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iFGcCGcs_]t>iVN@?cCGar>iNjH[nJTBGHKTHKTBG;u>;mE W\\KW<@_MFdZWcTEmnFg?Uk@qrcLoPH?\\tbLqeniZpn_V_eV_ef>dMWdmlOmSp[HR sir?ibOGEH[ES[EVsBAgdOgDSsBVsbhebNswICMlPkDxMslUsoZ? F;Ed;u>;u>cs>iv>cs_]DR;MTXsb^`M^`K_`nV\\K_`KkbZbDdcCRiFR;ERiv>iFG]t>cS NL=\\dnvLiFGK_bTN@oEMmbLMTBG]lE?O;EGiNTLMT@_K_@L]nV\\M^@cCGaBGckeLmbXO T@OjhMNimPLoFq\\N_dNGmN_dN_DDgShkS`dN?ouniQIdQqiHvZ`Px?gbianiantndTpdT PmTPmP_NowZoGHkCHkCSCEVCEUebhychSuIyCiDoYLpDlpYPkYP;SESSU?;Dis_OvaysIy U?PkDxmDx=CEUSESSEUSEUC=S]^CHeBSeGUCESL@SMpLPpLPplpYlpYPKXyMcluDluDPKc xmYluKEoKmPplPp:x]fcuysIycNyCi_tIycNxMp^gfGjOfhf?Z:V<fuBR]lE]dQ@fQ@Far>cSNV\\K;cs_aZZ?fbxln>O]t_LmbVfn>fZ_`Z_`MBWc;LMtL]nVAkcNkCHCESCESejDPKVPKV\\mDPkDPkY\\MlPKl^g^_ffCUC]VjYx mYlPlLPr^f^_FlKHLPlLPlxMpLpYl`f_g^waLg\\LwiLg\\yNdyrID_yvaLWnfwafKpxMp BUxMqEoYx]yvimyaOvat`ff_gfSWSEU;B:AJ:>:1:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 288 "" 0 "" {TEXT 276 33 "Something about how t he bird work" }{TEXT 277 1 "s" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 846 "The bird is basically a baromometer. \+ It contains a liquid (freon-11) which has a very low boiling point. At a fixed temperature at which there are both liquid and gassous \+ freon present the pressure of the gas phase, the vapor pressure, is pr oportional to the temperature. Heat entering the bird increases the t emperature of the liquid, raising the vapor pressure of the gas. This exerts a downward force on the liquid in the body which attempts to \+ force the liquid up the \"neck.\" However if the bird is at a uniform temperature throughout, the vapor pressure above the liquid in the \+ neck is the same as that in the body so nothing happens. On the other \+ hand, if the bird has \"been drinking\" its head is wet with water w hich cools the top of the bird, lowering the pressure of the vapor in \+ the upper chamber (recall the gas law, " }{XPPEDIT 18 0 "PV=kT" "/ %#PVG%#kTG" }{TEXT 261 957 " which says that at constant volume the pr essure of a gas is proportional to temperature). This imbalance forc es the liquid up the bird's \"neck\" to a height required for weight of the liquid in the column to exactly offset the difference in pre ssure. If that equilibrium point is sufficient for the center of grav ity of the bird to be above its \"perch\" then the bird tips over and \+ \"takes a drink\". If the level of inclination of the drinking bird is such that the bottom of the \"neck\" is no longer beneath the surfac e of the liquid then the higher pressure gas on the bottom forces its \+ way up the tube in the form of bubbles, causing pressure to rise in th e upper chamber, forcing liquid back down the neck and raising the te mperature of the head. If this is sufficient to lower the center of g ravity below the \"perch\" then the bird's head rises from the liquid \+ it moves to minimize the height of the center of gravity, and the cycl e starts again." }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 259 16 "General Question" }{TEXT 271 210 ": If all of the fluid is in the body the bird \"sits\" on its perch. However when the fluid rises to a certain level the bird tips over and \"drinks\". How far up the \"neck\" is the fluid when the bird tips over?\n\n " } {TEXT 273 16 "Specific Problem" }{TEXT 272 333 ": Suppose the bird's n eck is a a cylinder .8 cm in diameter and 8 cm long and that the head \+ and body are spheres 2 cm in diameter. It initially has all of the li quid in the lower body which is completely full and the perch is 3.5 c m above the top of the lower body. How high is the liquid when the cen ter of gravity is at the perch?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 279 44 "Starting the model - simplifying assumptions" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 260 840 "As stated there is insufficient inform ation to solve the problem since the answer clearly depends on the wei ght of the liquid, the density and thickness of the glass and the den sity of the liquid vapor. If we assume that the liquid is weightless \+ then the center of gravity will already be above the perch, thus we mu st have a non-negative weight for the liquid in order to have an inter esting problem. The vapor consists of such a small amount of the liq uid that it can reasonably be assumed to be weightless in a first atte mpt. If the glass is sufficiently thin its contribution is small comp ared to that of the liquid so that the limiting case we can assume tha t the glass is weighless and of zero thickness. Solve the problem un der the assumption that the liquid has positive weight and the glass \+ is weightless and dimensionless." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT 268 29 "Some Maple-generated pictures" } {TEXT 274 144 ". These graphics are an introduction to things that can be done and may run slowly on some machines. They are not necessary a nd may be bypassed." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " } {MPLTEXT 1 0 20 "body:=x^2+y^2+z^2=1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "BODY:=plots[implicitplot3d](body,x=-1..1,y=-1..1,z=-1..1):\nRED BODY:=plots[implicitplot3d](body,x=-1..1,y=-1..1,z=-1..1,style=wirefra me,color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "head:=x^2+y^2+(z- 8)^2=(1/2)^2:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "HEAD:=plots[implic itplot3d](head,x=-1/2..1,y=-1/2..1,z= 7.5 ..8.5): " }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 112 "perch:=y^2+(z-2.5)^2=.1:\nneck:=x^2+y^2=.1^2:\nNEC K:=plots[implicitplot3d](neck,x=-.1.. .1,y=-.1.. .1,z=-.8 .. 8):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "PERCH:=plots[implicitplot3d](perch, x=-1..1,y=-.1...1,z=2.4..2.6):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 119 " plots[display]([HEAD,BODY,REDBODY,NECK,PERCH],scaling=constrained, tit lefont=[TIMES,BOLD,28],title = `Idealized Bird`);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 262 171 "Since we have are assuming that glass is weightl ess we don't really need the head. What we need is the fluid in the b ody and neck. The more idelaized diagram looks like:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 90 " \+ \nLIQUIDINBODY:=plots[implicitplot3d](body,x=-1..1,y=-1..1,z=-1..0,sty le=patch,color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "perch:=y^2+(z-2.5)^2=.1:\nneck:=x^2+y^2= .1^2:\nLIQUIDINNECK:=plots[implicitplot3d](neck,x=-.1.. .1,y=-.1.. .1, z=-.8 .. 4, style=wireframe, color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "PERCH:=plots[implicitplot3d](perch,x=-1..1,y=-.1...1, z=2.4..2.6):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "plots[disp lay]([ LIQUIDINBODY,LIQUIDINNECK,PERCH],scaling=constrained, titlefont =[TIMES,BOLD,28],orientation=[90,90],title = `Just the fluid`);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT 269 15 "Getting Started" }{TEXT 275 231 ": A general introduct ion to the problem for the student. NOTE: this is not a complete solut ion, nor is it guaranteed that the specific numbers used in it are tho se that apply to the problem stated. Some modifications may be needed. " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 168 " The problem is to calculate how high the fluid is when the center of gravity is exactly at the pe rch. If the center is higher than the perch the bird will tip over. \+ " }}{PARA 0 "" 0 "" {TEXT -1 723 "Choose coordinates so that the x-axi s is vertical and the body is the unit sphere centered at the origin. \+ Look at the corss-section in the x-y plane of the liquid. It is symmet ric with respect to the x-axis. It consists of the portion of the u nit circle from x= -1 to FL, the level of the liquid in the body, to gether with the rectangle with vertices ( FL,+-.4) and (LIQTOP, +-.4), where LIQTOP is the top of the liquid in the neck. For instance, if \+ when FL=-.4, LIQTOP=6, the edge above the x-axis is the graph of the f unction. The Heaviside function allows one to construct piecewise-defi ned functions in Maple. With it you can make a function which represen ts the cross-sectional view of the \"mathematical\" bird. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " \+ readlib(Heaviside):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 268 "With the \+ x-axis representing the vertical axis of the bird, if the fluid level \+ in the body is at x=.4 and the fluid extends to x=6 then the fluid in \+ the body and neck will be the solid of revolution generated by revolci ng the following function, f, about the x-axis. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "f:=x->(1-He aviside(x-(-.4)))*sqrt(1-x^2)+ Heaviside(x-(-.4)) *.4;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot(f(x),x=-1..6,scaling=constrained,axes=boxed);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 420 "If the fluid is at different levels in \+ the body then there will be a different cross-section, given by a diff erent function like the one above. These other functions would be crea ted by varying the fluid level, \"FL\" (in the function f above, FL= \+ .4) and and the top of the liquid in the neck., \" LIQTOP\" ( in t he function \"f\" above LIQTOP=6). We can make a better \"f\" which p ermits us to vary the fluid level with:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "liquid:=(x,FL)->(1-Heaviside(x-FL))*sqrt(1-x^2)+( Hea viside(x-FL))*.4 ; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "plot (liquid(x,.2), x=-1..6, scaling=constrained, axes=boxed);" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 264 340 " Since the total volume of the liquid i s constant (this isn't absolutely correct and is an implied assumption which should be justified) LIQTOP and FL determine each other. The t otal volume of the liquid is equal to the volume of the fluid in the body plus that in the neck. Thus we have the following relation which should be justified." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "EQ N:=Int(Pi*(sqrt(1-x^2)^2),x=-1..FL)+Pi*.4^2*(LIQTOP-FL)=FLUIDVOLUME;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 590 "which expresses LIQTOP as a s imple function of FL. By changing \"Int\" to \"int\" we get an equati on which can be solved for LIQTOP in terms of FL. This can be done by \+ hand or with the Maple \"solve\" command. One can then calculate LIQ TOP for any particular FL and (depending on the total volume which you must provide) make a plot of the cross-section of the liquid dependin g only on the value of FL. An expression for LIQTOP depending only on \+ FL and the FLUIDVOLUME can be substituted for LIQTOP in the followin g will then allow one to graph the fluid cross-section for any value \+ of FL: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "plot(liquid(x,FL ), x=-1..LIQTOP, scaling=constrained, axes=boxed);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 267 115 "For any fixed FL between -1 an d 1 the cross-sectional area of the liquid at x will be A(x,FL) =Pi* liquid(x,FL)^2" }}{PARA 284 "" 0 "" {TEXT 266 125 "The moment of that cross-section against the perch will be A(x,FL)*(x-PRCH), where PRC H is the x-coordinate of the perch." }}{PARA 285 "" 0 "" {TEXT -1 133 "Finally, for a fixed FL the total moment of the liquid against th e perch will be the integral of this function from -1 to LIQTOP(FL) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "TOTALMOMENT:=Int('A(x,FL)*(x-PRCH )',x=-1..LIQTOP);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 286 "" 0 "" {TEXT -1 824 "This is a function only of FL. The b ird is exactly balanced at the value of FL for which the total moment \+ is zero. Properly done the right side of the above equation depends on ly on the know values of FLUIDVOLUME and PRCH, and the variable FL. Th us commands like \"solve(TOTALMOMENT=0,\{FL\})\" or \"fsolve(TOTALMOME NT=0, \{FL\})\", or \"plot(TOTALMOMENT, FL = ?..?)\" will calculate th e value of FL at which the (idealized) bird is unstable. One should d o calculations for different, reasonable values of the fluid volume an d height of the bird's perch and make some observations and even edjus tments. One should certainly plot the TOTALMOMENT as a function of FL \+ for various values of FLUIDVOLUME and/or PRCH and interpret the differ ences in the graphs. Questions like \" are there always solutions to t he problem?\" are important." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}} {MARK "7 0 1" 224 }{VIEWOPTS 1 1 0 1 1 1803 }