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{SECT 0 {PARA 258 "" 0 "" {TEXT -1 15 "Linear Pendulum" }}{PARA 256 "
" 0 "" {TEXT -1 10 "A. J. Meir" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 257 "" 0 "" {TEXT -1 46 "Copyright (C) A. J. Meir. All rights re
served." }}{PARA 257 "" 0 "" {TEXT -1 492 "This worksheet is for educa
tional use only. No part of this publication may be reproduced or tran
smitted for profit in any form or by any means, electronic or mechanic
al, including photocopy, recording, or any information storage and ret
rieval system without prior written permission from the author. Not fo
r profit distribution of the software is allowed without prior written
 permission, providing that the worksheet is not modified in any way a
nd full credit to the author is acknowledged." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 17 "Example: Pendulum" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 283 "An ordin
ary differential equation is an equation which involves an unknown fun
ction and its derivatives with respect to one independent variable. (A
 partial differential equation involves an unknown function and its pa
rtial derivatives with respect to several independent variables.)" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "The diffe
rential equation (initial value problem) describing a pendulum" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "m;" "
6#%\"mG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d^2/(dt^2);" "6#*&%\"dG\"\"#*
$%#dtGF%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }
{TEXT -1 3 " + " }{XPPEDIT 18 0 "c/l;" "6#*&%\"cG\"\"\"%\"lG!\"\"" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "d/dt;" "6#*&%\"dG\"\"\"%#dtG!\"\"" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 3 " + " }
{XPPEDIT 18 0 "m*g/l;" "6#*(%\"mG\"\"\"%\"gGF%%\"lG!\"\"" }{TEXT -1 1 
" " }{XPPEDIT 18 0 "sin(phi);" "6#-%$sinG6#%$phiG" }{TEXT -1 13 " = F(
t)      " }{XPPEDIT 18 0 "phi(0) = phi[0];" "6#/-%$phiG6#\"\"!&F%6#F'
" }{TEXT -1 6 "      " }{XPPEDIT 18 0 "d/dt;" "6#*&%\"dG\"\"\"%#dtG!\"
\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "phi(0) = phi[1];" "6#/-%$phiG6#\"
\"!&F%6#\"\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "we usual
ly approximate this nonlinear equation, by the linear equation" }}
{PARA 259 "" 0 "" {XPPEDIT 18 0 "m;" "6#%\"mG" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "d^2/(dt^2);" "6#*&%\"dG\"\"#*$%#dtGF%!\"\"" }{TEXT -1 
1 " " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 3 " + " }{XPPEDIT 18 
0 "c/l;" "6#*&%\"cG\"\"\"%\"lG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d
/dt;" "6#*&%\"dG\"\"\"%#dtG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "phi;
" "6#%$phiG" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "m*g/l;" "6#*(%\"mG\"\"
\"%\"gGF%%\"lG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "phi;" "6#%$phiG" 
}{TEXT -1 14 " = F(t)       " }{XPPEDIT 18 0 "phi(0) = phi[0];" "6#/-%
$phiG6#\"\"!&F%6#F'" }{TEXT -1 6 "      " }{XPPEDIT 18 0 "d/dt;" "6#*&
%\"dG\"\"\"%#dtG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "phi(0) = phi[1]
;" "6#/-%$phiG6#\"\"!&F%6#\"\"\"" }}{PARA 0 "" 0 "" {TEXT -1 45 "which
 is a good approximation when the angle " }{XPPEDIT 18 0 "phi;" "6#%$p
hiG" }{TEXT -1 16 " is small. Here " }{XPPEDIT 18 0 "m;" "6#%\"mG" }
{TEXT -1 25 " is the mass of the bob, " }{XPPEDIT 18 0 "c;" "6#%\"cG" 
}{TEXT -1 24 " the amount of damping, " }{XPPEDIT 18 0 "l;" "6#%\"lG" 
}{TEXT -1 25 " length of pendulum, and " }{XPPEDIT 18 0 "g;" "6#%\"gG
" }{TEXT -1 48 " acceleration of gravity. The initial condition " }
{XPPEDIT 18 0 "phi[0];" "6#&%$phiG6#\"\"!" }{TEXT -1 26 " is the initi
al angle and " }{XPPEDIT 18 0 "phi[1];" "6#&%$phiG6#\"\"\"" }{TEXT -1 
38 "the initial velocity. Below we assume " }{XPPEDIT 18 0 "F(t) = 0;
" "6#/-%\"FG6#%\"tG\"\"!" }{TEXT -1 1 "." }}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 0 "" }{TEXT 256 0 "" }{TEXT 257 14 "Maple commends" }{TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart:with(plott
ools):with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name c
hangecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "pendulum:=proc(m,c,l,g,T,phi0,phi1)" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 60 "local eq, ic, soln, p, L, C, N, seq1, seq2, h, s
upport, bob:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "global n, x, y, gra
ph:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "eq:=m*diff(phi(t),t,t)+(c/l)
*diff(phi(t),t)+(m*g/l)*phi(t)=0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
31 "ic:=phi(0)=phi0,D(phi)(0)=phi1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
34 "soln:=rhs(dsolve(\{eq,ic\},phi(t))):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "graph:=plot(\{soln,diff(soln,t)\},t=0..T,thickness=2)
:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p:=unapply(soln,t):" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 50 "L:=(x,y)->line([0,0],[x,y],thickness=3,co
lor=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "C:=(x,y)->disk([x,y],0
.3,color=blue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "N:=200:" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 43 "x:=l*sin(p((T/N)*n)):y:=-l*cos(p((T/N)*n)
):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "seq1:=seq(L(x,y),n=0..N):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "seq2:=seq(C(x,y),n=0..N):" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 46 "h:=line([-4,0],[4,0],thickness=5,color=gr
een):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "support:=display(seq1,inse
quence=true):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "bob:=display(seq2,
insequence=true):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "display(h,supp
ort,bob,scaling=constrained):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end
:" }}}}{PARA 0 "" 0 "" {TEXT -1 43 "We now prescribe values for the pa
rameters " }{XPPEDIT 18 0 "m;" "6#%\"mG" }{TEXT -1 2 ", " }{XPPEDIT 
18 0 "c;" "6#%\"cG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "l;" "6#%\"lG" }
{TEXT -1 6 ", and " }{XPPEDIT 18 0 "g;" "6#%\"gG" }{TEXT -1 154 ", giv
e the final time (how long the simulation should run, note longer run \+
times require more computer resources), the initial angle and initial \+
velocity " }{XPPEDIT 18 0 "phi[0];" "6#&%$phiG6#\"\"!" }{TEXT -1 5 " a
nd " }{XPPEDIT 18 0 "phi[1];" "6#&%$phiG6#\"\"\"" }{TEXT -1 15 ", resp
ectively." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "Mass:=1/20:Damp
ing:=0.03:Length:=4:G:=10:Time:=20:\nInitAngle:=1:InitVelocity:=-1:" }
}}{PARA 0 "" 0 "" {TEXT -1 112 "Below we obtain a simulation of the li
near pendulum and also a plot of the angle and angular velosity of the
 bob" }}{PARA 0 "" 0 "" {TEXT -1 22 "as a function of time." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "pendulum(Mass,Damping,Length,G,Time
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