1G/,&-%%diffG6$-%\"xG6#%
\"tG-%\"$G6$F-\"\"#\"\"\"*&*&%\"gGF2F*F2F2%\"LG!\"\"F2,$*&%\"wGF2-F(6$
-%\"yGF,F-F2#\"\"$F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "DE2
:=diff(y(t),t$2)+g/L*y(t)=-2*w*sf*diff(x(t),t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$DE2G/,&-%%diffG6$-%\"yG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*
&*&%\"gGF2F*F2F2%\"LG!\"\"F2,$*&%\"wGF2-F(6$-%\"xGF,F-F2#!\"$F1" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Here " }{TEXT 257 1 "w" }{TEXT -1 
37 " is the rotation speed of the Earth, " }{TEXT 259 1 "L" }{TEXT -1 
33 " is the pendulum arm length, and " }{TEXT 258 1 "g" }{TEXT -1 35 "
 is the gravitational acceleration." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "w:=evalf(2*Pi/(24*60*60));
 # in inverse seconds" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG$\"+=_?
ss!#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "L:=69; #69 meters
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG\"#p" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 7 "g:=9.8;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"
gG$\"#)*!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "We pick some ini
tial conditions: an initial displacement in " }{TEXT 261 1 "x" }{TEXT 
-1 16 ", we are on the " }{TEXT 260 1 "y" }{TEXT -1 54 "-axis, and the
 pendulum bob starts with zero velocity." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "IC:=x(0)=L/10,D(x)(0)=0,y(0)=0,D(y)(0)=0;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#ICG6&/-%\"xG6#\"\"!#\"#p\"#5/--%\"DG6#F(F
)F*/-%\"yGF)F*/--F16#F5F)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
38 "sol:=dsolve(\{DE1,DE2,IC\},\{x(t),y(t)\});" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%$solG<$/-%\"xG6#%\"tG,&*&,&*$-%%sqrtG6#\";*3$RSrz6**Q
&f6o&\"\"\"#\"+\\B\\sK\";g[DudZvbH(Q,%\\#\"#p\"#?F3F3-%$cosG6#,&F*#!+$
yI34\"\"/++++++?*(#F3F@F3F*F3F/F3F3F3F3*&,&F7F3*&#\"+\\B\\sKF6F3F.F3!
\"\"F3-F;6#,&F*#\"+$yI34\"F@*(FBF3F*F3F/F3F3F3F3/-%\"yGF),&*&F-F3-%$si
nGF<F3F3*&FDF3-FUFJF3FH" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "
assign(sol);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "We have a simple \+
analytic solution for the two coupled equations!" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 19 "evalf([x(t),y(t)]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7$,&-%$cosG6#,$%\"tG$\"+Rz7oP!#5$\"+'H*\\]M!\"**&$\"+/2
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5F3!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "We can plot the two c
omponents side-by-side after one half hour for about three minutes:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "plot([x(t),y(t)],t=1800..2
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0 "" 0 "" {TEXT -1 27 "The initially non-existent " }{TEXT 263 1 "y" }
{TEXT -1 48 "-component has picked up over 10 percent of the " }{TEXT 
262 1 "x" }{TEXT -1 124 "-amplitude. Let us graph the actual trajector
y at the very beginning emphasizing the beginning of the rotation (pre
cession):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot([x(t),y(t
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