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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
256 59 "Example:  Pendulum animation using the equations of motion:" }
}{PARA 0 "" 0 "" {TEXT -1 55 "                             mu'' + g * \+
 u' + k u =f(t)" }}{PARA 0 "" 0 "" {TEXT -1 93 "(Note:  g is usually r
eserved for gravity, but Maple has \"gamma\" reserved for something el
se," }}{PARA 0 "" 0 "" {TEXT -1 58 "which is why we're using \"g\" for
 the friction coefficient)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f:=
0;    #This is the forcing function" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "m:=1; g:=0;  k:=1;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "de:=m*diff(u(t),t$2)+g*diff(u(t),t)+k*u(t)=f;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "U:=rhs(dsolve(\{de,u(0)=1, D(u)(0)=-1\},u(t)));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "dU:=diff(U,t);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 16 "with(plottools):" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "Here, we show the \"Actu
al\" Pendulum-  For the animation, I took the length of the pendulum t
o be fixed at 1." }}{PARA 0 "" 0 "" {TEXT -1 107 "Remember that the \"
u\" we solved for is the angle the pendulum makes with the vertical ax
is-  An interesting" }}{PARA 0 "" 0 "" {TEXT -1 109 "question is:  Wha
t are the (x,y) coordinates of the pendulum's bob?  It's answered belo
w in the animation set" }}{PARA 0 "" 0 "" {TEXT -1 87 "of commands.  Y
ou will not have to change anything here, except for \"noffm\"  and \"
divs\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "numsteps:=100; divs:=5;
   #The time values go from 0 to numsteps/5 = 20" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 32 "for i from 0 by 1 to numsteps do" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 24 "theta:=subs(t=i/divs,U);" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 46 "x1:=evalf(sin(theta)); y1:=evalf(-cos(theta));" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "rod[i]:=curve([[0,0],[x1,y1]]):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ms[i]:=disk([x1,y1],0.02,color=red)
:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "anima[i]:=display(\{rod[i],ms[
i]\});" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 86 "display([seq(anima[i],i=0..numsteps)],insequen
ce=true,scaling=constrained, axes=none);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "Here is the \"Phase Plan
e\" plot of u(t) versus u'(t)-  Same behavior, just a different visual
ization." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "plot([U,dU,t=0..20]);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Here is the standard visu
alization of u(t) and u'(t):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot([U,dU],t=0..20,color=[r
ed,black]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "
17" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }