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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 260 37 "Rolls and Strings and Yo-
Yo Solutions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 15 "A roll (radius " }{TEXT 258 1 "R" }{TEXT -1 7 ", mass " }{TEXT 
257 1 "M" }{TEXT -1 62 ") has rope wound around it which leads via a p
ulley to a mass " }{TEXT 256 1 "m" }{TEXT -1 26 " that falls under gra
vity." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 675 
"The point of this worksheet is to explain how the motion of the cente
r of mass (CM) is affected by the net force that acts on the roll: the
 string force, which applies where the string is unwound, and the stat
ic friction force which acts at the point of contact (assuming that th
ere is rolling without slipping). Both these forces contribute torques
 about the CM of the roll. This work shows that the direction of the f
riction force is difficult to predict. There are situations where it h
elps to push the CM forward, and acts against the rolling, and there a
re situtations where it helps the rolling, but acts against the motion
 of the CM. This is a consequence of applying:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 283 3 "(i)" }{TEXT -1 156 " the \+
CM theorem: all force vectors acting on the body are applying at the l
ocation of the CM, and determine the rate of change of the total linea
r momentum;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
284 4 "(ii)" }{TEXT -1 543 " the angular momentum theorem: the rate of
 change of angular momentum is controlled by the net torque that appli
es; one can use always the quantities about the CM, or about the point
 of contact (in which case friction does not contribute a torque), how
ever, only in the case of pure rolling (no slipping) is the angular ve
locity about the CM the same as the angular velocity about the point o
f contact. In the case of pure rolling the part of the roll that touch
es the ground at the point of contact has no motion, and static fricti
on applies." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
285 5 "(iii)" }{TEXT -1 142 " the case of pure rolling is characterize
d by a fixed relationship between the CM motion and the angular veloci
ty that describes the rotation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 286 4 "(iv)" }{TEXT -1 148 " the mass that falls
 under gravity undergoes a motion that is composed of motion of the CM
 and the additional effect of the unwinding of the string." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rest
art;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 149 "We define moments of ine
rtia for a disk about the CM, as well as about the point of contact O;
 the latter is obtained from the parallel-axis theorem." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "I_CM:=M/2*R^2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%I_CMG,$*&%\"MG\"\"\")%\"RG\"\"#F(#F(F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "I_O:=I_CM+M*R^2;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$I_OG,$*&%\"MG\"\"\")%\"RG\"\"#F(#\"\"$F+" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Motion of the center of mass: appl
y the forces at the CM, namely the friction force " }{TEXT 19 1 "f" }
{TEXT -1 23 ", and the string force " }{TEXT 19 1 "S" }{TEXT -1 115 ".
 We choose the same sign for both forces and let the calculationd deci
de which way the friction force will point.." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 15 "eq1:=M*Xdd=S+f;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%$eq1G/*&%\"MG\"\"\"%$XddGF(,&%\"SGF(%\"fGF(" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 94 "For pure rolling we can apply the torque about the
 contact point. The arm length is given by 2" }{TEXT 287 1 "R" }{TEXT 
-1 2 ". " }{TEXT 19 3 "Tdd" }{TEXT -1 59 " is the 2nd derivative of th
e rotation angle. The friction " }{TEXT 19 1 "f" }{TEXT -1 58 " contri
