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"" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 2" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 30 "Classical Mechanics with \+ Maple" }}{PARA 256 "" 0 "" {TEXT -1 62 "Section 2.2: Motions in Relati ve Coordinates in Two Dimensions" }}{PARA 19 "" 0 "" {TEXT -1 41 "Dr. \+ Harald Kammerer\nmaple@jademountain.de" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 14 "Initialisation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "libname:=\"C :/mylib/m6dynlib\",\"C:/mylib/m6dynfig\",libname:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 80 "with(linalg):with(plots):with(plottools):wit h(dynamics);with(figures_chapter_2);" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 37 "2.2.1 Motions in Relative Coordinates" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 386 "Sometimes it is suitable to describe motions not \+ in an inertial system but in a moving coordinate system, as we had don e above, when we described the motion in polar coordinates or in natur al coordinates. At this point, we want to survey some important relati ons, without deriving them in detail. (To show the exact derivation re quires some descriptions that are not so easy in Maple.) " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "In this section w e consider the special case of a planar system. The inertial system is given by the coordinates " }{TEXT 256 2 "x " }{TEXT -1 4 "and " } {TEXT 257 1 "y" }{TEXT -1 51 ". Additionally we have the moving system , given by " }{TEXT 261 1 "X" }{TEXT -1 5 " and " }{TEXT 262 1 "Y" } {TEXT -1 6 ". The " }{TEXT 263 4 "X-Y-" }{TEXT -1 63 "system is moving with translational motion with respect to the " }{TEXT 258 3 "x-y" } {TEXT -1 128 "-system. Translation means that the axes don't change th eir direction, but the origin of the system is moving. Additionally th e " }{TEXT 264 4 "X-Y-" }{TEXT -1 106 "system in general makes rotatio nal motions around its origin. Because we consider only planar motion, the " }{TEXT 259 1 "z" }{TEXT -1 10 "- and the " }{TEXT 266 1 "Z" } {TEXT -1 59 "-axes are always parallel. We describe the rotation of th e " }{TEXT 265 4 "X-Y-" }{TEXT -1 33 "system with the angular velocity " }{TEXT 260 1 "w" }{TEXT -1 15 " by the vector " }{XPPEDIT 18 0 "Ome ga;" "6#%&OmegaG" }{TEXT -1 56 " (Omega), which points in the directio n of the positive " }{TEXT 267 1 "Z" }{TEXT -1 20 "-axis. (See Fig 14. )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "display(Fig_2_8(),scal ing=constrained,axes=none,title=\"Figure 14\");" }}}{PARA 0 "" 0 "" {TEXT -1 121 "To define vectors in both coordinate systems we need uni t vectors. So we define the unit vectors in the direction of the " } {TEXT 364 2 "x-" }{TEXT -1 9 " and the " }{TEXT 365 2 "y-" }{TEXT -1 7 "axis as" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "ex:=vector(3,[1, 0,0]);" "6#>%#exG-%'vectorG6$\"\"$7%\"\"\"\"\"!F+" }}}{EXCHG {PARA 0 " > " 0 "" {XPPEDIT 19 1 "ey:=vector(3,[0,1,0]);" "6#>%#eyG-%'vectorG6$ \"\"$7%\"\"!\"\"\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 41 "The unit vectors in the direction of the " }{TEXT 366 1 "X" }{TEXT -1 10 "- and the " }{TEXT 367 2 "Y-" }{TEXT -1 40 "axis of the moving system are defined \+ by" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "eX:=vector(3,[eXx,eXy,0] );" "6#>%#eXG-%'vectorG6$\"\"$7%%$eXxG%$eXyG\"\"!" }}}{EXCHG {PARA 0 " > " 0 "" {XPPEDIT 19 1 "eY:=vector(3,[eYx,eYy,0]);" "6#>%#eYG-%'vector G6$\"\"$7%%$eYxG%$eYyG\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 89 "They depe nd on time, but we don't write this explicitly for the sake of clarity . Because " }{TEXT 368 1 "e" }{TEXT 396 3 "x, " }{TEXT 397 1 "e" } {TEXT 398 3 "y, " }{TEXT 399 1 "e" }{TEXT 400 1 "X" }{TEXT -1 4 " and " }{TEXT 370 1 " " }{TEXT 401 1 "e" }{TEXT 402 1 "Y" }{TEXT -1 42 " ar e unit vectors, they all have length 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "In the following, the derivative wit h respect to time is described by appending \"" }{TEXT 375 2 "_d" } {TEXT -1 103 "\" to the variable. For example: The derivative of the v ector rho with respect to time is described by \"" }{TEXT 373 1 "r" } {TEXT 374 2 "_d" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 77 "The \+ current position and the motion of the point Q is described by the vec tor" }{TEXT 369 3 " rQ" }{TEXT -1 106 ", which is a combination of the current position of the moving coordinate system, described by the ve ctor " }{TEXT 371 2 "rP" }{TEXT -1 118 ", and the relative position of the point Q with respect to the origin P of the moving system, descri bed by the vector " }{TEXT 372 1 "r" }{TEXT -1 7 " (rho)." }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "rP := xP*ex+yP*ey:" "6#>%#rPG,&*&%#xP G\"\"\"%#exGF(F(*&%#yPGF(%#eyGF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalm(rP);" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "rho:=Xrho*eX+Yrho*eY:" "6#>%$rhoG,&*&%%XrhoG\"\"\"%#eXGF(F(*&%%Yrho GF(%#eYGF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalm(rho); " }}}{PARA 0 "" 0 "" {TEXT -1 5 "Here " }{XPPEDIT 18 0 "Xrho*eXx;" "6# *&%%XrhoG\"\"\"%$eXxGF%" }{TEXT -1 52 " is the x-component of the proj ection of the vector " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 48 " \+ on the X-axis of the moving coordinate system, " }{XPPEDIT 18 0 "Yrho* eYx;" "6#*&%%YrhoG\"\"\"%$eYxGF%" }{TEXT -1 53 " is the x-component o f the projection of the vector " }{XPPEDIT 18 0 "rho" "6#%$rhoG" } {TEXT -1 60 " on the Y-axis of the moving coordinate system. According ly " }{XPPEDIT 18 0 "Xrho*eXy;" "6#*&%%XrhoG\"\"\"%$eXyGF%" }{TEXT -1 52 " is the y-component of the projection of the vector " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 48 " on the X-axis of the moving coord inate system, " }{XPPEDIT 18 0 "Yrho*eYy;" "6#*&%%YrhoG\"\"\"%$eYyGF% " }{TEXT -1 53 " is the y-component of the projection of the vector \+ " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 48 " on the Y-axis of the \+ moving coordinate system. " }}{PARA 0 "" 0 "" {TEXT -1 37 "For clarity we make the substitutions" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 " subs1:=Xrho*eXx=XQx:" "6#>%&subs1G/*&%%XrhoG\"\"\"%$eXxGF(%$XQxG" }}} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "subs2:=Yrho*eYx=YQx:" "6#>%&su