{VERSION 3 0 "IBM RISC UNIX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 }{CSTYLE "" -1 256 " " 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 8 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 256 23 "CONSTRUCTING SOLUTIONS\n" }{TEXT 257 13 "Tevian Dray \n\n" }{TEXT 258 32 "copyright 1998-2000 Tevian Dray\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Choose arbitrary linear functions for the " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y" "6#%\"yG" } {TEXT -1 69 " coordinates of the pen. In other words, choose arbitrar y values of " }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "y[0]" "6#&%\"yG6#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "v[x]" "6#&%\"vG6#%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "v[y]" " 6#&%\"vG6#%\"yG" }{TEXT -1 20 ", in the expressions" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=x[0]+v[x]*t" "6#/%\"xG,&&F$6#\"\"! \"\"\"*&&%\"vG6#F$F)%\"tGF)F)" }{TEXT -1 1 " " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=y[0]+v[y]*t" "6#/%\"yG,&&F$6#\"\"!\" \"\"*&&%\"vG6#F$F)%\"tGF)F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 93 "and enter the resulting formulas below. (Don't forget to end e ach command with a semicolon!)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "x:=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "y:=" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Now choose an angular velocity " } {XPPEDIT 18 0 "Omega;" "6#%&OmegaG" }{TEXT -1 10 ", so that " } {XPPEDIT 18 0 "theta = Omega*t;" "6#/%&thetaG*&%&OmegaG\"\"\"%\"tGF'" }{TEXT -1 1 "." }}}{EXCHG {PARA 265 "" 0 "" {TEXT -1 56 "You may find \+ it helpful to first record your values of " }{XPPEDIT 18 0 "Omega;" " 6#%&OmegaG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" } {TEXT -1 6 " here:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Omega: =" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "theta:=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Now transform to rotating coordinates (" }{XPPEDIT 18 0 "X,Y;" "6$%\"XG%\"YG" }{TEXT -1 38 ") using the formula s derived in class:" }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "X=x*cos(theta)+y*sin(theta)-R" "6#/%\"XG,(*&%\"xG\"\"\"-%$cosG6#%&t hetaGF(F(*&%\"yGF(-%$sinG6#F,F(F(%\"RG!\"\"" }{TEXT -1 1 " " }}{PARA 260 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Y=-x*sin(theta)+y*cos(theta )" "6#/%\"YG,&*&%\"xG\"\"\"-%$sinG6#%&thetaGF(!\"\"*&%\"yGF(-%$cosG6#F ,F(F(" }{TEXT -1 1 " " }}{PARA 264 "" 0 "" {TEXT -1 32 "(It is safe to assume here that " }{XPPEDIT 18 0 "R=0" "6#/%\"RG\"\"!" }{TEXT -1 46 ", i.e. that both frames have the same origin.)" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 3 "X:=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "Y:=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Verify that these expre ssions solve the differential equations derived in class:" }}{PARA 262 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "diff(X,t,t)-2*Omega*diff(Y, t)-Omega^2*(X+R)=0" "6#/,(-%%diffG6%%\"XG%\"tGF)\"\"\"*(\"\"#F*%&Omega GF*-F&6$%\"YGF)F*!\"\"*&F-\"\"#,&F(F*%\"RGF*F*F1\"\"!" }{TEXT -1 1 " \+ " }}{PARA 263 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "diff(Y,t,t)+2*Ome ga*diff(X,t)-Omega^2*Y=0" "6#/,(-%%diffG6%%\"YG%\"tGF)\"\"\"*(\"\"#F*% &OmegaGF*-F&6$%\"XGF)F*F**&F-\"\"#F(F*!\"\"\"\"!" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 1 "(" }{TEXT 260 43 "Second derivatives can b e entered as either" }{TEXT -1 1 " " }{MPLTEXT 0 21 13 " diff(X,t,t) \+ " }{TEXT 259 2 "or" }{TEXT -1 3 " " }{MPLTEXT 0 21 12 "diff(X,t$2) \+ " }{TEXT -1 2 ")." }}}{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 "2 0 0" 8 }{VIEWOPTS 1 1 0 3 2 1804 }