butes no torque, given that the arm length is zero." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 19 "eq2:=I_O*Tdd=2*R*S;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$eq2G/,$*(%\"MG\"\"\")%\"RG\"\"#F)%$TddGF)#\"\"$F,,$*
&F+F)%\"SGF)F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "The height of t
he mass falling under gravity is given by " }{TEXT 19 1 "Y" }{TEXT -1 
29 ". We choose a different mass " }{TEXT 261 2 "m " }{TEXT -1 33 "for
 the bob attached to the rope." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 17 "eq3:=m*Ydd=m*g-S;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq3G/*
&%\"mG\"\"\"%$YddGF(,&*&F'F(%\"gGF(F(%\"SG!\"\"" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 14 "The CM motion " }{TEXT 19 1 "X" }{TEXT -1 67 " is li
nked to the angular rotation in perfect rolling without slip:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eq4:=R*Tdd=Xdd;" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%$eq4G/*&%\"RG\"\"\"%$TddGF(%$XddG" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 58 "We need to connect the fall with the unra
velling of wire (" }{TEXT 19 5 "R*Tdd" }{TEXT -1 33 "), and with the m
otion of the CM." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "eq5:=Yd
d=Xdd+R*Tdd;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq5G/%$YddG,&%$XddG
\"\"\"*&%\"RGF)%$TddGF)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "How \+
many equations/unknowns? The unknowns are: " }{TEXT 19 19 "Ydd, Xdd, T
dd, S, f" }{TEXT -1 416 " - so we should use five equations. Equation \+
one is redundant for the determination of the motion, because we use t
he rotation equation about the point of contact. If we were to conside
r the Newton equation (angular momentum theorem) about the CM of the r
oll, the frictional torque would enter, and we would need to determine
 it as well. In the present case we use the first equation to determin
e the friction force." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "so
l:=solve(\{eq1,eq2,eq3,eq4,eq5\},\{Tdd,Xdd,Ydd,S,f\});" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$solG<'/%$TddG,$*&*&%\"mG\"\"\"%\"gGF,F,*&%\"RGF
,,&F+\"\")*&\"\"$F,%\"MGF,F,F,!\"\"\"\"%/%$YddG,$*&*&F+F,F-F,F,F0F5F1/
%\"SG,$*&*(F4F,F+F,F-F,F,F0F5F3/%\"fGF?/%$XddG,$F:F6" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 120 "Note that the friction force has the same sign
 as the string force, i.e., it is helping to propel the roll's CM forw
ard." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "assign(sol);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 24 "m:=1; M:=2; g:=10; R:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"
mG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG\"\"#" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"gG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"RG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "Ydd;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6##\"#S\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 4 "Xdd;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#?\"\"(" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "Tdd;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6##\"#?\"\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "We w
ant a graph of a circle with a radius vector indicating the location o
f a marker (" }{TEXT 19 3 "Tdd" }{TEXT -1 21 " is the acceleration)" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "theta:=unapply(1/2*Tdd*t^2
,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&thetaGR6#%\"tG6\"6$%)operat
orG%&arrowGF(,$*$)9$\"\"#\"\"\"#\"#5\"\"(F(F(F(" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 26 "X:=unapply(1/2*Xdd*t^2,t);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"XGR6#%\"tG6\"6$%)operatorG%&arrowGF(,$*$)9$\"\"#
\"\"\"#\"#5\"\"(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Y
:=unapply(1/2*Ydd*t^2,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YGR6#
%\"tG6\"6$%)operatorG%&arrowGF(,$*$)9$\"\"#\"\"\"#\"#?\"\"(F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with(plottools): with(plots)
:" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has b
een redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "arrowsca
le:=0.25:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 566 "c := t-> [cir
cle([X(t),R], R, color=red,thickness=3),line([X(t),R],[X(t)+R*cos(thet
a(t)+Pi/2),R-R*sin(theta(t)+Pi/2)],color=blue,thickness=3),line([X(t),
2*R],[8.2,2*R],color=grey,thickness=3),line([8.2,2*R],[8.2,-Y(t)],colo
r=grey,thickness=3),circle([8.2,-Y(t)],0.15,color=red,thickness=3),arr
ow([X(t),0], [X(t)+f*arrowscale,0], .2, .4, .1, color=magenta),arrow([
X(t),2*R], [X(t)+S*arrowscale,2*R], .2, .4, .1, color=cyan),arrow([8.2
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-Y(t)], [8.2,-Y(t)-m*g*arrowscale], .2, .4, .1, color=green)]:\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "PL:=[seq(display(c(2.2*i/60)
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "plots[display](op(PL),insequence=tr
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$F_al$!+Un)yO\"F`[l7$FdalFgb]p7$FgalFdb]p7$FjalFab]p7$F_blF^b]p7$FdblF
[b]p7$FiblFha]p7$F^clFea]p7$FcclFba]p7$FhclF_a]p7$F]dlF\\a]p7$FbdlFi`]
p7$Fgdl$!+Xr(4Q\"F`[l7$F\\elF``]p7$Fgdl$!+Urt%Q\"F`[l7$Fbdl$!+\"\\(e'Q
\"F`[l7$F]dl$!+6!z$)Q\"F`[l7$Fhcl$!+[M3!R\"F`[l7$Fccl$!+ARn\"R\"F`[l7$
F^cl$!+]`7$R\"F`[l7$Fibl$!+U[T%R\"F`[l7$Fdbl$!+i?_&R\"F`[l7$F_bl$!+\\&
HkR\"F`[l7$Fjal$!+\"*H7(R\"F`[l7$Fgal$!+_9f(R\"F`[l7$Fdal$!+Wv#yR\"F`[
l7$F_alFje]p7$Fj`lFge]p7$Fe`lFde]p7$F``lFae]p7$F[`lF^e]p7$Ff_lF[e]p7$F
a_lFhd]p7$F\\_lFed]p7$Fg^lFbd]p7$Fb^lF_d]p7$F]^lF\\d]p7$Fh]l$!+Trt%Q\"
F`[lFd`]pFjzFb[l-Fhil6&7&7$Fb_]pF\\jl7$Fb_]pF_jl7$$\"+&G9dB(F5F_jl7$F`
g]pF\\jl7%7$F`g]pFgjl7$$\"+r&G9F(F5Fa[l7$F`g]pF][mF_[mFc[m-F(6#7*F]g]p
F^g]pF_g]pFdg]pFeg]pFhg]pFbg]pF]g]p-Fhil6&7&7$Fb_]pF\\\\m7$Fb_]pF_\\m7
$$\"+G9dyyF5F_\\m7$Fbh]pF\\\\m7%7$Fbh]pFg\\m7$$\"+&G9d)zF5Fa\\l7$Fbh]p
F]]mF_[mF_]m-F(6#7*F_h]pF`h]pFah]pFfh]pFgh]pFjh]pFdh]pF_h]p-Fhil6&7&7$
Fh]mF``]p7$F[^mF``]p7$F[^m$!+s&GkG\"F`[l7$Fh]mFdi]p7%7$Fa^mFdi]p7$Fd\\
l$!+'G9dF\"F`[l7$Fe^mFdi]pF_[mF_]m-F(6#7*Fai]pFbi]pFci]pFhi]pFii]pF\\j
]pFfi]pFai]p-Fhil6&7&Fbi]pFai]p7$Fh]m$!+Vr&yg\"F`[l7$F[^mFdj]p7%7$Fe^m
Fdj]p7$Fd\\l$!+Vr&Gj\"F`[l7$Fa^mFdj]pF_[mFi_m-F(6#7*Fbi]pFai]pFcj]pFhj
]pFij]pF\\[^pFfj]pFbi]pF^`mFb`mF^`m" 1 2 0 1 10 0 2 9 1 4 1 1.000000 
47.000000 46.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve
 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur
ve 12" "Curve 13" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 289 "The frictio
n force points forward, i.e., it acts against the rotational torque ab
out the CM provided by the string force, but it helps to move the CM t
o the right. The system adjusts itself such that the condition of pure
 rolling is met, i.e., such that the winding of theta (acceleration " 
}{TEXT 19 3 "Tdd" }{TEXT -1 50 ") and the movement of the CM (linear a
cceleration " }{TEXT 19 3 "Xdd" }{TEXT -1 19 ") are synchronized." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "What is t
he change in string length after, say, 2.2 seconds?" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 21 "evalf(Y(2.2)-X(2.2));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#$\"+;dG9p!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
237 "Now let us re-do the problem using rotation about the center of m
ass. In this case it is evident how the friction force helps to propel
 the CM forward, but at the same time it acts as an opposite torque to
 the string torque about the CM." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 
"I_CM:=M/2*R^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%I_CMG,$*&%\"MG\"
\"\")%\"RG\"\"#F(#F(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "I
_O:=I_CM+M*R^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$I_OG,$*&%\"MG\"
\"\")%\"RG\"\"#F(#\"\"$F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Moti