bs2G/*&%%YrhoG\"\"\"%$eYxGF(%$YQxG" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "subs3:=Xrho*eXy=XQy:" "6#>%&subs3G/*&%%XrhoG\"\"\"%$eXy GF(%$XQyG" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "subs4:=Yrho*eYy= YQy:" "6#>%&subs4G/*&%%YrhoG\"\"\"%$eYyGF(%$YQyG" }}}{PARA 0 "" 0 "" {TEXT -1 34 "Now we can write for the vector rQ" }}{EXCHG {PARA 0 "> \+ " 0 "" {XPPEDIT 19 1 "rQ := rP+rho;" "6#>%#rQG,&%#rPG\"\"\"%$rhoGF'" } }}{PARA 0 "" 0 "" {TEXT -1 21 "This yields by using " }{TEXT 377 5 "su bs1" }{TEXT -1 4 " to " }{TEXT 376 5 "subs4" }{TEXT -1 29 " (we use th e appended letter " }{TEXT 378 1 "s" }{TEXT -1 61 " to distinguish the expression with and without substutution)" }}{EXCHG {PARA 0 "> " 0 " " {XPPEDIT 19 1 "rQs:=subs(\{subs1,subs2,subs3,subs4\},evalm(rQ));" "6 #>%$rQsG-%%subsG6$<&%&subs1G%&subs2G%&subs3G%&subs4G-%&evalmG6#%#rQG" }}}{PARA 0 "" 0 "" {TEXT -1 57 "The rotation of the X-Y-system is desc ribed by the vector" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "Omega:= vector(3,[0,0,omega]);" "6#>%&OmegaG-%'vectorG6$\"\"$7%\"\"!F*%&omegaG " }}}{PARA 0 "" 0 "" {TEXT -1 16 "The velocity of " }{TEXT 383 1 "Q" } {TEXT -1 74 " is compounded by three parts. At first we have the veloc ity of the point " }{TEXT 382 1 "P" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "vP:=vector(3,[xP_d,yP_d,0]);" "6#>%#vPG-%'vectorG6$\"\" $7%%%xP_dG%%yP_dG\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 46 "Next there is \+ the changing of the position of " }{TEXT 384 1 "Q" }{TEXT -1 27 " insi de the relative system" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "vr els:=vector(3,[XQx_d+YQx_d,XQy_d+YQy_d,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 41 "We write for the derivatives (remember: \"" }{TEXT 381 2 "_d" } {TEXT -1 44 "\" means the derivation with respect to time)" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "subs5:=XQx_d=Xrho_d*eXx:" "6#>%&subs5 G/%&XQx_dG*&%'Xrho_dG\"\"\"%$eXxGF)" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "subs6:=YQx_d=Yrho_d*eYx:" "6#>%&subs6G/%&YQx_dG*&%'Yrho _dG\"\"\"%$eYxGF)" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "subs7:=X Qy_d=Xrho_d*eXy:" "6#>%&subs7G/%&XQy_dG*&%'Xrho_dG\"\"\"%$eXyGF)" }}} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "subs8:=YQy_d=Yrho_d*eYy:" "6#> %&subs8G/%&YQy_dG*&%'Yrho_dG\"\"\"%$eYyGF)" }}}{PARA 0 "" 0 "" {TEXT -1 31 "Using this substitutions we get" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "vrel:=subs(\{subs5,subs6,subs7,subs8\},eval(vrels)); " }}}{PARA 0 "" 0 "" {TEXT -1 52 "Because we consider the changing of \+ the position of " }{TEXT 385 1 "Q" }{TEXT -1 73 " inside the relative \+ system there are no derivatives of the unit vectors " }{TEXT 386 2 "eX " }{TEXT -1 5 " and " }{TEXT 387 2 "eY" }{TEXT -1 168 " in this compon ent of the velocity. This is given by the third component which descri bes the additional velocity caused by the rotations of the moving coor dinate system" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "vrot:=crosspr od(Omega,rho);" "6#>%%vrotG-%*crossprodG6$%&OmegaG%$rhoG" }}}{PARA 0 " " 0 "" {TEXT -1 38 "or with use of the substitutions above" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "vrots:=subs(\{subs1,subs2,subs3,subs4 \},evalm(vrot));" "6#>%&vrotsG-%%subsG6$<&%&subs1G%&subs2G%&subs3G%&su bs4G-%&evalmG6#%%vrotG" }}}{PARA 0 "" 0 "" {TEXT -1 47 "caused by the \+ rotation of the relative system. " }}{PARA 0 "" 0 "" {TEXT -1 7 "We ca ll" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "v_g:=vP+vrot;" "6#>%$v_g G,&%#vPG\"\"\"%%vrotGF'" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "ev alm(v_g);" "6#-%&evalmG6#%$v_gG" }}}{PARA 0 "" 0 "" {TEXT -1 34 "and w ith use of the substitutions " }{TEXT 379 6 "subs1 " }{TEXT -1 2 "to" }{TEXT 380 6 " subs4" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "v_gs:= subs(\{subs1, subs2, subs3, subs4\},evalm(v_g));" "6#>%%v_gsG-%%subsG6 $<&%&subs1G%&subs2G%&subs3G%&subs4G-%&evalmG6#%$v_gG" }}}{PARA 0 "" 0 "" {TEXT -1 4 "the " }{TEXT 268 14 "guide velocity" }{TEXT -1 140 ", b ecause this is the velocity of the point which is fixed in the moving \+ system and has currently the same position as the moving particle. " } }{PARA 0 "" 0 "" {TEXT -1 97 "The total velocity is given by the guide velocity plus the velocity of Q inside the moving system" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "vQ := v_g+vrel;" "6#>%#vQG,&%$v_gG\" \"\"%%vrelGF'" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "evalm(vQ);" "6#-%&evalmG6#%#vQG" }}}{PARA 0 "" 0 "" {TEXT -1 24 "or written as com ponents" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "vx:=evalm(vQ)[1];" "6#>%#vxG&-%&evalmG6#%#vQG6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "vy:=evalm(vQ)[2];" "6#>%#vyG&-%&evalmG6#%#vQG6#\"\"#" } }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "vz:=evalm(vQ)[3];" "6#>%#vzG &-%&evalmG6#%#vQG6#\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 108 "Next we con sider the accelerations. Here we denote the second derivative with res pect to time by appending \"" }{TEXT 269 3 "_dd" }{TEXT -1 16 "\", for example \"" }{TEXT 271 1 "r" }{TEXT 270 3 "_dd" }{TEXT -1 2 "\"." }} {PARA 0 "" 0 "" {TEXT -1 65 "The acceleration of the origin of the mov ing system P is given by" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "aP := vector(3,[xP_dd, yP_dd, 0]);" "6#>%#aPG-%'vectorG6$\"\"$7%%&xP_ddG %&yP_ddG\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 52 "The acceleration of Q i nside the relative system is " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "arels := vector(3,[XQx_dd+YQx_dd, XQy_dd+YQy_dd, 0]);" }}}{PARA 0 "" 0 "" {TEXT -1 41 "As above we can write for the derivatives" }} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "subs9 := XQx_dd = Xrho_dd*eXx: " "6#>%&subs9G/%'XQx_ddG*&%(Xrho_ddG\"\"\"%$eXxGF)" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "subs10:=YQx_dd=Yrho_dd*eYx:" "6#>%'subs10G/%' YQx_ddG*&%(Yrho_ddG\"\"\"%$eYxGF)" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "subs11:=XQy_dd=Xrho_dd*eXy:" "6#>%'subs11G/%'XQy_ddG*&% (Xrho_ddG\"\"\"%$eXyGF)" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "su bs12:=YQy_dd=Yrho_dd*eYy:" "6#>%'subs12G/%'YQy_ddG*&%(Yrho_ddG\"\"\"%$ eYyGF)" }}}{PARA 0 "" 0 "" {TEXT -1 30 "With this substitutions we get " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "arel:=subs(\{subs9,subs1 0,subs11,subs12\},eval(arels));" }}}{PARA 0 "" 0 "" {TEXT -1 129 "Now \+ we consider the fixed point in the moving system which has currently t he same position as the mass. This point is called the " }{TEXT 404 16 "coinciding point" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 194 " Caused by the rotation of the moving coordinate system we get addition ally an acceleration for this coinciding point, same as described abov e for the description of motions in polar coordinates." }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "O_d:=vector(3,[0,0,o_d]);" "6#>%$O_dG -%'vectorG6$\"\"$7%\"\"!F*%$o_dG" }}}{PARA 0 "" 0 "" {TEXT -1 53 "We g et for this acceleration the tangential component" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "arot_t:=crossprod(O_d,rho);" "6#>%'arot_tG-%*cr ossprodG6$%$O_dG%$rhoG" }}}{PARA 0 "" 0 "" {TEXT -1 5 "From " }{TEXT 388 5 "subs1" }{TEXT -1 4 " to " }{TEXT 389 5 "subs4" }{TEXT -1 7 " we get" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "arot_ts:=subs(\{subs 1,subs2,subs3,subs4\},evalm(arot_t));" }}}{PARA 0 "" 0 "" {TEXT -1 30 "In the normal direction we get" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "arot_n:=crossprod(Omega,crossprod(Omega,rho));" "6#>%'arot_nG-%* crossprodG6$%&OmegaG-F&6$F(%$rhoG" }}}{PARA 0 "" 0 "" {TEXT -1 22 "And again with use of " }{TEXT 390 5 "subs1" }{TEXT -1 4 " to " }{TEXT 391 5 "subs4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "arot_ns:=sub s(\{subs1,subs2,subs3,subs4\},evalm(arot_n));" }}}{PARA 0 "" 0 "" {TEXT -1 35 "At last we have in the general the " }{TEXT 272 21 "corio lis acceleration" }{TEXT 406 0 "" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "acs := 2*crossprod(Omega,vrels);" "6#>%$acsG*&\"\"#\"\"\"-%*cros sprodG6$%&OmegaG%&vrelsGF'" }}}{PARA 0 "" 0 "" {TEXT -1 12 "Now we use " }{TEXT 392 5 "subs5" }{TEXT -1 4 " to " }{TEXT 393 5 "subs8" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "ac:=subs(\{subs5,subs6,subs7 ,subs8\},evalm(acs));" }}}{PARA 0 "" 0 "" {TEXT -1 92 "This part is al ways non-zero when a mass particle is moving in a rotating coordinate \+ system." }}{PARA 0 "" 0 "" {TEXT -1 7 "We call" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "a_g:=aP+arot_t+arot_n:" "6#>%$a_gG,(%#aPG\"\"\"%'a rot_tGF'%'arot_nGF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "eval m(a_g);" }}}{PARA 0 "" 0 "" {TEXT -1 34 "and with use of the substitut ions " }{TEXT 394 6 "subs1 " }{TEXT -1 2 "to" }{TEXT 395 6 " subs4" }} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "a_gs:=subs(\{subs1, subs2, sub s3, subs4\},evalm(a_g));" "6#>%%a_gsG-%%subsG6$<&%&subs1G%&subs2G%&sub s3G%&subs4G-%&evalmG6#%$a_gG" }}}{PARA 0 "" 0 "" {TEXT -1 4 "the " } {TEXT 273 18 "guide acceleration" }{TEXT -1 84 ", because this is the \+ acceleration of the point which is fixed in the moving system." }} {PARA 0 "" 0 "" {TEXT -1 34 "The total acceleration is given by" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "aQ:=evalm(a_g+ac+arel);" }}} {PARA 0 "" 0 "" {TEXT -1 24 "or written as components" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "ax:=evalm(aQ)[1];" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 17 "ay:=evalm(aQ)[2];" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "az:=evalm(aQ)[3];" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 43 "Example: Straight Motion on a Rolling Wheel" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "unassign( 'v','v0','a','R','phi','omega','rho','vg','Pg','V','V0','vP','vrot','v Q','arot_t','arot_n','ac');" }}}{PARA 0 "" 0 "" {TEXT -1 37 "Here we c onsider a wheel with radius " }{TEXT 282 1 "R" }{TEXT -1 62 " rolling \+ on the ground. On the wheel there is a mass particle " }{TEXT 281 1 "m " }{TEXT -1 215 " moving from the center in radial direction to the ri m of the wheel. The velocity of the wheel should be constant. The mass particle has a constant velocity relative to the wheel. The situation is shown in Fig. 13 (" }{TEXT 407 63 "click the figure and press \"pl ay\" on the button in the menu bar" }{TEXT -1 2 ")." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "display(Fig_2_9(),insequence=true,scaling=c onstrained,axes=none, title=\"Figure 15\");" }}}{PARA 0 "" 0 "" {TEXT -1 62 "The inertial system to describe the situation is given by the \+ " }{TEXT 274 3 "x-y" }{TEXT -1 30 "-system with the unit vectors " } {TEXT 290 1 "e" }{TEXT -1 0 "" }{TEXT 291 1 "x" }{TEXT -1 5 " and " } {TEXT 292 1 "e" }{TEXT -1 0 "" }{TEXT 293 1 "y" }{TEXT -1 1 "." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "ex:=vector(3,[1,0,0]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "ey:=vector(3,[0,1,0]);" }}} {PARA 0 "" 0 "" {TEXT -1 47 "The relative motion is described by the m oving " }{TEXT 275 4 "X-Y-" }{TEXT -1 44 "system with the time depende nt unit vectors " }{TEXT 294 1 "e" }{TEXT -1 0 "" }{TEXT 295 1 "X" } {TEXT -1 5 " and " }{TEXT 296 1 "e" }{TEXT 297 1 "Y" }{TEXT -1 1 "." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eX:=vector(3,[eXx,eXy,0]); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eY:=vector(3,[eYx,eYy,0 ]);" }}}{PARA 0 "" 0 "" {TEXT -1 18 "The origin of the " }{TEXT 276 3 "X-Y" }{TEXT -1 252 "-system is fixed at the center of the wheel, and \+ the system rotates with the wheel.. The moving mass particle is shown \+ as a blue dot in the figure. It moves with constant velocity in the di rection of the X-axis. The fixed inertial system is called the " } {TEXT 277 6 "global" }{TEXT -1 40 " system, the moved system is called the " }{TEXT 278 5 "local" }{TEXT -1 8 " system." }}{PARA 0 "" 0 "" {TEXT -1 79 "The velocity of the center of the wheel is constant and p oints in the positive " }{TEXT 280 2 "x-" }{TEXT -1 35 "direction. The velocity is given by" }{TEXT 279 0 "" }{TEXT -1 11 " the vector" }} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "v(t):=v0*ex:" "6#>-%\"vG6#%\"t G*&%#v0G\"\"\"%#exGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "ev alm(v(t));" }}}{PARA 0 "" 0 "" {TEXT -1 87 "We know that the velocity \+ of the wheel is constant. That yields for the position vector" }} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "P(t) := v0*t*ex:" "6#>-%\"PG6# %\"tG*(%#v0G\"\"\"F'F*%#exGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalm(P(t));" }}}{PARA 0 "" 0 "" {TEXT -1 56 "The roation of the wheel will be described by the angle " }{XPPEDIT 18 0 "phi" "6#%$phiG " }{TEXT -1 47 ". At this point we ignore the relation between " } {XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT -1 45 " and the position. We wil l consider it later." }}{PARA 0 "" 0 "" {TEXT -1 22 "The local coordin ates " }{TEXT 283 1 "X" }{TEXT -1 5 " and " }{TEXT 284 1 "Y" }{TEXT -1 31 " are also rotated by the angle " }{XPPEDIT 18 0 "phi" "6#%$phiG " }{TEXT -1 36 " from the orientation of the global " }{TEXT 285 4 "x- y-" }{TEXT -1 59 "system. This yields for the components of the unit v ectors " }{TEXT 286 1 "e" }{TEXT -1 0 "" }{TEXT 287 4 "X(t)" }{TEXT -1 5 " and " }{TEXT 288 1 "e" }{TEXT 289 4 "Y(t)" }}{EXCHG {PARA 0 "> \+ " 0 "" {XPPEDIT 19 1 "eXx:=cos(phi(t)):" "6#>%$eXxG-%$cosG6#-%$phiG6#% \"tG" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "eXy:=sin(phi(t)):" "6 #>%$eXyG-%$sinG6#-%$phiG6#%\"tG" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "eYx:=-sin(phi(t)):" "6#>%$eYxG,$-%$sinG6#-%$phiG6#%\"tG!\"\"" }} }{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "eYy:=cos(phi(t)):" "6#>%$eYyG -%$cosG6#-%$phiG6#%\"tG" }}}{PARA 0 "" 0 "" {TEXT -1 42 "and for the u nit vectors themselves we get" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "eX:=map(eval,eX);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " eY:=map(eval,eY);" }}}{PARA 0 "" 0 "" {TEXT -1 18 "The mass particle \+ " }{TEXT 299 1 "m" }{TEXT -1 39 " moves with constant relative velocit y " }{TEXT 403 4 "V(t)" }{TEXT -1 31 " in the direction of the local \+ " }{TEXT 300 1 "X" }{TEXT -1 12 "-coordinate." }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "V(t) := V0*eX:" "6#>-%\"VG6#%\"tG*&%#V0G\"\"\"%#eX GF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalm(V(t));" }}} {PARA 0 "" 0 "" {TEXT -1 51 "The position of the mass with respect to \+ the local " }{TEXT 298 3 "X-Y" }{TEXT -1 46 "-coordinate system is des cribed by the vector " }{XPPEDIT 18 0 "rho(t);" "6#-%$rhoG6#%\"tG" } {TEXT -1 18 " which is given by" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "rho(t) := V0*t*eX:" "6#>-%$rhoG6#%\"tG*(%#V0G\"\"\"F'F*%#eXGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalm(rho(t));" }}}{PARA 0 "" 0 "" {TEXT -1 95 "This vector is drawn in blue color in the figur e above. The results of the above relations for " }{XPPEDIT 18 0 "V(t) " "6#-%\"VG6#%\"tG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "rho(t)" "6#-%$ rhoG6#%\"tG" }{TEXT -1 42 " which are activated by the Maple-command \+ " }{TEXT 408 5 "evalm" }{TEXT -1 104 " are the descriptions of the vec tors in the global coordinate system. The position of the mass particl e " }{TEXT 301 1 "m" }{TEXT -1 34 " in global coordinates is given by " }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "Q(t) := P(t)+rho(t):" "6#> -%\"QG6#%\"tG,&-%\"PG6#F'\"\"\"-%$rhoG6#F'F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Q(t):=evalm(Q(t));" }}}{PARA 0 "" 0 "" {TEXT -1 64 "For the components of the position we have in global coordinates" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "xQ(t):=Q(t)[1];" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "yQ(t):=Q(t)[2];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "zQ(t):=Q(t)[3];" }}}{PARA 0 "" 0 " " {TEXT -1 26 "The rotation of the moved " }{TEXT 302 3 "X-Y" }{TEXT -1 45 "-coordinate system is described by the angle " }{XPPEDIT 18 0 " phi(t)" "6#-%$phiG6#%\"tG" }{TEXT -1 52 ", and the angle velocity of t he rotation is given by" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "o mega:=diff(phi(t),t);" }}}{PARA 0 "" 0 "" {TEXT -1 17 "We use the vect or" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Omega:=vector(3,[0,0,o mega]);" }}}{PARA 0 "" 0 "" {TEXT -1 83 "which describes the rotation \+ of the local system with respect to the global system." }}{PARA 0 "" 0 "" {TEXT -1 34 "The velocity of the mass particle " }{TEXT 303 1 "m " }{TEXT -1 45 " is compounded by the velocity of the origin " }{TEXT 304 2 "P " }{TEXT -1 30 "of the local coordinate system" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalm(v(t));" }}}{PARA 0 "" 0 "" {TEXT -1 42 "the relative velocity of the mass particle" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalm(V(t));" }}}{PARA 0 "" 0 "" {TEXT -1 85 "and additionally the velocity caused by the rotation of t he local coordinate system. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "vrot(t) := crossprod(Omega,rho(t));" }}}{PARA 0 "" 0 "" {TEXT -1 25 "We get the guide velocity" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "v_g:=v(t)+vrot(t):" "6#>%$v_gG,&-%\"vG6#%\"tG\"\"\"-%%vrotG6#F)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalm(v_g);" }}}{PARA 0 " " 0 "" {TEXT -1 80 "The total velocity is the guide velocity plus the \+ velocity of the mass particle " }{TEXT 305 1 "m" }{TEXT -1 44 " with r espect to the local coordinate system" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "vQ := evalm(v_g)+evalm(V(t));" "6#>%#vQG,&-%&evalmG6#%$ v_gG\"\"\"-F'6#-%\"VG6#%\"tGF*" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "evalm(vQ);" "6#-%&evalmG6#%#vQG" }}}{PARA 0 "" 0 "" {TEXT -1 24 "or written as components" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "vx:=evalm(vQ)[1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "vy:=e valm(vQ)[2];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "vz:=evalm(v Q)[3];" }}}{PARA 0 "" 0 "" {TEXT -1 99 "Comparison with the derivative of the components of the position vector with respect to time yields " }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "diff(xQ(t),t);" "6#-%%diff G6$-%#xQG6#%\"tGF)" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "diff(yQ (t),t);" "6#-%%diffG6$-%#yQG6#%\"tGF)" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "diff(zQ(t),t);" "6#-%%diffG6$-%#zQG6#%\"tGF)" }}}{PARA 0 "" 0 "" {TEXT -1 56 "Of course, there is no difference between the s olutions." }}{PARA 0 "" 0 "" {TEXT -1 55 "Next we consider the acceler ation of the mass particle." }}{PARA 0 "" 0 "" {TEXT -1 75 "The accele ration of the origin of the local coordinate system P is given by" }} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "a(t) := vector(3,[diff(v(t)[1] ,t), diff(v(t)[2],t), diff(v(t)[3],t)]);" "6#>-%\"aG6#%\"tG-%'vectorG6 $\"\"$7%-%%diffG6$&-%\"vG6#F'6#\"\"\"F'-F.6$&-F26#F'6#\"\"#F'-F.6$&-F2 6#F'6#F+F'" }}}{PARA 0 "" 0 "" {TEXT -1 71 "This result was clear beca use the velocity of the origin P is constant." }}{PARA 0 "" 0 "" {TEXT -1 47 "The relative acceleration of the mass particle " }{TEXT 307 1 "m" }{TEXT -1 12 " is given by" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "A(t) := vector(3,[diff(V(t)[1],t),diff(V(t)[2],t),dif f(V(t)[3],t)]);" }}}{PARA 0 "" 0 "" {TEXT -1 100 "Now we consider the \+ acceleration of the coinciding point caused by the rotation of the loc al system." }}{PARA 0 "" 0 "" {TEXT -1 80 "The changing of the angular velocity with respect to time is given by the vector" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "O_d:=vector(3,[diff(Omega[1],t),diff(Omega[ 2],t),diff(Omega[3],t)]);" "6#>%$O_dG-%'vectorG6$\"\"$7%-%%diffG6$&%&O megaG6#\"\"\"%\"tG-F+6$&F.6#\"\"#F1-F+6$&F.6#F(F1" }}}{PARA 0 "" 0 "" {TEXT -1 83 "With this we get for the tangential part of the accelerat ion caused by the rotation" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 " arot_t(t) := crossprod(O_d,rho(t));" "6#>-%'arot_tG6#%\"tG-%*crossprod G6$%$O_dG-%$rhoG6#F'" }}}{PARA 0 "" 0 "" {TEXT -1 35 "and in the norma l direction we have" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "arot_n( t) := crossprod(Omega,crossprod(Omega,rho(t)));" "6#>-%'arot_nG6#%\"tG -%*crossprodG6$%&OmegaG-F)6$F+-%$rhoG6#F'" }}}{PARA 0 "" 0 "" {TEXT -1 91 "In this example the angular velocity is constant and so its der ivative is zero. Then we get" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "arot_t(t):=subs(diff(phi(t),`$`(t,2))=0,arot_t(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "arot_n(t):=subs(diff(phi(t),`$`(t,2 ))=0,arot_n(t));" }}}{PARA 0 "" 0 "" {TEXT -1 77 "We see that only the acceleration in normal direction is different from zero." }}{PARA 0 " " 0 "" {TEXT -1 41 "At last we have the coriolis acceleration" }} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "ac(t) := 2*crossprod(Omega,V(t ));" "6#>-%#acG6#%\"tG*&\"\"#\"\"\"-%*crossprodG6$%&OmegaG-%\"VG6#F'F* " }}}{PARA 0 "" 0 "" {TEXT -1 37 "We collect now the guide acceleratio n" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "a_g:=a(t)+arot_t(t)+arot_ n(t):" "6#>%$a_gG,(-%\"aG6#%\"tG\"\"\"-%'arot_tG6#F)F*-%'arot_nG6#F)F* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalm(a_g);" }}}{PARA 0 "" 0 "" {TEXT -1 124 "Of course, it is the same as the rotational ac celeration in the normal direction, because all the other components a re zero." }}{PARA 0 "" 0 "" {TEXT -1 34 "The total acceleration is giv en by" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "aQ:=evalm(a_g+ac(t) +A(t));" }}}{PARA 0 "" 0 "" {TEXT -1 24 "or written as components" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "ax:=evalm(aQ)[1];" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "ay:=evalm(aQ)[2];" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "az:=evalm(aQ)[3];" }}}{PARA 0 "" 0 "" {TEXT -1 100 "Comparison with the derivative of the componen ts of the position vector in global coordinates yields" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "diff(xQ(t),t$2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "diff(yQ(t),t$2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "diff(zQg(t),t$2);" }}}{PARA 0 "" 0 "" {TEXT -1 13 "We know that " }{XPPEDIT 18 0 "diff(phi(t),`$`(t,2)) = 0;" "6#/-%% diffG6$-%$phiG6#%\"tG-%\"$G6$F*\"\"#\"\"!" }{TEXT -1 30 " and so the r esults are equal." }}{PARA 0 "" 0 "" {TEXT -1 30 "Now we want to have \+ a look at " }{TEXT 355 4 "some" }{TEXT 356 1 " " }{TEXT 353 13 "specia l cases" }{TEXT 354 1 "." }}{PARA 0 "" 0 "" {TEXT 306 35 "Case 1. The \+ wheel makes no rotation" }{TEXT -1 2 ". " }{TEXT 409 27 "It slides alo ng the ground." }}{PARA 0 "" 0 "" {TEXT -1 28 "Then we get for the pos ition" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "x_slip:=subs(phi(t) =phi0,xQ(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "y_slip:=su bs(phi(t)=phi0,yQ(t));" }}}{PARA 0 "" 0 "" {TEXT -1 23 "For the veloci ty we get" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "vx_slip:=eval(s ubs(phi(t)=phi0,vx));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "vy _slip:=eval(subs(phi(t)=phi0,vy));" }}}{PARA 0 "" 0 "" {TEXT -1 24 "An d for the acceleration" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ax _slip:=eval(subs(phi(t)=phi0,ax));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ay_slip:=eval(subs(phi(t)=phi0,ay));" }}}{PARA 0 "" 0 "" {TEXT -1 127 "In this case we wave the sum of two translational m otion with constant velocities. Then the general motion has no acceler ation." }}{PARA 0 "" 0 "" {TEXT 308 91 "2. The center of the wheel doe sn't make any translational motion, and the angular velocity " } {XPPEDIT 18 0 "omega0" "6#%'omega0G" }{TEXT 357 13 " is constant:" }} {PARA 0 "" 0 "" {TEXT -1 49 "Then we get for the position of the mass \+ particle" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "x_rot:=subs(\{ph i(t)=omega0*t,v0=0\},xQ(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "y_rot:=subs(\{phi(t)=omega0*t,v0=0\},yQ(t));" }}}{PARA 0 "" 0 " " {TEXT -1 23 "For the velocity we get" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "vx_rot:=eval(subs(\{phi(t)=omega0*t,v0=0\},vx));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "vy_rot:=eval(subs(\{phi(t)=o mega0*t,v0=0\},vy));" }}}{PARA 0 "" 0 "" {TEXT -1 24 "And for the acce leration" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "ax_rot:=eval(sub s(\{phi(t)=omega0*t,v0=0\},ax));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "ay_rot:=eval(subs(\{phi(t)=omega0*t,v0=0\},ay));" }}} {PARA 0 "" 0 "" {TEXT -1 81 "Here the acceleration is not 0, so we con sider the components of the acceleration" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 324 59 "acceleration of the origin of the local coordi nate system P" }{TEXT -1 8 " is here" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "a_rot_ori:=eval(subs(\{phi(t)=omega0*t,v0=0\},a(t))); " }}}{PARA 0 "" 0 "" {TEXT -1 98 "Of course, there is no acceleration \+ of point P, because we specified that this point doesn't move." }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 327 21 "relative acceleration " }{TEXT -1 22 " of the mass particle " }{TEXT 323 1 "m" }{TEXT -1 45 " is in global coordinates given by the vector" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "a_rot_rel:=map(eval,(subs(\{phi(t)=omega0*t,v0=0 \},A(t))));" }}}{PARA 0 "" 0 "" {TEXT -1 184 "Notice that the relative coordinates are defined by unambiguous relations with respect to the \+ global coordinates. So here all results are given with respect to the global coordinates." }}{PARA 0 "" 0 "" {TEXT -1 14 "Caused by the " } {TEXT 326 39 "rotation of the local coordinate system" }{TEXT -1 24 " \+ we get the acceleration" }}{PARA 0 "" 0 "" {TEXT -1 3 "in " }{TEXT 325 10 "tangential" }{TEXT -1 10 " direction" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "a_rot_tan:=map(eval,(subs(\{phi(t)=omega0*t,v0=0\} ,arot_t(t))));" }}}{PARA 0 "" 0 "" {TEXT -1 11 "and in the " }{TEXT 328 6 "normal" }{TEXT -1 18 " direction we have" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 62 "a_rot_nor:=map(eval,(subs(\{phi(t)=omega0*t,v0 =0\},arot_n(t))));" }}}{PARA 0 "" 0 "" {TEXT -1 20 "At last we have th e " }{TEXT 329 21 "coriolis acceleration" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "a_rot_cor:=map(eval,(subs(\{phi(t)=omega0*t,v0=0\},ev alm(ac(t)))));" }}}{PARA 0 "" 0 "" {TEXT -1 16 "We have now the " } {TEXT 330 18 "guide acceleration" }{TEXT -1 25 " as the sum of the par ts " }{TEXT 358 9 "a_rot_tan" }{TEXT -1 5 "(t), " }{TEXT 359 12 "a_rot _nor(t)" }{TEXT -1 5 " and " }{TEXT 360 4 "a(t)" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 84 "a_rot_gui:=map(eval,(subs(\{phi(t)=omega0*t,v0 =0\},evalm(a(t)+arot_t(t)+arot_n(t)))));" }}}{PARA 0 "" 0 "" {TEXT 309 34 "3. The wheel rolls without sliding" }}{PARA 0 "" 0 "" {TEXT -1 121 "This case can be sonsidered as a combination of the two situat ions above. The rotation of the wheel is given by the angle" }{TEXT 310 1 " " }{XPPEDIT 18 0 "phi(t) = -P(t)[1]/R;" "6#/-%$phiG6#%\"tG,$*& &-%\"PG6#F'6#\"\"\"F/%\"RG!\"\"F1" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "wh" }{TEXT 311 4 "ere " }{XPPEDIT 18 0 "P(t)[1];" "6#&-%\" PG6#%\"tG6#\"\"\"" }{TEXT 312 3 " is" }{TEXT -1 70 " the current posit ion of the origin of the local coordinate system in " }{TEXT 313 1 "x " }{TEXT -1 60 "-direction. Then we get for the postion of the mass pa rticle" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "subs13:=phi(t)=-ev alm(P(t))[1]/R;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "x_reel:= subs(subs13,xQ(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "y_re el:=subs(subs13,yQ(t));" }}}{PARA 0 "" 0 "" {TEXT -1 23 "For the veloc ity we get" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "vx_reel:=eval( subs(subs13,vx));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "vy_ree l:=eval(subs(subs13,vy));" }}}{PARA 0 "" 0 "" {TEXT -1 24 "And for the acceleration" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ax_reel:=ev al(subs(subs13,ax));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ay_ reel:=eval(subs(subs13,ay));" }}}{PARA 0 "" 0 "" {TEXT 314 7 "Notice: " }{TEXT -1 11 " The angle " }{XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT -1 171 " is choosen to start with the value 0 at the begining of the c onsidered period. The negative sign is used, because usually the angle is counter clockwise defined positive." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 316 59 "acceleration of the origin of the local coordina te system P" }{TEXT -1 20 " is in the last case" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 36 "a_reel_ori:=eval(subs(subs13,a(t)));" }}} {PARA 0 "" 0 "" {TEXT -1 55 "Of cause this acceleration is zero, bacau se the origin " }{TEXT 361 1 "P" }{TEXT -1 60 " makes a linear transla tional motion with constant velocity." }}{PARA 0 "" 0 "" {TEXT -1 4 "T he " }{TEXT 319 21 "relative acceleration" }{TEXT -1 22 " of the mass \+ particle " }{TEXT 315 1 "m" }{TEXT -1 24 " is in the global system" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "a_reel_rel:=subs(subs13,A(t) );" }}}{PARA 0 "" 0 "" {TEXT -1 14 "Caused by the " }{TEXT 318 39 "rot ation of the local coordinate system" }{TEXT -1 24 " we get the accele ration" }}{PARA 0 "" 0 "" {TEXT -1 3 "in " }{TEXT 317 14 "the tangenti al" }{TEXT -1 10 " direction" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "a_reel_tan:=subs(subs13,arot_t(t));" }}}{PARA 0 "" 0 "" {TEXT -1 11 "and in the " }{TEXT 320 6 "normal" }{TEXT -1 18 " direction we hav e" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "a_reel_nor:=map(eval,(s ubs(subs13,arot_n(t))));" }}}{PARA 0 "" 0 "" {TEXT -1 63 "This is also the same result as the corresponding component in " }{TEXT 362 6 "cas e 2" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "omega0=-v0/R" "6#/%'omega0G,$ *&%#v0G\"\"\"%\"RG!\"\"F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "At last we have the " }{TEXT 321 21 "coriolis acceleration" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "a_reel_cor:=map(eval,(subs(s ubs13,evalm(ac(t)))));" }}}{PARA 0 "" 0 "" {TEXT -1 56 "This is also t he same result as the accordingly part in " }{TEXT 363 6 "case 2" } {TEXT -1 5 " for " }{XPPEDIT 18 0 "omega0=-v0/R" "6#/%'omega0G,$*&%#v0 G\"\"\"%\"RG!\"\"F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 16 "We have now the " }{TEXT 322 18 "guide acceleration" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "a_reel_gui:=map(eval,(subs(subs13,evalm(a(t)+ arot_t(t)+arot_n(t)))));" }}}{PARA 257 "" 0 "" {TEXT -1 43 "At last we consider the situation for some " }{TEXT 343 15 "concrete values" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 331 30 "1. The w heel makes no rotation" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 3 " Let" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "R:=1;v0:=1;V0:=0.1;ph i0:=0;" }}}{PARA 0 "" 0 "" {TEXT -1 28 "Then we get for the position" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(x_slip);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(y_slip);" }}}{PARA 0 "" 0 "" {TEXT -1 23 "For the velocity we get" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf(vx_slip);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf(vy_slip);" }}}{PARA 0 "" 0 "" {TEXT -1 24 "And \+ for the acceleration" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eval f(ax_slip);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf(ay_sli p);" }}}{PARA 0 "" 0 "" {TEXT -1 117 "In this case the mass particle m oves with constant velocity translational in one direction. There is n o acceleration." }}{PARA 0 "" 