{VERSION 4 0 "SUN SPARC SOLARIS" "4.0" } {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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 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 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 3 0 }1 0 0 -1 -1 -1 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 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 0 }3 3 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 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 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 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT -1 38 " masses, springs, washboarding cars. " }}{PARA 257 "" 0 "" {TEXT -1 11 "Math 2280-2" }}{PARA 258 "" 0 "" {TEXT -1 14 " March 6, 2001" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 7 "Part I:" }{TEXT -1 58 " two equal masses suspended with three identical spring s:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg): " }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and trace have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "A:=matrix(2,2,[-2*k/m, k/m,k/m,-2*k/m]);\n #th is should be the \"A\" matrix you get for\n #our two-mass, three-spri ng system. " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$ ,$*&%\"kG\"\"\"%\"mG!\"\"!\"#F+7$F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvects(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%, $*&%\"kG\"\"\"%\"mG!\"\"!\"$F'<#-%'vectorG6#7$F)F'7%,$F%F)F'<#-F-6#7$F 'F'" }}}{PARA 0 "" 0 "" {TEXT -1 84 "Here is the data from our experim ent..... we'll recheck the timing data in class. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 154 "(1) The \"in-phase\" \+ characteristic motion oscillated at 50 cycles per 30 seconds. \n(2) T he \"out of phase\" oscillation was still 100 cycles in 33 seconds." } }{PARA 0 "" 0 "" {TEXT -1 67 "(3) The mass of each ball 23 grams. ( I checked this with Ziggy.)" }}{PARA 0 "" 0 "" {TEXT -1 39 "(4) The m ass of each spring is 9 grams" }}{PARA 0 "" 0 "" {TEXT -1 62 "(4) A m ass of 20 grams stretches a single spring by 5.9 cm. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "Notice, our theory \+ predicts natural angular frequencies of sqrt(3k/m) and sqrt(k/m), so t he ratio of our experimental frequencies should be sqrt(3):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f1:=100/33.0;\nf2:=50/30.0;\nf1/f2; \nsqrt(3.0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G$\"+IIIII!\"*" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2G$\"+nmmm;!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+=====!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" +330K " 0 "" {MPLTEXT 1 0 25 "m:=.023 ;\nk:=.02*9.8/.059;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG$\"#B!\"$ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG$\"+)*Q.AL!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "evalf(sqrt(k/m)/(2*Pi));\nevalf(sqr t(3*k/m)/(2*Pi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+V,v7>!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+K/)HJ$!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 12 "EXPLANATION:" }{TEXT -1 906 " The springs actually have mass, equal to 9 grams each, which yo u did not take into account. This is on the same order of magnitude a s the ball masses, and causes the actual experiment to run more slowly than our model predicts. In order to be more accurate the total ener gy of our model must account for the kinetic energy of the springs. \+ You actually have the tools to model this more-complicated situation, \+ using the ideas of total energy discussed in section 5.6, and a little Calculus. I carried out such an analysis, assuming that the spring v elocity at a point on the spring linearly interpolates the velocity o f the wall and mass (or mass and mass) which bounds it. Based on this analysis, the improved model predicts the two fundamental frequencies to be 1.68 cycles per second and 2.99 cycles per second. This agrees with our experimental results to within 1%, which is pretty amazing. \+ " }{TEXT 258 160 " I offer a prestigeous Math Department T-shirt to t he first student or group of students to submit this \"corrected\" mod el and results, including its derivation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Part 2: Washboarding revis ited. (The two-axle automobile page 332)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "restart:with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and trace have been redefined an d unprotected\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Here is the ma trix A which we deduce from equation 40 on page 332." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "A:=matrix(2,2,[-(k1+k2)/m, (k1*L1 - k2*L2 )/m, \n (k1*L1-k2*L2)/In, -(k1*L1^2+k2*L2^2)/In]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$,$*&,&%#k1G\"\"\"%#k2GF.F.% \"mG!\"\"F1*&,&*&F-F.%#L1GF.F.*&F/F.%#L2GF.F1F.F0F17$*&F3F.%#InGF1,$*& ,&*&F-F.)F5\"\"#F.F.*&F/F.)F7F@F.F.F.F:F1F1" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 56 "Let's do the system which results from the data in #24) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "m:=100;\nIn:=1000;\nL1 :=6;\nL2:=4;\nk1:=2000;\nk2:=2000;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"mG\"$+\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#InG\"%+5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#L1G\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#L2G\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G\"%+?" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G\"%+?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvects(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ 7%,&!#s\"\"\"*&\"\"%F&-%%sqrtG6#\"#uF&F&F&<#-%'vectorG6#7$F&,&#!\"%\" \"&F&*&#F&\"#5F&F)F&F&7%,&F%F&*&F(F&F)F&!\"\"F&<#-F/6#7$F&,&F3F&*&#F&F 8F&*$F)F&F&F<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%$!+$*)p!fP!\")$\"\"\"\"\"!<#-%'vec torG6#7$F'$\"*n_K-'!#57%$!+6I4k5!\"(F'<#-F,6#7$F'$!+FDBg;!\"*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "pi:=evalf(Pi);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#piG$\"+aEfTJ!\"*" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 59 "f1:=sqrt(37.59069893)/(2*pi);\nf2:=sqrt(106.4093011 )/(2*pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G$\"+v\"*)zv*!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2G$\"+q4wT;!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "fforce:=v/L;\n #forcing requency is spe ed divided\n #by wavelength of washboard (more like freeway\n #syndr ome than off-road case here..." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'f forceG*&%\"vG\"\"\"%\"LG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "L:=40;\nsolve(f1=v/L,v);\nsolve(f2=v/L,v);\n #critical car spee ds in feet/sec" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG\"#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+qc>.R!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!)Q/nl!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "39. 03195670*(1/5280)*3600;\n65.67043880*(1/5280)*3600;\n #speeds in mile s/hour\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+v(p7m#!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+?*HvZ%!\")" }}}}{MARK "19 1" 55 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }