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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "table.ms   " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "A mass m on a fricti
onless tabletop is connected by a string of length H through" }}{PARA 
0 "" 0 "" {TEXT -1 56 "a hole in the table to a mass M hanging below t
he table." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
29 "Done via lagrange's equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 27 "Students might be asked to " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "a) Identify the tw
o constants of the motion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 87 "b) Given m and M and  initial r and phi dot values
, predict the maximum r analytically." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 44 "c) Check the answer with the maple so
lution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
81 "d) Determine the maximum tension in the cord from the conditions o
f the problem. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart;with(plots)
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "alias(r=r(t),phi=phi(t
)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "x1:=r*cos(phi): y1:=
r*sin(phi): z:=H-r:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "KE:=
simplify(m/2*(diff(x1,t)^2+diff(y1,t)^2)+M/2*diff(z,t)^2);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "PE:=-M*g*z;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 9 "L:=KE-PE;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 60 "valuesin:=\{diff(r,t)=rdot,diff(phi,t)=phidot,r=ra,ph
i=phia\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Lsvel:=subs(va
luesin,L); # L with dummy variables" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 96 "pr:=diff(Lsvel,rdot); pphi:=diff(Lsvel,phidot); # gen
eralized momenta w/ symbols (pj = dLdqjdot)" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 56 "dLdr:=diff(Lsvel,ra);dLdphi:=diff(Lsvel,phia); #  \+
dL/dqj" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "valuesback:=\{rdo
t=diff(r,t),phidot=diff(phi,t),ra=r,phia=phi\};" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 29 "pra:=    subs(valuesback,pr);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "pphia:=subs(valuesback,pphi); # gen
eralized p's 'live'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "dLdr
a:=subs(valuesback,dLdr);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
79 "dLdphia:=subs(valuesback,dLdphi); # zero, so pphia is a constant o
f the motion." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "eqr:= diff
(pra,t)-dLdra = 0; # lagrange's eqn in r." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 50 "eqphi:=diff(pphia,t)-dLdphia = 0; # ditto for phi.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "vals:=\{r=rs(t),phi=phi
s(t),m=1,M=6,g=98/10\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "
eqra:=    subs(vals,eqr);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
49 "eqphia:=subs(vals,eqphi); # new set of named vars" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "The squa
re brackets are important in the following dsolve command to keep the \+
solutions in correct order" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 94 "sol:=dsolve(\{eqra,rs(0)=1,D(rs)(0)=0,eqphi
a,phis(0)=0,D(phis)(0)=30\},[rs(t),phis(t)],numeric);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "sol(1);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 53 "rp:= z -> rhs(sol(z)[2]); phip:= z -> rhs(sol(z)[4]
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "phidotp:=z->rhs(sol(z
)[5]); # phi dot for plotting" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 55 "plot(phidotp,0..2,color=black,title=` d/dt phi  vs t`);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "h:=plot(rp,0..2,color=blue):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "j:=plot(phip,0..2,color
=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 50 "display(\{h,j\},title=` r(blue) and phi(r
ed) vs t`);" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 72 "plot([rp,phip,0..6],coords=polar,title=`polar pl
ot; dphi/dt=30 @ r=1`); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{MARK "30 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }