{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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 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 "2 D Output" 2 20 "" 0 1 0 0 255 1 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 24 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 10 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 258 "" 1 18 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 18 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 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 1 18 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 "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 } 1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 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 "" 1 12 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 "" 0 261 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 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 7 "RECORDS" }}{PARA 258 "" 0 "" {TEXT 259 11 "Tevian Dray " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 257 34 "co pyright 1998-2000 by Tevian Dray" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 259 "" 0 "" {TEXT 258 36 "What does the phonograph record see?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "restart:with(linalg):with(plots):with(plottools):" }}{PARA 7 " " 1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 " " {TEXT -1 33 "Warning, new definition for trace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "read(`/usersB/tevian/maple/ph429/Record`):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 15 "Getting started" }}{PARA 0 "" 0 "" {TEXT -1 216 "Straig ht line motion in the plane traces out a complicated path as seen by a rotating observer. The needle of a phonograph moves radially towards the center of the record, yet traces out a long spiral on the record! " }}{PARA 0 "" 0 "" {TEXT -1 223 "Replace the record by a blank sheet \+ of paper, and the phonograph needle by a red pen. Furthermore, allow \+ the needle to start anywhere, and move in any fixed direction with con stant speed. What curve is drawn on the paper?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "We showed in class that t he equation of this curve is" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "r[rel]:=rsol2[1]*r[hat]+rsol2[2]*theta[hat];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#%$relG,&*&,**&-%$cosG6#*&%&OmegaG\"\"\"%\"tGF1 F1&%\"xG6#\"\"!F1F1*(F,F1F2F1&%\"vG6#F4F1F1*&-%$sinGF.F1&%\"yGF5F1F1*( FF1F1*(F,F1F2F1FAF1F1F1&%&thetaGFDF1F1" }}}{PARA 0 "" 0 "" {TEXT -1 40 "where the initial position of the pen is" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "r[0]:=[x[0],y[0]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"!7$&%\"xGF&&%\"yGF&" }}}{PARA 0 "" 0 "" {TEXT -1 27 "and its initial velocity is" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "v[fix]:=[v[x],v[y]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"vG6#%$fixG7$&F%6#%\"xG&F%6#%\"yG" }}}{PARA 0 "" 0 "" {TEXT -1 35 "(A derivation of this equation for " }{XPPEDIT 18 0 "r[rel]" "6#&% \"rG6#%$relG" }{TEXT -1 80 ", as well as the differential equation it \+ solves, can be found in the worksheet " }{HYPERLNK 17 "2D Rotations" 1 "/usersB/tevian/maple/ph429/2Drot.mws" "" }{TEXT -1 49 "; the equati on is also derived in Class Notes 1.)" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "v[x]" "6# &%\"vG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v[y]" "6#&%\"vG6#% \"yG" }{TEXT -1 49 " denote the initial velocity with respect to the \+ " }{TEXT 260 11 "nonrotating" }{TEXT -1 37 " frame. The initial veloc ity in the " }{TEXT 261 8 "rotating" }{TEXT -1 18 " frame is given by " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "v[rel]:=eval(subs(t=0,di ff(r[rel],t)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"vG6#%$relG,&*& ,&&F%6#%\"xG\"\"\"*&%&OmegaGF.&%\"yG6#\"\"!F.F.F.&%\"rG6#%$hatGF.F.*&, &*&F0F.&F-F3F.!\"\"&F%6#F2F.F.&%&thetaGF7F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 5 "Demos" }}{PARA 0 "" 0 "" {TEXT -1 151 "The following demos graph the path of the pen, both as seen by the nonrot ating observer (blue vector), and as seen by the rotating observer (re d line)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "First, we set define some parameters. The angular velocity of the record (in radians/second " }{TEXT 263 16 "counterclockwise" }{TEXT -1 14 ") is given by " }{XPPEDIT 18 0 "Omega" "6#%&OmegaG" }{TEXT -1 21 "; a typical value is " }{XPPEDIT 18 0 "Omega=1" "6#/%&OmegaG\"\"\" " }{TEXT -1 46 ". The number of seconds the demo will run is " } {XPPEDIT 18 0 "T" "6#%\"TG" }{TEXT -1 21 ", a typical value is " } {XPPEDIT 18 0 "T=5" "6#/%\"TG\"\"&" }{TEXT -1 44 ". The total number \+ of pictures is given by " }{XPPEDIT 18 0 "N" "6#%\"NG" }{TEXT -1 39 ", so that the time between pictures is " }{XPPEDIT 18 0 "T" "6#%\"TG" } {TEXT -1 1 "/" }{XPPEDIT 18 0 "N" "6#%\"NG" }{TEXT -1 27 ". A reasona ble choice for " }{XPPEDIT 18 0 "N" "6#%\"NG" }{TEXT -1 4 " is" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "N:=5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 55 "(A smoothe r animation is obtained for larger values of " }{XPPEDIT 18 0 "N" "6#% \"NG" }{TEXT -1 2 "; " }{XPPEDIT 18 0 "N=10" "6#/%\"NG\"#5" }{TEXT -1 41 " works fairly well. But large values of " }{XPPEDIT 18 0 "N" "6#% \"NG" }{TEXT -1 74 " may also take significantly longer to run. See w hat works well for you.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 90 "Finally, you should make a choice of how you want \+ the graphics to be displayed, using any " }{TEXT 262 3 "one" }{TEXT -1 25 " of the following 3 lines" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plotsetup(inline):" }{TEXT -1 26 " (graphs in worksheet)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plotsetup(window):" } {TEXT -1 29 " (graphs in Maple window)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 15 "plotsetup(x11):" }{TEXT -1 41 " (graphs \+ in separate X window)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 378 "In the figures you will construct, the blue arrow represents the linear moti on seen by an inertial observer, while the red curve represents the mo tion seen by the rotating observer. (The green arrow is to help you k eep track of the rotation.) Think of the tip of the blue arrow as con taining a red pen which writes on the record as the record rotates ben eath the (moving) pen." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 166 "The demo consists of 2 functions, one from the poin t of view of each reference frame. The first of these shows what the \+ nonrotating observer sees, and takes the form" }}{PARA 261 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fix(start,velocity,vframe,T,N,Omega" "6 #-%$FixG6(%&startG%)velocityG%'vframeG%\"TG%\"NG%&OmegaG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "Here, " }{XPPEDIT 18 0 "N" "6#%\"NG " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "T" "6#%\"TG" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "Omega" "6#%&OmegaG" }{TEXT -1 23 " are as defined above , " }{XPPEDIT 18 0 "start" "6#%&startG" }{TEXT -1 31 " denotes the ini tial position, " }{XPPEDIT 18 0 "velocity" "6#%)velocityG" }{TEXT -1 35 " denotes the initial velocity, and " }{XPPEDIT 18 0 "vframe" "6#%' vframeG" }{TEXT -1 79 " specifies whether the initial velocity is with respect to the fixed observer (" }{MPLTEXT 0 21 13 "vframe:=`FIX`" } {TEXT -1 28 ") or the rotating observer (" }{MPLTEXT 0 21 13 "vframe:= `ROT`" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "For example, try" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Fix([-1,0],[1/4,0],`FIX`,4,N,-10);" }}}{PARA 0 "" 0 " " {TEXT -1 55 "which is roughly what happens when you play a record. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT -1 242 "(T o run the animations, use the buttons at the top; for inline display, \+ you will first need to click on the figure. For either inline or Mapl e window display, you can find out what the buttons do by turning on B alloon Help in the Help menu.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 78 "To see this from the reference frame of t he rotating observer, use the command" }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "Rot(start,velocity,vframe,T,N,Omega" "6#-%$RotG6(%&startG%)veloc ityG%'vframeG%\"TG%\"NG%&OmegaG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 67 "which takes the same parameters. Compare your previous o utput with" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Rot([-1,0],[1/ 4,0],`FIX`,4,N,-10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "You can reproduce the results in Figure 10-4 on page 3 89 of Marion and Thornton using " }{XPPEDIT 18 0 "Omega=1" "6#/%&Omega G\"\"\"" }{TEXT -1 25 ", the initial positition" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "MTstart:=[-1/2,0]:" }}}{PARA 0 "" 0 "" {TEXT -1 46 "one of the initial velocities (and total time " }{XPPEDIT 18 0 "T" "6#%\"TG" }{TEXT -1 15 " as indicated):" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "MTv[a]:=[0,1]*3/2:" }{TEXT -1 31 " \+ (T=0.86)" }{MPLTEXT 1 0 19 "\nMTv[b]:=[0,1]*4/5:" }{TEXT -1 30 " (T=2.9)" }{MPLTEXT 1 0 20 "\nMTv[c]:=[0,1]*9/20 :" }{TEXT -1 29 " (T=17.3)" }{MPLTEXT 1 0 21 "\nMT v[d]:=[0,1]*0.328:" }{TEXT -1 26 " (T=5.0)" } {MPLTEXT 1 0 28 "\nMTv[e]:=[1,1]*0.47/sqrt(2):" }{TEXT -1 12 " (T=3 .83)" }{MPLTEXT 1 0 29 "\nMTv[f]:=[1,1]*0.283/sqrt(2):" }{TEXT -1 9 " \+ (T=3.3)" }}}{PARA 0 "" 0 "" {TEXT -1 15 "and specifying " }{MPLTEXT 0 21 5 "`ROT`" }{TEXT -1 5 " for " }{MPLTEXT 0 21 6 "vframe" }{TEXT -1 30 ". Try it! Try other choices!" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Fix(MTstart,MTv[e],`ROT`,3.83,N,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}} {MARK "2 0 0" 49 }{VIEWOPTS 1 1 0 1 1 1803 }