on of the center of mass: apply the forces at the CM:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eq1:=M*Xdd=S+f;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$eq1G/*&%\"MG\"\"\"%$XddGF(,&%\"SGF(%\"fGF(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "We use the angular momentum theore
m for angular momentum " }{TEXT 262 1 "L" }{TEXT -1 12 " and torque " 
}{TEXT 263 1 "N" }{TEXT -1 237 " defined about the CM: we have two tor
que contributions that oppose each other. Tdd is the 2nd derivative of
 the rotation angle.For rotation with slipping the angular velocity ab
out the CM and about the point of contact are not the same." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "eq2:=I_CM*Tdd=R*S-R*f;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/,$*(%\"MG\"\"\")%\"RG\"\"#F)%$
TddGF)#F)F,,&*&F+F)%\"SGF)F)*&F+F)%\"fGF)!\"\"" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 17 "eq3:=m*Ydd=m*g-S;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$eq3G/*&%\"mG\"\"\"%$YddGF(,&*&F'F(%\"gGF(F(%\"SG!\"
\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eq4:=R*Tdd=Xdd;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq4G/*&%\"RG\"\"\"%$TddGF(%$XddG" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "eq5:=Ydd=Xdd+R*Tdd;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq5G/%$YddG,&%$XddG\"\"\"*&%\"RGF)%
$TddGF)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "sol:=solve(\{e
q1,eq2,eq3,eq4,eq5\},\{Tdd,Xdd,Ydd,S,f\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$solG<'/%$XddG,$*&*&%\"mG\"\"\"%\"gGF,F,,&F+\"\")*&\"
\"$F,%\"MGF,F,!\"\"\"\"%/%$YddG,$F)F//%\"SG,$*&*(F2F,F+F,F-F,F,F.F3F1/
%\"fGF;/%$TddG,$*&*&F+F,F-F,F,*&%\"RGF,F.F,F3F4" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 375 "Evidently the answer is the same as before! This fo
rmulation is more flexible, since it allows to discuss the case when t
he surface cannot provide the friction torque/force required due to an
 insufficient value of static friction constant mu. In this case the d
isk is not at rest with respect to the surface at the point of contact
. In particular, we can discuss the case of " }{TEXT 19 3 "f=0" }
{TEXT -1 42 " (negligible static and kinetic friction)." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eq1:=M
*Xdd=S;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/*&%\"MG\"\"\"%$XddG
F(%\"SG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "We use the angular mom
entum theorem for angular momentum " }{TEXT 264 1 "L" }{TEXT -1 12 " a
nd torque " }{TEXT 265 1 "N" }{TEXT -1 80 " defined about the CM: we h
ave two torque contributions that oppose each other. " }{TEXT 19 3 "Td
d" }{TEXT -1 155 " is the 2nd derivative of the rotation angle. For ro
tation with slipping the angular velocity about the CM and about the p
oint of contact are not the same." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "eq2:=I_CM*Tdd=R*S;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%$eq2G/,$*(%\"MG\"\"\")%\"RG\"\"#F)%$TddGF)#F)F,*&F+F)%\"SGF)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "eq3:=m*Ydd=m*g-S;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$eq3G/*&%\"mG\"\"\"%$YddGF(,&*&F'F(%\"gGF(
F(%\"SG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The CM motion " }
{TEXT 19 1 "X" }{TEXT -1 253 " is unrelated to the angular rotation in
 rolling with total slip [rolling doesn't help at all to move forward \+
- spinning car wheels on ice; but in contrast to that problem, where t
orque is the only source of motion we are pulling the disk in this cas
e]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eq4:='eq4': #eq4:=R*
Tdd=Xdd;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "We need to connect th
e fall with the unravelling of wire (" }{TEXT 19 5 "R*Tdd" }{TEXT -1 
33 "), and with the motion of the CM." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "eq5:=Ydd=Xdd+R*Tdd;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%$eq5G/%$YddG,&%$XddG\"\"\"*&%\"RGF)%$TddGF)F)" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 47 "How many equations/unknowns? The unknowns are: " }
{TEXT 19 16 "Ydd, Xdd, Tdd, S" }{TEXT -1 35 " - so we should use four \+
equations." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "sol:=solve(\{
eq1,eq2,eq3,eq5\},\{Tdd,Xdd,Ydd,S\});" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%$solG<&/%$TddG,$*&*&%\"mG\"\"\"%\"gGF,F,*&%\"RGF,,&%\"MGF,*&\"\"$
F,F+F,F,F,!\"\"\"\"#/%$YddG,$*&*&F+F,F-F,F,F0F4F3/%\"SG*&*(F1F,F+F,F-F
,F,F0F4/%$XddGF9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "The mass " }
{TEXT 259 1 "m" }{TEXT -1 182 " accelerates on account of translation \+
and rotation as before, the effect from the rotation (wire unravels) i
s twice as important as the effect of pulling the CM by the string for
ce " }{TEXT 266 1 "S" }{TEXT -1 3 ". (" }{TEXT 19 3 "Ydd" }{TEXT -1 
28 " is three times as large as " }{TEXT 19 3 "Xdd" }{TEXT -1 78 "; in
 the roll without slip they are contributing equally due to the condit
ion " }{TEXT 19 11 "Xdd = R*Tdd" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 275 11 "Exercise 1:" }}{PARA 0 "
" 0 "" {TEXT -1 53 "Explore the above solution for a simple case such \+
as " }{TEXT 277 1 "m" }{TEXT -1 1 "=" }{TEXT 276 1 "M" }{TEXT -1 69 ":
 how does the loss of friction manifest iteself in the fall of mass " 
}{TEXT 278 1 "m" }{TEXT -1 1 "?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 279 9 "Question:" }{TEXT -1 36 " How woul
d we solve a problem where " }{TEXT 267 1 "f" }{TEXT -1 124 " is not z
ero, but less than the amount required to keep the rotation locked wit
h the CM translation (no-slip)? We would add " }{TEXT 268 1 "f" }
{TEXT -1 61 " as a degree of freedom, and we would need a replacement \+
for " }{TEXT 19 3 "eq4" }{TEXT -1 26 ", which would specify how " }
{TEXT 19 3 "Tdd" }{TEXT -1 5 " and " }{TEXT 19 3 "Xdd" }{TEXT -1 13 " \+
are related." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 200 " Now we consider another, related problem: A yo-yo is \+
being pulled in the horizontal direction. Let us assume that the radiu
s at which the string is wound up is one-half the radius of the disk, \+
i.e., " }{TEXT 269 1 "R" }{TEXT -1 37 "/2. We go back to no-slip condi
tions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 
"I_CM:=M/2*R^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%I_CMG,$*&%\"MG\"
\"\")%\"RG\"\"#F(#F(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "I
_O:=I_CM+M*R^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$I_OG,$*&%\"MG\"
\"\")%\"RG\"\"#F(#\"\"$F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
15 "eq1:=M*Xdd=S+f;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/*&%\"MG
\"\"\"%$XddGF(,&%\"SGF(%\"fGF(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 
"The radius at which the yo-yo unwinds the string: (we pick a value ve
ry close to " }{TEXT 290 1 "R" }{TEXT -1 53 "/2, but not exactly to av
oid plotting problems below)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 12 "R1:=19*R/40;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#R1G,$%\"RG#\"
#>\"#S" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "The torque from the pul
ling string has arm length " }{TEXT 19 2 "R1" }{TEXT -1 56 ", and appl
ies on the same side as in the example before:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 23 "eq2:=I_CM*Tdd=R1*S-R*f;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$eq2G/,$*(%\"MG\"\"\")%\"RG\"\"#F)%$TddGF)#F)F,,&*&F+
F)%\"SGF)#\"#>\"#S*&F+F)%\"fGF)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "eq3:=m*Ydd=m*g-S;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%$eq3G/*&%\"mG\"\"\"%$YddGF(,&*&F'F(%\"gGF(F(%\"SG!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eq4:=R*Tdd=Xdd;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$eq4G/*&%\"RG\"\"\"%$TddGF(%$XddG" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 20 "eq5:=Ydd=Xdd+R1*Tdd;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$eq5G/%$YddG,&%$XddG\"\"\"*(#\"#>\"#SF)%\"RGF)%$TddGF
)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "sol:=solve(\{eq1,eq2
,eq3,eq4,eq5\},\{Tdd,Xdd,Ydd,S,f\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "
6#>%$solG<'/%$TddG,$*&*&%\"mG\"\"\"%\"gGF,F,*&%\"RGF,,&F+\"%\"[$*&\"%+
CF,%\"MGF,F,F,!\"\"\"%gB/%\"SG,$*&*(F4F,F+F,F-F,F,F0F5F3/%\"fG,$F:!#S/
%$YddG,$*&*&F+F,F-F,F,F0F5F1/%$XddG,$FCF6" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 130 "Interesting coincidence: there will be no static frictio
n force! When the armlength where the string force applies is larger t
han " }{TEXT 270 1 "R" }{TEXT -1 30 "/2, a positive friction force " }
{TEXT 271 1 "f" }{TEXT -1 39 " results, for arm lengths smaller than \+
" }{TEXT 272 1 "R" }{TEXT -1 30 "/2 it will be negative, i.e., " }
{TEXT 273 1 "f" }{TEXT -1 3 "<0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 274 11 "Exercise 2:" }}{PARA 0 "" 0 "" {TEXT -1 
91 "Vary the inner radius of the yo-yo, and observe how the role of th
e friction force changes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "assign(sol);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "m:=1; M
:=2; g:=10; R:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG\"\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"gG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG\"
\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "Ydd;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6##\"&5[$\"%\"G)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 4 "Xdd;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"&+O#\"%\"G)
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "Tdd;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6##\"&+O#\"%\"G)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 30 "theta:=unapply(1/2*Tdd*t^2,t);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%&thetaGR6#%\"tG6\"6$%)operatorG%&arrowGF(,$*$)9$\"\"#\"\"\"#\"
&+=\"\"%\"G)F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "X:=un
apply(1/2*Xdd*t^2,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XGR6#%\"t
G6\"6$%)operatorG%&arrowGF(,$*$)9$\"\"#\"\"\"#\"&+=\"\"%\"G)F(F(F(" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Y:=unapply(1/2*Ydd*t^2,t);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YGR6#%\"tG6\"6$%)operatorG%&ar
rowGF(,$*$)9$\"\"#\"\"\"#\"&0u\"\"%\"G)F(F(F(" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 29 "with(plottools): with(plots):" }}{PARA 7 "" 1 "
" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "arrowscale:=0.25:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 615 "c := t-> [circle([X(t),R], \+
R, color=red,thickness=3),circle([X(t),R], R1, color=red,thickness=3),
line([X(t),R],[X(t)+R*cos(theta(t)+Pi/2),R-R*sin(theta(t)+Pi/2)],color
=blue,thickness=3),line([X(t),R+R1],[8.2,R+R1],color=grey,thickness=3)
,line([8.2,R+R1],[8.2,-Y(t)],color=grey,thickness=3),circle([8.2,-Y(t)
],0.15,color=red,thickness=3),arrow([X(t),R+R1], [X(t)+S*arrowscale,R+
R1], .2, .4, .1, color=cyan),arrow([X(t),0], [X(t)+f*arrowscale,0], .2
, .4, .1, color=magenta),arrow([8.2,-Y(t)], [8.2,-Y(t)+S*arrowscale], \+
.2, .4, .1, color=cyan),arrow([8.2,-Y(t)], [8.2,-Y(t)-m*g*arrowscale],
 .2, .4, .1, color=green)]:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 81 "PL:=[seq(display(c(2.2*i/60),scaling=constrained,view=[-2..8.4,-
8..2]),i=0..60)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "plots[
display](op(PL),insequence=true,scaling=constrained);" }}{PARA 13 "" 
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{TEXT -1 381 "This calculation shows that forward motion without any f
riction provided by the surface is possible. In fact, we can see from \+
the equations that the case when the string force applies at one half \+
the radius corresponds to the situation when no friction is required t
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282 1 "R" }{TEXT -1 11 "/2; choose " }{TEXT 19 7 "r:=R/4:" }{TEXT -1 
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brF\\ibrFaibrFbibrFeibrF_ibrF[hbrFa^nFe^nFa^n" 1 2 0 1 10 0 2 9 1 4 1 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve
 11" "Curve 12" "Curve 13" "Curve 14" }}}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 443 "Note that the torque from the string force acts against \+
the rotation, and consequently agains the forward motion of the disk d
ue to rotation, but that the force applies at the CM, and therefore th
e disk is going to accelerate to the right! In this example (winding o
rientation) the string winds itself around the yo-yo as the roll accel
erates to the right. The friction force adjusts itself (if possible, i
.e., if the static friction constant " }{XPPEDIT 18 0 "mu;" "6#%#muG" 
}{TEXT -1 25 " is large enough so that " }{TEXT 288 1 "f" }{TEXT -1 
17 " does not exceed " }{XPPEDIT 18 0 "mu*M*g;" "6#*(%#muG\"\"\"%\"MGF
%%\"gGF%" }{TEXT -1 333 " ) in such a way that the the condition of pu
re rolling is satisfied. This raises the question why pure rolling is \+
desirable. The origin of this pure rolling (and of static friction in \+
general) are the molecular forces of electrostatic origin that act bet
ween the particles which make up the cylinder and the table. The norma
l force " }{TEXT 289 2 "Mg" }{TEXT -1 127 " helps the molecules from t
he different materials to get closer, so that the bonds occur more oft
en, and the material constant " }{XPPEDIT 18 0 "mu;" "6#%#muG" }{TEXT 
-1 70 " describes how effective they are. Anything that is within this
 limit " }{XPPEDIT 18 0 "mu*M*g;" "6#*(%#muG\"\"\"%\"MGF%%\"gGF%" }
{TEXT -1 388 " is allowed to occur. Compare this condition to static f
riction when trying to push a block resting on a surface. The block wo
n't move until the pushing force exceeds the static friction force. In
 the same sense pure rolling is bound to occur until the friction requ
ired for pure rolling motion (forward or backward, depending on the ca
se as demonstrated above) exceeds the allowed limit." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "The yo-yo is winding it
self up by the amount:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "e
valf(Y(2.2)-X(2.2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+h%Q:'=!\"*
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 500 "We can ask another question \+
to understand the role of friction, namely how it helps the rotation a
nd resists the forward motion of the center of mass, and vice versa: w
e can calculate the amount of work performed for the rotation and the \+
translation. We integrate the frictional torque about the CM times the
 angular velocity, and also the linear friction force times the linear
 velocity. We use the sign for the friction force and the frictional t
orque with which they enter the equations of motion:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 35 "W_CM:=int(f*diff(X(t),t),t=0..2.2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%W_CMG$!+gU7\"H#!\")" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "W_rot:=int(-R*f*diff(theta(t),t),t=
0..2.2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&W_rotG$\"+gU7\"H#!\")" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 457 "The contributions have to canc
el, because static friction implies that there is no net motion carrie
d out [the forward translation of the CM is perfectly balanced by the \+
tangential velocity at the contact point due to the rotation]. Also we
 were able to show that one can calculate the entire motion without co
nsidering the friction at all [the rotation of the disk about the cont
act point about which friction contributes no torque due to zero arm l
ength]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 280 11 "Exercise 4:" }}{PARA 0 "" 0 "" {TEXT -1 130 "Change the \+
inner yo-yo radius and find out whether there are characteristic chang
es in the behaviour of the static friction force." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "119 0 0" 455 }{VIEWOPTS 1 1 0 1 
1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }