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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;" }}}
{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);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "DE
2:=diff(y(t),t$2)+g/L*y(t)=-2*w*sf*diff(x(t),t);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 5 "Here " }{TEXT 257 1 "w" }{TEXT -1 37 " is the rotat
ion speed of the Earth, " }{TEXT 259 1 "L" }{TEXT -1 33 " is the pendu
lum arm length, and " }{TEXT 258 1 "g" }{TEXT -1 35 " is the gravitati
onal acceleration." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 47 "w:=evalf(2*Pi/(24*60*60)); # in inverse sec
onds" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "L:=69; #69 meters" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "g:=9.8;" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 60 "We pick some initial conditions: an initial dis
placement 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;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "sol:=dsolve(\{DE1,DE2,IC\},\{x(t),y(t)\});" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assign(sol);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 65 "We have a simple analytic solution for th
e two coupled equations!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 
"evalf([x(t),y(t)]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "We can pl
ot the two components 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..2000,color=[red,blue],thickness=3);" }}}{EXCHG {PARA 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
),t=0..25],thickness=3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{MARK "0 0 0" 17 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
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