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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 63 "First we i
nitialize all of the packages we'll use in this study" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "with(VariationalCalculu
s):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(ODEtools):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "with( plots ): with( plottools ): \+
" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has be
en redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 59 "Warning, the names arro
w and translate have been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "PDEtools[declare]( (theta,phi)(t), prime=t):" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%J~theta(t)~will~now~be~displayed~as~t
heta~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%F~phi(t)~will~now~be~displa
yed~as~phi~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%dpderivatives~with~re
spect~to:~t~of~functions~of~one~variable~will~now~be~displayed~with~'~
G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}
{PARA 7 "" 1 "" {TEXT -1 45 "Warning, the name adjoint has been redefi
ned\n" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names nor
m and trace have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }{TEXT 257 51 "Here we define our position vector
s and velocities." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "x[1]:
=l[1]*sin(theta(t)):\ny[1]:=-l[1]*cos(theta(t)):\nx[2]:=-l[2]*sin(thet
a(t)):\ny[2]:=l[2]*cos(theta(t)):\nx[4]:=x[1]-l[4]*sin(theta(t)+phi(t)
):\ny[4]:=y[1]+l[4]*cos(theta(t)+phi(t)):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 86 "R[1]:=vector(2,[x[1],y[1]]):\nR[2]:=vector(2,[x[2],
y[2]]):\nR[4]:=vector(2,[x[4],y[4]]):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 107 "`R[1]'` := map(diff,R[1],t):\n`R[2]'` := map(diff,R[
2],t):\n`R[4]'` := map(diff,R[4],t):#the Velocity vectors" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 92 "Next, we calculate the ki
netic and potential energies of the system and find the Lagrangian." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "T := collect(simplify(m[1
]/2*innerprod(`R[4]'`,`R[4]'`)),[m[1],l[1],l[4]])+\ncollect(simplify(m
[2]/2*innerprod(`R[2]'`,`R[2]'`)),[m[1],m[2],l[2]]):\nT:=combine(T,tri
g):\n#kinetic energy" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "U:
=m[1]*g*y[4]+m[2]*g*y[2]:\nU:=collect(expand(U,trig),[m[1],m[2],l[1],l
[4],l[2]]):\nU:=collect(expand(U,trig),[m[1],m[2],l[1],l[4],l[2]]): \n
#potential energy" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "L:=co
llect(simplify(expand(T-U,trig)),[l[1],l[2],l[4],m[1],m[2],g,diff(thet
a(t),t),diff(diff(theta(t),t),t)]):\n#lagrangian" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }{TEXT 259 64 "Using the EulerLagrange package to
 find our equations for motion" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 42 "EuL:=EulerLagrange(L,t,[theta(t),phi(t)]):" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 26 "eqs:=remove(has,EuL,K[1]):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "eq1:=factor(simplify(op(select(has,
eqs,l[2]^2)))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "eq2:=fac
tor(simplify(op(remove(has,eqs,l[2]^2)))):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 15 "sys:=\{eq1,eq2\}:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }{TEXT 260 49 "Setting up the initial conditions and Parameter
s." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "INITS:=\{theta(0)=3*P
i/4,D(theta)(0)=0,phi(0)=Pi/4,D(phi)(0)=0\}:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 57 "PARAM:=[g=9.8,l[1]=10,l[2]=100,l[4]=21,m[1]=1000
,m[2]=1]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "sys2:=subs((
\{theta(t)=th,diff(theta(t),t)=dth,diff(diff(theta(t),t),t)=ddth,phi(t
)=ph,diff(phi(t),t)=dph,diff(diff(phi(t),t),t)=ddph\},sys)):" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 12 "" 1 "" {TEXT -1 0 "" 
}{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "ddphi:
=solvefor[ddph](op(op(remove(has,eliminate(sys2,ddth),ddth)))):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "ddthe:=solvefor[ddth](op(op(
remove(has,eliminate(sys2,ddph),ddph)))):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 20 "sys3:=\{ddphi,ddthe\}:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 139 "sys3:=subs(\{th=theta(t),dth=diff(theta(t),t),ddth
=diff(diff(theta(t),t),t),ph=phi(t),dph=diff(phi(t),t),ddph=diff(diff(
phi(t),t),t)\},sys3):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
261 74 "Evaluating the general equation for motion with respect to our
 parameters." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "sys4:=eval(
sys3,PARAM):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 84 "W
e use a numeric ODE solver to approximate the change in theta and thet
a' over time." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "sol:=dsolv
e(sys4 union INITS, numeric,output=listprocedure):" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 0 "" }{TEXT 264 51 "The animation and/or the strobe p
lot for the model." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "noffm
:=30:" }}}{PARA 13 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "divs:=10:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 625 "for i from 0 to noffm do\nx1:=eval
(l[1]*sin(rhs(sol[4](i/divs))),PARAM):\ny1:=eval(-l[1]*cos(rhs(sol[4](
i/divs))),PARAM):\nx2:=eval(-l[2]*sin(rhs(sol[4](i/divs))),PARAM):\ny2
:=eval(l[2]*cos(rhs(sol[4](i/divs))),PARAM):\nx4:=eval(x1-l[4]*sin(rhs
(sol[4](i/divs))+rhs(sol[2](i/divs))),PARAM):\ny4:=eval(y1+l[4]*cos(rh
s(sol[4](i/divs))+rhs(sol[2](i/divs))),PARAM): rod1[i]:=line([x2,y2],[
x1,y1]):\nrod2[i]:=line([x1,y1],[x4,y4]):\nstructure[i]:=polygon([[-20
,-70],[0,0],[20,-70],[-20,-70]]):\nma1[i]:=disk([x4,y4],2,colour=red):
\nma2[i]:=disk([x2,y2],1,colour=blue):\nanima[i]:=display(\{structure[
i],rod1[i],rod2[i],ma1[i],ma2[i]\}):\nend do:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 80 "display(seq(anima[i],i=0..noffm),insequence=true
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{TEXT -1 0 "" }{TEXT 263 38 "Plotting time versus angular velocity." }
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inged Counter-Weight\"):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 265 63 "Calculating the optimal release angle and the associated
 range." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "divs:=100:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "x_0:=seq(eval(-l[2]*sin(rhs(
sol[4](i/divs))),PARAM),i=150..250):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 63 "y_0:=seq(eval(l[2]*cos(rhs(sol[4](i/divs))),PARAM),i=
150..250):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "Dx:=seq(eval(
-l[2]*sin(rhs(sol[4](i/divs)))*rhs(sol[5](i/divs)),PARAM),i=150..250):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "Dy:=seq(eval(-l[2]*cos(
rhs(sol[4](i/divs)))*rhs(sol[5](i/divs)),PARAM),i=150..250):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "y_proj:=seq(-4.9*(t)^2+Dy[i-
149]*(t)+y_0[i-149],i=150..250):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "T:=seq(solvefor[t](y_proj[i]=0),i=1..100):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Ti:=seq(op(select(type,[rhs(
T[i][1]),rhs(T[i][2])],positive)),i=1..100):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "x_
proj:=seq(Dx[i]*Ti[i]+x_0[i],i=1..100):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 115 "theta_r:=seq(fnormal(180/evalf(Pi)*rhs(sol[4]((i+150
)/divs)),4),i=1..100):\nrng:=seq(fnormal(x_proj[i],4),i=1..100):" }}}
{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "expr:=seq([theta_r[i],rng[i]],i=1..100):\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "45 1 0" 0 }
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }