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{SECT 0 {PARA 261 "" 0 "" {TEXT -1 28 "PENDULUM PERIOD CALCULATIONS" }
}{PARA 261 "" 0 "" {TEXT 261 16 "PH 421 Fall 2002" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning,
 the name changecoords has been redefined\n" }}}{PARA 257 "" 0 "" 
{TEXT -1 55 "The period of a pendulum oscillating to a maxmum angle " 
}{XPPEDIT 18 0 "theta[max]" "6#&%&thetaG6#%$maxG" }{TEXT -1 19 " can b
e written as:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{XPPEDIT 18 0 "T=sqrt(2)/Pi*T[0]*Int(1/sqrt(cos(theta)-cos(theta[max])
),theta=0..theta[max])" "6#/%\"TG**-%%sqrtG6#\"\"#\"\"\"%#PiG!\"\"&F$6
#\"\"!F*-%$IntG6$*&F*F*-F'6#,&-%$cosG6#%&thetaGF*-F86#&F:6#%$maxGF,F,/
F:;F/&F:6#F?F*" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" 
{TEXT -1 6 "where " }{XPPEDIT 18 0 "T[0]" "6#&%\"TG6#\"\"!" }{TEXT -1 
271 " is the period for small angle.  Maple can calculate this integra
l exactly as shown below, but it can be quite slow.  If we recognize i
t as an elliptical integral, then we can use the more efficient call t
o the complete ellpitic integral of the first kind K(k) defined by" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "Ellip
ticK(k) = Int(1/sqrt(1-t^2)/sqrt(1-k^2*t^2),t=0..1)" "6#/-%*EllipticKG
6#%\"kG-%$IntG6$*(\"\"\"F,-%%sqrtG6#,&F,F,*$%\"tG\"\"#!\"\"F4-F.6#,&F,
F,*&F'F3F2F3F4F4/F2;\"\"!F," }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
256 6 "where " }{XPPEDIT 257 0 "k = sin(theta[max]/2)" "6#/%\"kG-%$sin
G6#*&&%&thetaG6#%$maxG\"\"\"\"\"#!\"\"" }{TEXT 258 48 " in our case.  \+
The period can then be written as" }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "
T=2* T[0] /Pi*EllipticK(sin(theta[max]/2)" "6#/%\"TG**\"\"#\"\"\"&F$6#
\"\"!F'%#PiG!\"\"-%*EllipticKG6#-%$sinG6#*&&%&thetaG6#%$maxGF'F&F,F'" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 10 "A:=1.0472:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "evalf(sqrt(2)/P
i*Int(1/sqrt(cos(x)-cos(A)),x=0..A));" }}{PARA 8 "" 1 "" {TEXT -1 51 "
Error, (in evalf/int) unable to handle singularity\n" }}}{PARA 265 "" 
0 "" {TEXT -1 53 "Note the small imaginary piece, but we can ignore it
." }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "evalf(2/
Pi*EllipticK(sin(A/2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+sB=t5!
\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 101 
"Now make a list of the values of the ratio of the period to the small
 angle period for use in a plot." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 166 "for A from 10 to 170 by 10 do\n    printf(`Amplitude = %g deg
 (%g rad),   Period Ratio = %g`,A,evalf(A*Pi/180),evalf(2/Pi*EllipticK
(sin(A*Pi/360)))):\n    lprint():\nod:" }}{PARA 6 "" 1 "" {TEXT -1 70 
"Amplitude = 10        deg (.174533 rad),   Period Ratio = 1.001907NUL
L" }}{PARA 6 "" 1 "" {TEXT -1 70 "Amplitude = 20        deg (.349066 r
ad),   Period Ratio = 1.007669NULL" }}{PARA 6 "" 1 "" {TEXT -1 70 "Amp
litude = 30        deg (.523599 rad),   Period Ratio = 1.017409NULL" }
}{PARA 6 "" 1 "" {TEXT -1 70 "Amplitude = 40        deg (.698132 rad),
   Period Ratio = 1.031341NULL" }}{PARA 6 "" 1 "" {TEXT -1 70 "Amplitu
de = 50        deg (.872665 rad),   Period Ratio = 1.049783NULL" }}
{PARA 6 "" 1 "" {TEXT -1 71 "Amplitude = 60        deg (1.047198 rad),
   Period Ratio = 1.073182NULL" }}{PARA 6 "" 1 "" {TEXT -1 71 "Amplitu
de = 70        deg (1.22173  rad),   Period Ratio = 1.102145NULL" }}
{PARA 6 "" 1 "" {TEXT -1 71 "Amplitude = 80        deg (1.396263 rad),
   Period Ratio = 1.137493NULL" }}{PARA 6 "" 1 "" {TEXT -1 71 "Amplitu
de = 90        deg (1.570796 rad),   Period Ratio = 1.180341NULL" }}
{PARA 6 "" 1 "" {TEXT -1 72 "Amplitude = 100        deg (1.745329 rad)
,   Period Ratio = 1.232229NULL" }}{PARA 6 "" 1 "" {TEXT -1 72 "Amplit
ude = 110        deg (1.919862 rad),   Period Ratio = 1.29534 NULL" }}
{PARA 6 "" 1 "" {TEXT -1 72 "Amplitude = 120        deg (2.094395 rad)
,   Period Ratio = 1.372881NULL" }}{PARA 6 "" 1 "" {TEXT -1 72 "Amplit
ude = 130        deg (2.268928 rad),   Period Ratio = 1.469819NULL" }}
{PARA 6 "" 1 "" {TEXT -1 72 "Amplitude = 140        deg (2.443461 rad)
,   Period Ratio = 1.594446NULL" }}{PARA 6 "" 1 "" {TEXT -1 72 "Amplit
ude = 150        deg (2.617994 rad),   Period Ratio = 1.762204NULL" }}
{PARA 6 "" 1 "" {TEXT -1 72 "Amplitude = 160        deg (2.792527 rad)
,   Period Ratio = 2.007507NULL" }}{PARA 6 "" 1 "" {TEXT -1 72 "Amplit
ude = 170        deg (2.96706  rad),   Period Ratio = 2.439363NULL" }}
}{PARA 0 "" 0 "" {TEXT 260 57 "Or make an array of these values for pl
otting right here." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "period
_ratio:=[seq(evalf(2/Pi*EllipticK(sin(i*6*Pi/360))),i=1..29)]:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "exact_plot:=pointplot([seq([
n*6,period_ratio[n]],n=1..27)],connect=true,color=black,thickness=3):
" }}}{PARA 262 "" 0 "" {TEXT -1 44 "The approximate expression for the
 period is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" 
{XPPEDIT 18 0 "T=T[0]*(1+theta[max]^2/16)" "6#/%\"TG*&&F$6#\"\"!\"\"\"
,&F)F)*&&%&thetaG6#%$maxG\"\"#\"#;!\"\"F)F)" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "approx_plot:=plot(
1+(x*Pi/180  )^2/16,x=0..180,color=black,thickness=1):" }}}{PARA 264 "
" 0 "" {TEXT -1 106 "Maple printouts have trouble with line thicknesse
s, so note that the upper curve is the exact calcutation." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "display(exact_plot,approx_plot,lab
els=[`Angle(deg)`,`Ratio`],view=[0..180,0.95..1.8],font=[COURIER,12]);
" }}{PARA 13 "" 1 "" {GLPLOT2D 686 303 303 {PLOTDATA 2 "6'-%'CURVESG6%
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