0 "" {TEXT 332 100 "2. The center of the wheel don't make any translational motion and the angular velocity is constant:" }}{PARA 0 "" 0 "" {TEXT -1 3 "Let" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "R:=1;v0:=0;V0:=0.1;omega0:=-1;" }}}{PARA 0 "" 0 "" {TEXT -1 49 "Then we get for the position of the mass particle" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(x_rot);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(y_rot);" }}}{PARA 0 "" 0 "" {TEXT -1 23 "For the velocity we get" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(vx_rot);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(vy_rot);" }}}{PARA 0 "" 0 "" {TEXT -1 24 "And for the ac celeration" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(ax_rot); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(ay_rot);" }}} {PARA 0 "" 0 "" {TEXT -1 93 "The time histories of this solution is sh own in the following figures Fig. 16 - Fig. 18. The " }{TEXT 349 1 "x " }{TEXT -1 46 "-component is allways drawn in red color, the " } {TEXT 350 1 "y" }{TEXT -1 20 "-component in green." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 70 "Px2:=plot(evalf(x_rot),t=0..10,color=red, th ickness=3, legend=\"x(t)\"):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Py2:=plot(evalf(y_rot),t=0..10,color=green, thickness=3, legend= \"y(t)\"):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "display(\{Px2 ,Py2\},title=\"Figure 16 - Displacement\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "Pvx2:=plot(evalf(vx_rot),t=0..10,color=red, thickn ess=3, legend=\"vx(t)\"):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Pvy2:=plot(evalf(vy_rot),t=0..10,color=green, thickness=3, legend= \"vy(t)\"):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "display(\{Pv x2,Pvy2\},title=\"Figure 17 - Velocity\");" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 73 "Pax2:=plot(evalf(ax_rot),t=0..10,color=red, thickne ss=3, legend=\"ax(t)\"):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Pay2:=plot(evalf(ay_rot),t=0..10,color=green, thickness=3, legend=\"a y(t)\"):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "display(\{Pax2, Pay2\},title=\"Figure 18 - Acceleration\");" }}}{PARA 0 "" 0 "" {TEXT -1 52 "Here we consider the components of the acceleration." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 336 59 "acceleration of the origin \+ of the local coordinate system P" }{TEXT -1 20 " is in the last case" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "evalm(a_rot_ori);" }}} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 339 21 "relative acceleration " }{TEXT -1 22 " of the mass particle " }{TEXT 335 1 "m" }{TEXT -1 24 " is in the global system" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf(a_rot_rel)[1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "ev alf(a_rot_rel)[2];" }}}{PARA 0 "" 0 "" {TEXT -1 14 "Caused by the " } {TEXT 338 39 "rotation of the local coordinate system" }{TEXT -1 24 " \+ we get the acceleration" }}{PARA 0 "" 0 "" {TEXT -1 7 "in the " } {TEXT 337 10 "tangential" }{TEXT -1 10 " direction" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 20 "evalf(a_rot_tan[1]);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "evalf(a_rot_tan[2]);" }}}{PARA 0 "" 0 "" {TEXT -1 11 "and in the " }{TEXT 340 6 "normal" }{TEXT -1 18 " directi on we have" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf(a_rot_no r[1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf(a_rot_nor[2 ]);" }}}{PARA 0 "" 0 "" {TEXT -1 20 "At last we have the " }{TEXT 341 21 "coriolis acceleration" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf(a_rot_cor[1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "ev alf(a_rot_cor[2]);" }}}{PARA 0 "" 0 "" {TEXT -1 16 "We have now the " }{TEXT 342 18 "guide acceleration" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf(a_rot_gui[1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf(a_rot_gui[2]);" }}}{PARA 0 "" 0 "" {TEXT 333 35 "3. The wheel rolls without sliding " }{XPPEDIT 18 0 "phi(t) = -P(t )[1]/R;" "6#/-%$phiG6#%\"tG,$*&&-%\"PG6#F'6#\"\"\"F/%\"RG!\"\"F1" }} {PARA 0 "" 0 "" {TEXT -1 6 "Let be" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "R:=1;v0:=1;V0:=0.1;" }}}{PARA 0 "" 0 "" {TEXT -1 42 " Now we get for the postion of the particle" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "evalf(x_reel);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(y_reel);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "For the velocity we get" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf(vx_reel);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eval f(vy_reel);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "And for the accele ration" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf(ax_reel);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf(ay_reel);" }}} {PARA 0 "" 0 "" {TEXT -1 93 "The time histories of this solution is sh own in the following figures Fig. 19 - Fig. 21. The " }{TEXT 351 1 "x " }{TEXT -1 46 "-component is allways drawn in red color, the " } {TEXT 352 1 "y" }{TEXT -1 20 "-component in green." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 71 "Px3:=plot(evalf(x_reel),t=0..10,color=red, t hickness=3, legend=\"x(t)\"):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "Py3:=plot(evalf(y_reel),t=0..10,color=green, thickness=3, legend =\"y(t)\"):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "display(\{Px 3,Py3\},title=\"Figure 19 - Displacement\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "Pvx3:=plot(evalf(vx_reel),t=0..10,color=red, thi ckness=3, legend=\"vx(t)\"):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "Pvy3:=plot(evalf(vy_reel),t=0..10,color=green, thickness=3, lege nd=\"vy(t)\"):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "display( \{Pvx3,Pvy3\},title=\"Figure 20 - Velocity\");" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 74 "Pax3:=plot(evalf(ax_reel),t=0..10,color=red, t hickness=3, legend=\"ax(t)\"):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "Pay3:=plot(evalf(ay_reel),t=0..10,color=green, thickness=3, le gend=\"ay(t)\"):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "display (\{Pax3,Pay3\},title=\"Figure 21 - Acceleration\");" }}}{PARA 0 "" 0 " " {TEXT -1 4 "The " }{TEXT 334 59 "acceleration of the origin of the l ocal coordinate system P" }{TEXT -1 20 " is in the last case" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalf(a_reel_ori)[1];" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalf(a_reel_ori)[2];" }}} {PARA 0 "" 0 "" {TEXT -1 14 "Caused by the " }{TEXT 345 39 "rotation o f the local coordinate system" }{TEXT -1 24 " we get the acceleration " }}{PARA 0 "" 0 "" {TEXT -1 3 "in " }{TEXT 344 10 "tangential" } {TEXT -1 10 " direction" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "e valf(a_reel_tan[1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eva lf(a_reel_tan[2]);" }}}{PARA 0 "" 0 "" {TEXT -1 11 "and in the " } {TEXT 346 6 "normal" }{TEXT -1 18 " direction we have" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalf(a_reel_nor[1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalf(a_reel_nor[2]);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 86 "We see the time history of the components of this \+ part of the acceleration in Fig. 22." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "Pax3_nor:=plot(evalf(a_reel_nor)[1],t=0..10,color=red , thickness=3, legend=\"ax_normal(t)\"):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "Pay3_nor:=plot(evalf(a_reel_nor)[2],t=0..10,color=gre en, thickness=3, legend=\"ay_normal(t)\"):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 47 "display(\{Pax3_nor,Pay3_nor\},title=\"Figure 22\"); " }}}{PARA 0 "" 0 "" {TEXT -1 20 "At last we have the " }{TEXT 347 21 "coriolis acceleration" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ev alf(a_reel_cor[1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eval f(a_reel_cor[2]);" }}}{PARA 0 "" 0 "" {TEXT -1 86 "We see the time his tory of the components of this part of the acceleration in Fig. 23." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "Pax3_cor:=plot(evalf(a_reel _cor)[1],t=0..10,color=red, thickness=3, legend=\"ax_coriolis(t)\"):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "Pay3_cor:=plot(evalf(a_re el_cor)[2],t=0..10,color=green, thickness=3, legend=\"ay_coriolis(t)\" ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "display(\{Pax3_cor,Pa y3_cor\},title=\"Figure 23\");" }}}{PARA 0 "" 0 "" {TEXT -1 16 "We hav e now the " }{TEXT 348 18 "guide acceleration" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalf(a_reel_gui[1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalf(a_reel_gui[2]);" }}}{PARA 0 "" 0 "" {TEXT -1 204 "Like we have seen above in the general consideration of this e xample this is the same as the normal component of the acceleration ca used by the rotation of the local system, because all other parts are \+ 0." }}{PARA 0 "" 0 "" {TEXT 410 31 "Finally we give some animations" } {TEXT -1 84 ". But at first we define some simplified expressions for \+ the further presentations. " }}{PARA 0 "" 0 "" {TEXT -1 72 "We have th e position of the mass particle in the local coordinate system" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "X:=V0*t;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 5 "Y:=0;" }}}{PARA 0 "" 0 "" {TEXT -1 65 "Additi onally we have the position of the local coordinate system " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "x:=v0*t;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 5 "y:=0;" }}}{PARA 0 "" 0 "" {TEXT -1 30 "The local sys tem is rotated by" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "phi:=-v 0*t/R;" }}}{PARA 0 "" 0 "" {TEXT -1 53 "For the acceleration we have t he part of the guidance" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "a gx:=a_reel_gui[1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "agy:= a_reel_gui[2];" }}}{PARA 0 "" 0 "" {TEXT -1 41 "And for the coriolis a cceleration we have" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "acx:= a_reel_cor[1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "acy:=a_re el_cor[2];" }}}{PARA 0 "" 0 "" {TEXT -1 180 " At first we repeat the f igure from the beginning of this example (see the animation in Fig. 24 ). This shows the motion as people who are fixed in the global coordin ate system see." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "display(F ig_2_9(),insequence=true,scaling=constrained,axes=none, title=\"Figure 24\");" }}}{PARA 0 "" 0 "" {TEXT -1 152 "Interesting is also what peo ple see when they are moving themselves together with the local coordi nate system. This is shown in the animation in Fig. 25" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "display(Fig_2_10(R,x,y,X,Y,phi,t), insequence=true,scaling=constrained,axes=none, title=\"Figure 25\");" }}}{PARA 0 "" 0 "" {TEXT -1 386 "One of the most interesting things in this example are the accelerations. Fig 26 shows the guide accelerati on (blue) and the coriolis acceleration (red) which act on the mass pa rticle. The magenta arrow shows the sum of both parts. All the other p arts of the acceleration are zero, as we have seen above. Additionally we see the trajectory of the motion of the mass particle (red line). " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "display(Fig_2_11(R,x,y, X,Y,phi,agx,agy,acx,acy,t),insequence=true,scaling=constrained,axes=no ne, title=\"Figure 26\");" }}}{PARA 0 "" 0 "" {TEXT -1 139 "At last we see in Fig. 27 the same as above, but now from the point of view of s omeone who moves together with the local coordinate system." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "display(Fig_2_12(R,x,y,X,Y,phi,agx ,agy,acx,acy,t),insequence=true,scaling=constrained,axes=none, title= \"Figure 27\");" }}}{PARA 0 "" 0 "" {TEXT 405 10 "Conclusion" }}{PARA 0 "" 0 "" {TEXT -1 578 "In this example we can see the influence of th e coriolis acceleration. This acceleration is only different from zero for the situation that there is a motion inside a rotating system. In our example the angular velocity of the rotation is constant, also th e relative velocity is constant. This yields that the coriolis acceler ation also is constant in its absolute value. Only its orientation is \+ changing. The definition with use of the crossproduct yields that the \+ orientation of the coriolis acceleration always points orthogonal to t he plane, which is defined by the vectors " }{XPPEDIT 18 0 "Omega" "6# %&OmegaG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "V[rel]" "6#&%\"VG6#%$rel G" }{TEXT -1 1 "." }}}}}{MARK "0 0 0" 10 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }