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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 9 "Gyroscope" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "We visualize the motio
n of the simple gyroscope discussed in the text." }}{PARA 0 "" 0 "" 
{TEXT -1 516 "The objective is to explain how Euler's equations for th
e angular velocity vector which defines the motion of the gyroscope ar
e giving this quantity in terms of components in the (almost) body fra
me (a frame in which the moments of inertia don't change, and in which
 the external gravitational torque is still easily defined). The task \+
is then to connect these components to the laboratory frame. This is d
one through two rotation angles, theta and phi, as explained in Exampl
e 13.7 in J.M. Knudsen and P.G. Hjorth: " }{TEXT 257 31 "Elements of N
ewtonian Mechanics" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 88 "Thi
s version was corrected for a previous error, and is up to date as of \+
April 12, 2004." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 157 "An interesting extension for this worksheet would be the
 introduction of frictional torques to explain phenomena such as the s
leeper mode for a spinning top." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 22 "restart: with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning
, the protected names norm and trace have been redefined and unprotect
ed\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "We begin with the full an
gular momentum vector (bottom of page 338):" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 105 "Lv:=vector([-diff(theta(t),t)*I1,-diff(phi(t),t)*
sin(theta(t))*I2,(diff(phi(t),t)*cos(theta(t))+w0)*I3]);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#LvGK%'vectorG6#7%,$*&-%%diffG6$-%&thetaG6#%\"
tGF1\"\"\"%#I1GF2!\"\",$*(-F,6$-%$phiGF0F1F2-%$sinG6#F.F2%#I2GF2F4*&,&
*&F7F2-%$cosGF=F2F2%#w0GF2F2%#I3GF2Q)pprint106\"" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 97 "The angular velocity vector (big-Omega with the sp
in removed): it defines the 'almost body frame\"" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 91 "Wv:=vector([-diff(theta(t),t),-diff(phi(t),t)
*sin(theta(t)),diff(phi(t),t)*cos(theta(t))]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#WvGK%'vectorG6#7%,$-%%diffG6$-%&thetaG6#%\"tGF0!\"\"
,$*&-F+6$-%$phiGF/F0\"\"\"-%$sinG6#F-F8F1*&F4F8-%$cosGF;F8Q)pprint116
\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "The rate of change of the a
ngular momentum vector is given by the cross product of " }{TEXT 19 5 
"Omega" }{TEXT -1 5 " and " }{TEXT 19 1 "L" }{TEXT -1 64 " (page 339) \+
plus the rate of change in the 'almost body frame'. " }}{PARA 0 "" 0 "
" {TEXT -1 22 "Get the cross product:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "WcrL:=map(simplify,crossprod(Wv,Lv));" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#>%%WcrLGK%'vectorG6#7%,$*(-%%diffG6$-%$phiG6#%\"tG
F1\"\"\"-%$sinG6#-%&thetaGF0F2,(*(%#I3GF2F+F2-%$cosGF5F2F2*&F:F2%#w0GF
2F2*(F+F2F;F2%#I2GF2!\"\"F2FA,$*&-F,6$F6F1F2,(*(F+F2F;F2%#I1GF2F2F9FAF
=FAF2FA,$**FDF2F+F2F3F2,&F@FAFHF2F2FAQ)pprint126\"" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 104 "We define a vector with the time derivative of a
ngular momentum in the non-inertial 'almost-body' frame:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Ldotv:=map(expand,map(diff,op(Lv),t
));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&LdotvGK%'vectorG6#7%,$*&-%%d
iffG6$-%&thetaG6#%\"tG-%\"$G6$F1\"\"#\"\"\"%#I1GF6!\"\",&*(-F,6$-%$phi
GF0F2F6-%$sinG6#F.F6%#I2GF6F8**-F,6$F=F1F6-%$cosGFAF6-F,6$F.F1F6FBF6F8
,&*(%#I3GF6F;F6FFF6F6**FLF6FDF6F?F6FHF6F8Q)pprint136\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 40 "Now we have the total rate of change of \+
" }{TEXT 19 1 "L" }{TEXT -1 4 " ( (" }{TEXT 19 10 "dL/dt|_lab" }{TEXT 
-1 87 " which is equal to the torque) as expressed in unit vectors of \+
the (almost) body frame:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 
"LdotS:=map(simplify,evalm(Ldotv+WcrL));" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%&LdotSGK%'vectorG6#7%,**&-%%diffG6$-%&thetaG6#%\"tG-%\"$G6$F1
\"\"#\"\"\"%#I1GF6!\"\"**)-F,6$-%$phiGF0F1F5F6-%$sinG6#F.F6%#I3GF6-%$c
osGFAF6F8**F;F6F?F6FBF6%#w0GF6F8**F:F6FCF6F?F6%#I2GF6F6,,*(-F,6$F=F2F6
F?F6FHF6F8**F;F6FCF6-F,6$F.F1F6FHF6F8**F;F6FCF6FNF6F7F6F8**FNF6FBF6F;F
6FCF6F6*(FNF6FBF6FFF6F6,**(FBF6FKF6FCF6F6**FBF6F;F6F?F6FNF6F8**FNF6F;F
6F?F6FHF6F6**F;F6F?F6FNF6F7F6F8Q)pprint146\"" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 122 "Instead of Euler's equations we apply directly the ang
ular momentum theorem: the gravitational torque is always about the " 
}{TEXT 19 1 "x" }{TEXT -1 11 " (1-) axis:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 35 "eq1:=LdotS[1]=-M*g*l*sin(theta(t));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$eq1G/,**&-%%diffG6$-%&thetaG6#%\"tG-%\"$G6$F.\"
\"#\"\"\"%#I1GF3!\"\"**)-F)6$-%$phiGF-F.F2F3-%$sinG6#F+F3%#I3GF3-%$cos
GF>F3F5**F8F3F<F3F?F3%#w0GF3F5**F7F3F@F3F<F3%#I2GF3F3,$**%\"MGF3%\"gGF
3%\"lGF3F<F3F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eq2:=Ldot
S[2]=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/,,*(-%%diffG6$-%$ph
iG6#%\"tG-%\"$G6$F.\"\"#\"\"\"-%$sinG6#-%&thetaGF-F3%#I2GF3!\"\"**-F)6
$F+F.F3-%$cosGF6F3-F)6$F7F.F3F9F3F:**F<F3F>F3F@F3%#I1GF3F:**F@F3%#I3GF
3F<F3F>F3F3*(F@F3FEF3%#w0GF3F3\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "eq3:=LdotS[3]=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%$eq3G/,**(%#I3G\"\"\"-%%diffG6$-%$phiG6#%\"tG-%\"$G6$F0\"\"#F)-%$cosG
6#-%&thetaGF/F)F)**F(F)-F+6$F-F0F)-%$sinGF7F)-F+6$F8F0F)!\"\"**F?F)F;F
)F=F)%#I2GF)F)**F;F)F=F)F?F)%#I1GF)FA\"\"!" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 58 "Now we insert some values for the moments of inertia, etc
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "l:=1: g:=10: M:=1: I1:
=1/4+1: I2:=I1: I3:=1/2: w0:=20:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
54 "Initial condition: a horizontal gyro aligned with the " }{TEXT 19 
1 "x" }{TEXT -1 34 "-axis in the lab and being let go." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "th0:=Pi/2: ph0:=0: thdot0:=0: phido
t0:=0:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "The pair " }{TEXT 19 7 "
eq1/eq3" }{TEXT -1 31 " doesn't work! This is because " }{TEXT 19 3 "e
q3" }{TEXT -1 69 " represents just conservation of the 3-component of \+
angular momentum:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "eq3;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&#\"\"\"\"\"#F'*&-%%diffG6$-%$phi
G6#%\"tG-%\"$G6$F0F(F'-%$cosG6#-%&thetaGF/F'F'F'*&#F'F(F'*(-F+6$F-F0F'
-%$sinGF6F'-F+6$F7F0F'F'!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 151 "t:='t': sol:=dsolve(\{eq1,eq2,theta(0)=th0,D(theta)(
0)=thdot0,phi(0)=ph0,D(phi)(0)=phidot0\},\{theta(t),phi(t)\},numeric,m
ethod=gear,output=listprocedure);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%$solG7'/%\"tGf*6#F'6%%$resG%%dataG%)solnprocG6#%aoCopyright~(c)~1993~
by~the~University~of~Waterloo.~All~rights~reserved.G6\"C&>8%`6$%$GetG%
$SetGb6#%+thismoduleG6#%%DataGF06$%$GetG%$SetGF0F0F0F0%+_m38796472G6#%
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%#opG6#F\\p>FT-FD6#/Feo7$FjnF\\p@$FXOFhnFMF0F0F0/-%$phiGF)f+6#%\"tG6'F
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%%evalG6$FTFjn0FTQ)procnameF0OFT-FK6$FM7$FOFeoC$F`u@$FXO&FT6#\"\"#-FK6
$FM7$FPFfoC$FhtFbv3-FY6$FdrF^o-FK6$-Fbo6#FdrFdoC$@%-FY6$-F]pFawFen>FT-
FD6#/Feo7$Fcp-Fep6#Ffw>FT-FD6#/Feo7$FfvFfwFbv/FdrQ.solnprocedureF0O-Fi
u6#FD/FdrQ(sysvarsF0O7'F'F_q-%%diffG6$F_qF'-%&thetaGF)-F]y6$F_yF'@%09!
%(unknownGO-.FeyFcoC$>8(-%(pointtoG6#&-FF6#Q0soln_proceduresF0FevO-.F
\\zFcoZ%C$>FT-FDFawFdvF0YF0F0F0F0F0/F\\yf+6#%\"tG6'F+F,F-Feq%/diff(phi
(t),t)GFgqF0C)>F[rF\\r>F\\rF^r@%F`r>FdrFer>FdrFjr>F3F4>FDFE@$F^sC$@.Fe
sC$@%Fet>FTFit>FTFau@&FcuOFhuF[vOFTF^vC$F`\\l@$FXO&FT6#\"\"$FgvC$F_\\l
Fe\\lF[wC$@%Fdw>FTFhw>FT-FD6#/Feo7$Fi\\lFfwFe\\lFcxOFfxFhxOF[y@%FdyOFh
yC$>F\\z-F^z6#&FazFh\\lOFezZ%C$>FTFjzFg\\lF0YF0F0F0F0F0/F_yf+6#%\"tG6'
F+F,F-Feq%)theta(t)GFgqF0C)>F[rF\\r>F\\rF^r@%F`r>FdrFer>FdrFjr>F3F4>FD
FE@$F^sC$@.FesC$@%Fet>FTFit>FTFau@&FcuOFhuF[vOFTF^vC$Fe_l@$FXO&FT6#\"
\"%FgvC$Fd_lFj_lF[wC$@%Fdw>FTFhw>FT-FD6#/Feo7$F^`lFfwFj_lFcxOFfxFhxOF[
y@%FdyOFhyC$>F\\z-F^z6#&FazF]`lOFezZ%C$>FTFjzF\\`lF0YF0F0F0F0F0/Fayf+6
#%\"tG6'F+F,F-Feq%1diff(theta(t),t)GFgqF0C)>F[rF\\r>F\\rF^r@%F`r>FdrFe
r>FdrFjr>F3F4>FDFE@$F^sC$@.FesC$@%Fet>FTFit>FTFau@&FcuOFhuF[vOFTF^vC$F
jbl@$FXO&FT6#\"\"&FgvC$FiblF_clF[wC$@%Fdw>FTFhw>FT-FD6#/Feo7$FcclFfwF_
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0F0F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Theta:=subs(sol,
theta(t)): Phi:=subs(sol,phi(t)):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 82 "plot(['Theta(t)','Phi(t)'],t=0..20,color=[blue,green]
,numpoints=1000,thickness=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 934 362 
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"" 0 "" {TEXT -1 227 "For the plot of the gyro we need to connect to t
he laboratory frame. We simply define the Cartesian components of the \+
position vector in terms of arm length l, polar (nutation) angle theta
, and azimuthal (precession) angle phi." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "with(plottools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
139 "We record an animation, i.e., we define a time sequence of 3d plo
ts, and use the arrow command to depict the omega-vector in the lab fr
ame." }}{PARA 0 "" 0 "" {TEXT -1 44 "We use the numerical solution ava
ilable for " }{TEXT 19 8 "theta(t)" }{TEXT -1 5 " and " }{TEXT 19 6 "p
hi(t)" }{TEXT -1 74 " to construct the time evolution of the vector re
presenting the gyro axis." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
185 "N:=300: for i from 1 to N do: t:=0.03*i: Prec:=vector([l*sin(Thet
a(t))*cos(Phi(t)),l*sin(Theta(t))*sin(Phi(t)),l*cos(Theta(t))]) ; ll[i
] := arrow(vector([0,0,0]), Prec, .2, .4, .1):\nod:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 104 "i:='i': plots[display3d]([seq(ll[i],i=1.
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0 "" {TEXT -1 171 "Change the initial conditions and obtain different \+
motion: e.g., by pushing the gyro into pure precession without nutatio
n. What happens when you reverse the initial push?" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 11 "Exercise 2:" }}{PARA 0 "
" 0 "" {TEXT -1 188 "Consider a torque free gyroscope. Start the motio
n with a push in the nutational direction (hammer blow on the gyro axl
e). Observe the motion. You can make the gyro torque-free by setting \+
" }{TEXT 19 3 "g=0" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" 
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