{VERSION 5 0 "APPLE_PPC_MAC" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 0 "" 0 "" {TEXT -1 59 "We derive the equations for a pen dulum with spring support." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The position is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "x = l*sin(phi);" "6# /%\"xG*&%\"lG\"\"\"-%$sinG6#%$phiGF'" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "y = -l*cos(phi);" "6#/%\"yG,$*&%\"lG\"\"\"-%$cosG6#% $phiGF(!\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The velocity (the time rate of change of the po sition)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "diff(x,t) = sin(phi)*diff(l,t)+l*cos(phi)*diff(phi,t);" "6#/-%%d iffG6$%\"xG%\"tG,&*&-%$sinG6#%$phiG\"\"\"-F%6$%\"lGF(F/F/*(F2F/-%$cosG 6#F.F/-F%6$F.F(F/F/" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "diff(y,t) = -cos(phi)*diff(l,t)+l*s in(phi)*diff(phi,t);" "6#/-%%diffG6$%\"yG%\"tG,&*&-%$cosG6#%$phiG\"\" \"-F%6$%\"lGF(F/!\"\"*(F2F/-%$sinG6#F.F/-F%6$F.F(F/F/" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "The acc eleration (the time rate of change of the velocity)" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "diff(x,`$`(t,2)) = \+ sin(phi)*diff(l,`$`(t,2))+2*cos(phi)*diff(l,t)*diff(phi,t)-l*sin(phi)* diff(phi,t)^2+l*cos(phi)*diff(phi,`$`(t,2));" "6#/-%%diffG6$%\"xG-%\"$ G6$%\"tG\"\"#,**&-%$sinG6#%$phiG\"\"\"-F%6$%\"lG-F)6$F+F,F3F3**F,F3-%$ cosG6#F2F3-F%6$F6F+F3-F%6$F2F+F3F3*(F6F3-F06#F2F3-F%6$F2F+F,!\"\"*(F6F 3-F;6#F2F3-F%6$F2-F)6$F+F,F3F3" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "diff(y,`$`(t,2)) = -cos(phi)*diff(l,`$`( t,2))+2*sin(phi)*diff(l,t)*diff(phi,t)+l*cos(phi)*diff(phi,t)^2+l*sin( phi)*diff(phi,`$`(t,2));" "6#/-%%diffG6$%\"yG-%\"$G6$%\"tG\"\"#,**&-%$ cosG6#%$phiG\"\"\"-F%6$%\"lG-F)6$F+F,F3!\"\"**F,F3-%$sinG6#F2F3-F%6$F6 F+F3-F%6$F2F+F3F3*(F6F3-F06#F2F3-F%6$F2F+F,F3*(F6F3-F<6#F2F3-F%6$F2-F) 6$F+F,F3F3" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "We now write the d ifferential equations using Newton's second law." }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "m*sin(phi)*diff(l,`$`(t,2))+2*m*cos(phi)*diff(l,t)*diff (phi,t)-m*l*sin(phi)*diff(phi,t)^2+m*l*cos(phi)*diff(phi,`$`(t,2)) = - k*(l-l[0])*sin(phi);" "6#/,**(%\"mG\"\"\"-%$sinG6#%$phiGF'-%%diffG6$% \"lG-%\"$G6$%\"tG\"\"#F'F'*,F4F'F&F'-%$cosG6#F+F'-F-6$F/F3F'-F-6$F+F3F 'F'**F&F'F/F'-F)6#F+F')-F-6$F+F3F4F'!\"\"**F&F'F/F'-F76#F+F'-F-6$F+-F1 6$F3F4F'F',$*(%\"kGF',&F/F'&F/6#\"\"!FCF'-F)6#F+F'FC" }{TEXT -1 0 "" } }{PARA 256 "" 0 "" {XPPEDIT 18 0 "-m*cos(phi)*diff(l,`$`(t,2))+2*m*sin (phi)*diff(l,t)*diff(phi,t)+m*l*cos(phi)*diff(phi,t)^2+m*l*sin(phi)*di ff(phi,`$`(t,2)) = -mg+k*(l-l[0])*cos(phi);" "6#/,**(%\"mG\"\"\"-%$cos G6#%$phiGF'-%%diffG6$%\"lG-%\"$G6$%\"tG\"\"#F'!\"\"*,F4F'F&F'-%$sinG6# F+F'-F-6$F/F3F'-F-6$F+F3F'F'**F&F'F/F'-F)6#F+F')-F-6$F+F3F4F'F'**F&F'F /F'-F86#F+F'-F-6$F+-F16$F3F4F'F',&%#mgGF5*(%\"kGF',&%\"lGF'&FP6#\"\"!F 5F'-%$cosG6#%$phiGF'F'" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "Multiplying the fist equation by " } {XPPEDIT 18 0 "sin(phi);" "6#-%$sinG6#%$phiG" }{TEXT -1 19 " and the s econd by " }{XPPEDIT 18 0 "cos(phi);" "6#-%$cosG6#%$phiG" }{TEXT -1 37 " and subtracting the resulting second" }}{PARA 0 "" 0 "" {TEXT -1 30 "equation from the first we get" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {XPPEDIT 18 0 "diff(l,`$`(t,2)) = l*diff(phi,t)^2-k/ m*(l-l[0])+g*cos(phi);" "6#/-%%diffG6$%\"lG-%\"$G6$%\"tG\"\"#,(*&F'\" \"\"*$-F%6$%$phiGF+F,F/F/*(%\"kGF/%\"mG!\"\",&F'F/&F'6#\"\"!F7F/F7*&% \"gGF/-%$cosG6#F3F/F/" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "m ultiplying the fist equation by " }{XPPEDIT 18 0 "cos(phi);" "6#-%$cos G6#%$phiG" }{TEXT -1 19 " and the second by " }{XPPEDIT 18 0 "sin(phi) ;" "6#-%$sinG6#%$phiG" }{TEXT -1 19 " and adding the two" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "diff(phi,`$`(t ,2)) = -2/l*diff(l,t)*diff(phi,t)-g/l*sin(phi);" "6#/-%%diffG6$%$phiG- %\"$G6$%\"tG\"\"#,&**F,\"\"\"%\"lG!\"\"-F%6$F0F+F/-F%6$F'F+F/F1*(%\"gG F/F0F1-%$sinG6#F'F/F1" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "xv:=diff(l(t)*sin(phi(t)),t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xvG,&*&-%%diffG6$-%\"lG6#%\"tG F-\"\"\"-%$sinG6#-%$phiGF,F.F.*(F*F.-%$cosGF1F.-F(6$F2F-F.F." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "yv:=diff(-l(t)*cos(phi(t)),t );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#yvG,&*&-%%diffG6$-%\"lG6#%\"t GF-\"\"\"-%$cosG6#-%$phiGF,F.!\"\"*(F*F.-%$sinGF1F.-F(6$F2F-F.F." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "xa:=m*diff(l(t)*sin(phi(t)), t,t)=-k*(l-l_0)*sin(phi(t));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#xaG /*&%\"mG\"\"\",**&-%%diffG6$-%\"lG6#%\"tG-%\"$G6$F1\"\"#F(-%$sinG6#-%$ phiGF0F(F(**F5F(-F,6$F.F1F(-%$cosGF8F(-F,6$F9F1F(F(*(F.F(F6F()F@F5F(! \"\"*(F.F(F>F(-F,6$F9F2F(F(F(,$*(%\"kGF(,&F/F(%$l_0GFDF(F6F(FD" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "ya:=m*diff(-l(t)*cos(phi(t)) ,t,t)=-m*g+k*(l-l_0)*cos(phi(t));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#> %#yaG/*&%\"mG\"\"\",**&-%%diffG6$-%\"lG6#%\"tG-%\"$G6$F1\"\"#F(-%$cosG 6#-%$phiGF0F(!\"\"**F5F(-F,6$F.F1F(-%$sinGF8F(-F,6$F9F1F(F(*(F.F(F6F() FAF5F(F(*(F.F(F?F(-F,6$F9F2F(F(F(,&*&F'F(%\"gGF(F;*(%\"kGF(,&F/F(%$l_0 GF;F(F6F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "sin(phi(t))* xa-cos(phi(t))*ya;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,&*(-%$sinG6#-% $phiG6#%\"tG\"\"\"%\"mGF-,**&-%%diffG6$-%\"lGF+-%\"$G6$F,\"\"#F-F&F-F- **F9F--F26$F4F,F--%$cosGF(F--F26$F)F,F-F-*(F4F-F&F-)F?F9F-!\"\"*(F4F-F =F--F26$F)F6F-F-F-F-*(F=F-F.F-,**&F1F-F=F-FC**F9F-F;F-F&F-F?F-F-*(F4F- F=F-FBF-F-*(F4F-F&F-FEF-F-F-FC,&*()F&F9F-%\"kGF-,&F5F-%$l_0GFCF-FC*&F= F-,&*&F.F-%\"gGF-FC*(FPF-FQF-F=F-F-F-FC" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&% \"mG\"\"\",&-%%diffG6$-%\"lG6#%\"tG-%\"$G6$F.\"\"#F&*&F+F&)-F)6$-%$phi GF-F.F2F&!\"\"F&,(*&%\"kGF&F,F&F9*&F " 0 "" {MPLTEXT 1 0 30 "cos(phi(t))*xa+ sin(phi(t))*ya;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,&*(-%$cosG6#-%$ph iG6#%\"tG\"\"\"%\"mGF-,**&-%%diffG6$-%\"lGF+-%\"$G6$F,\"\"#F--%$sinGF( F-F-**F9F--F26$F4F,F-F&F--F26$F)F,F-F-*(F4F-F:F-)F?F9F-!\"\"*(F4F-F&F- -F26$F)F6F-F-F-F-*(F:F-F.F-,**&F1F-F&F-FC**F9F-F=F-F:F-F?F-F-*(F4F-F&F -FBF-F-*(F4F-F:F-FEF-F-F-F-,&**F&F-%\"kGF-,&F5F-%$l_0GFCF-F:F-FC*&F:F- ,&*&F.F-%\"gGF-FC*(FOF-FPF-F&F-F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&% \"mG\"\"\",&*&-%%diffG6$-%\"lG6#%\"tGF/F&-F*6$-%$phiGF.F/F&\"\"#*&F,F& -F*6$F2-%\"$G6$F/F4F&F&F&,$*(-%$sinG6#F2F&F%F&%\"gGF&!\"\"" }}}}{MARK "28 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }