Aufgabe 3.17 (Rollende Kugel auf schiefer Ebene)Erstellt: 21.2.2020
Ersteller: W. Kemmetm\303\274ller
\303\204nderungen:restart;
with(LinearAlgebra):
with(VectorCalculus,Jacobian):
with(VectorCalculus,SpaceCurve):
with(plots):
with(ColorTools):MFNWtKUb<ob<R=MDLCdNNZ@l:xT>D:;RLAwZU\\ZRLCTJcDXoHuuX;kD?qt@IBVowhqwhcR<EDN[BNCBkGT;SIkQu<KySiu\\;GRIcb;FDOTDKFc;G@CFLUSDuXqQDEyXZeiEQh>?YGKwwiIuagmWH=qrSIXyYX`msVWRXIuSIS@B:Gv@EB?=:;R^S:;B\\C=;bBm:;B>M@;BK_[:>ZZNl;ZB<;BJCs:B>J:<R^T:;BBZ:FLBJZ:N>BZvaeXpmXpv`qvV^]VnkHpKNc=Z<gZ??xQhnaxnYXaqosIysIfblVauViWgeWv`qvp[Hn[Nmc>^ZRsAB:<[ZygrXiqqw`isUXuTQu`ysYxicQt_FhkGxPplPViuWy=xZINiuWiu?ir?mTPm<G_;BjkI;IbeyiAytQyXwQyoAGIgDOcDYoIwMclAEGaf]Wfu;XZABTwuJAtjOV?QcDOcD;Gb>Z\\By=DDMvgUupGvawTioUGeROYwrarTgHe=tweXoehEWcE[dO<ZKdy;GxkIxsGisCycg[Gf;MhNcs;ybnYiquwZ_g:<rraJaQlyQlixUyPL`XLMTLeYweISP=VZ=VbXoeXypiwU\\lOBZviJof`o^[Oy_YsQ`_f^_^V_FqmwVmV`mvOwNIoJoDZZnhpoh<axcYtahdOgdyQrEphn_h^hn_H_b>_:Gy=BjKx;Ii^yi>?itYInYfMIXkEX`kF`KItKIKGdK?ysSyqywAQVkWRFYvwUYqqYEQs@OrA;JJFbIfgggg<ggo>ar>qYihhvex`q^GsLNr=QplPpSX`aodN_dNwdQW]SF]WFal_rDQsD?cB?kPplpyjivj=IvpywwyumyfXO`lOpJ?Y<MO<v:ixYQuqInZ@qoUwPpqxpYeUWeXoeXwXiqueR?eRCMO>yyuyuyqSV@Q;=mM<LJ<xditeHsMxN]HWnyTAykYtLJ<LryvYxYx=yTemWhmbtkwpmwTn[Hn;yUwa;;HlITXcIt]DDEviADKCDKAtJACF[CfggggxlQx\\Yv]Yv_iv_of`osBAsjMXlMUyoYwKg\\seysYX_eV_Eyoy>IcMCTW?EycY_QsOOytQymuvHIDPoCXwrIqexcRMQHlOHEyTf?bLwJ:Tn<DQtiJAPTrYmdTOedmWhmwdQwTTLMTBdLbeSGes=yjITMUTMqYMkXypiwMPlLPRShQNPykIjwUyvdQX@vYlsU=p:qu[QLNHQ:>]O>v:ixYqrCIss_nUWf]WfSGeSNm=PoEaxnaxwqoVauVaqsWx`W[EV[]V^]Hn^Ix[XZA>[s_kBIsCIb;XZ:>rbGZGYq@h[]V^]@iqHjlA]OynGisGypgWu@?s]aof`oUypiw[NxrAo[XNxjWhanmcV[gWQ<JKbbGgcGsFasFsupWg_gf?sb=iYtaxcEulgVTMUTey`IxKYrA]uXsI`CeiSFECdF^kT>hZQi>Pex`isawSYpOatNa_gf_mV`mFdOPq[Hn;IuSIeCGccxgXv]yNinIpeX_CaiPodPoaxnabFC<ByYQWeUWW[EV[mWhmSPiY;UxKWFA=hR]GTWHwMynqis]YlAykuTQwbwue>kcTUZJje`RuHLdTqHpkHXJgHYdarCtluyKtiU^TKgUq:<JZ]T;XXCYnRIo]yNX\\uuxQJQTs=wTQ=;l;yUjMy^IouyxhiKlPjf\\pgQ:[ZIWauOyniquvh@Fe;`mV`eTob@N[DJJDsZyKttYpawWXUqxJYuRR]R>Hj;^lOJ@ELbmuoLsvXkDPOhLOXXV`mTUUUMhmWhUR=TRywd=XNQ=;\\yVcUwbAfuBv[eV_dnpigupIx=yZLAc@X_Z@Z:>fnHj=vt]XeUWiuylrAinFfrVh`fZ?fZK_^=i\\JC:;R^NnepcH?efndF_eVIytipUwhiAtKItwwk^GeB`F;BZviEoeXouMy^X@g\\wthiuN_dNOimnfVOqVitc@[gf_g@oR@_Nx[:>Z:kYMXSIrkhqwhUHmUHAuNaJytVPHlKqReiSLMtLalN`lJma:>fnGj`ie]hd@ob@xmIx[T@usXu@@yfNxph]u>eR?eZ`I<Jo`Vjypd@sQ`Mj=QVajCXLFtVTLLIynaxnYApcULNxK:>JVblOhl?f:c`[WfawtqG?WfPAJ:L^rYchfer@khy^IGmlfdUW]DWZ:BB>xBhkCHkBv[AW\\pXifAnnJ:D\\FV]uY[Q>twpuFH]dXl`FbcC:;BK\\KI@oBAsIuMyDnChupmpJM=D<YfmWhMIrXqEG=EUOeb;GbPJ;B><PvLR<XRLlq^<TJ=TfauwXMDUJ:>B@x:Ivj=ydLxvijE\\N_dN[@R:<Z>O@xlxqyc=MFdPodxphMu<nn:fwgiwHxlV>t>Ql_Xc?hj?Hn:K:D<XXitoVamVLUTMuympgejLA:[:;UfktsQcMKeREhdeFisGi[DB<J:@c@xjaxjAGd[G^MV\\MqbeWgbf;<BAxbmWhmidawFoFPugWUr;Ir;cB<;JZjr=y:in\\`q<hlOhXWQyoATTJKJweUWioUtPJar`hsUHmS=UZ@b:>Z>[qwwVIogho_Qv`QekPg_^l@:FdOgdoAvXiinFp_WwPOitOib><<b@rrgX]qR=UR=?bZ?vLgVEAfn;B:\\Rl<MXdV_eVhpOIDt>qpWTucEWcEn:K\\jepSHAUflTF]UvEr<ijJM=D<lYxmqaxjALmRIquuLOdLO\\LBBZrjMivEigggGmSXmkcHONp:jKhlOhvmQruXqD`W=ER;tUJ<J:<BLXSIr[Xn]XV<arCljmdK[`@L>`tN`tQmyiDt>qWUUUbd[:>ZKl@IrXqMWySXPP?UUAuJDUJ::]rAir?ob>?fCSuwkXJM=C\\pnxQn_\\XiwH]m^dafc<PT>KNnYyqyA\\Wfk`nhSQu:GG^BTXODuRat>ETKELDU:ZTN\\li^xnm<Pj\\wvhiuJO]:jBsKIuSIci=WwsW;aYCESC=GJ>Z\\ZQoyiVdMOdcHocxpiauCfn;JBJTYuUyOuLQZHk_TSQPKETK=Ej:B\\BYsWiawf_WTMUTyYpgW?Ot:nrm>]rywIGr[AhR^[`ie^_tQaj>`uVaulQplNcdN_hogho_hn_sFisgocPeTNmThmWhmHokQHtDYsQoo^iv_ij<WZ:B\\<yWPmUXmYHYtmuuXXqtQytm^ETCEXkLqITKGdqxexdTMUTmn`ofER;EJMTLMqnGqtJmGByWdqYcQsVIueh?Gi?RMaDN[DV_UV=gx]Gi?fjWVQqGTMUTQK_ij?hjkDPktmx\\YfHocHyvIx]LUGmRPmJgLj\\XZ:><;ywKETKEpAuRqBwoYGofKeCrexTihEoDiKylivEqhymhnkHpKdF_hBoTx;iiueyweyKeHWcGWUnsY;WvOWslMxlATF?Z>CxES_eV_QuOUgMWdL?tngbOgfWquw_Gf[GbMioghOcuaiv`QfFkV\\MV\\[gYIuSIyhKwWKej;Hn[ynIw]mV@mrLir`yhiiuOaevCEDSD:;B@yFbEXwYENytQygTmiVcvkcIwciMihewFHee<yvkKFfKF<AsBAc`Ktj;E\\yIs=XiqYoeXosDqsd]WtOsWYuUcEES[TMouroY=MT\\MFNw>]FIgSGisGIIsCIsIysAg\\MyakeYcIgqUUUIFMG@ocHos_IdeGhCGSiOx_?fJGdOgRMCgcGg>GeocHosXiau;UDUmhheRckGdkFpavSYTNOsGcxoqtUsH[?g@gwL;B:kr[GHu=vtgiwgy;YbAsXhausaysYHDiRNeT?QVkOHlKHuuxhihn_hvici[xnIsM[ysYVAYGQmFcGgoiU;ER;EFxkGhcH_cbjsvPicULyD<KDlOSmpFtQAdtdxws`QR]PimYCEXkIv=tJKeTUlPG;gB_eV_Ux`isuAx[YBDOIquXqIimuv\\OGfMR<MbOuHQsDQsYyuYdMWdyUyEFNuuw?XhaIT?UmoWWGvfGewqg]sgZ?tOUSHEyXUSh?GisYrAr?cSfeuxgcrGfosIysy?ubEUGgodA[iteDsqHlMunIWjgX<VvDA]v>hWqtPqf`of@PxRfcYaqsWhCQl_>_IHtnajCV\\jIcV^\\N^hvAykyulQeIp]tpdQwdmWhmwwX>iJXfsgjhphNIwcGgWf]W>g<IuCIq_hn?_cV_i[Qn?_^eocOqtXhgf_uopsOX_jHg<B:DYNYqyuyxynIumBQqi`RT=qjE^?Of\\OftPqt@g\\QuoXgrqpJyqRhyhwkXFnCh[PYmbIwcyZTgc@odPPj<PZ?AhhochGf;GjaxjANmsXhqq?OgYWRNSitWfwkH`kF??w:uH:<JJDs]PtExLi<xM<oT]S=UR=MpNuRAuJWyTOeT_hn_hy\\YKwMyrIpf`_gf_gQfmWx;od=Gggn^hnbPihfQa?IyDX\\Ti\\JhlMX`kf`gyh_GcCGcvI[cydxy_[Ww>>Z:Hh__[XXcxQyoY_xOZEhrtYyuYlff_cF_;ycvWypicSGesydX?wW_dNNqQwtJA^VOxn>\\sFisawf`wbyxiymuGiogd?hgf?dwHqiQgr^]hv`avbLOd\\n_pnxg_b>hWqCp;T:<JK`QTepn`pNdLODvaXKDLKVeUWUyhXp^EsL\\TumTKETsirQQSD]wdAM^MlTanYloc@nb@kdeq[Dr;iMWdMMlNH=NnavOAuRAm@Hr^PNcDObql@HS=uwYtqbdQgdoghogypnHR=<J:<JK>]Jih=yZUPodPwMx^IoyVYgOn\\PNlwYuZPyAFlViqHGisGatxtUHnygsSIa_Fa_f^sFqYYo_hj?hjKaypqxpQjvaikG[C>u@G:;ZnPtLQtDV[EV^eqanfy=ilKO^tNatnnaxnMQtLqmxfiTHj;HZyvamV`MIsCIs;Ir[xu]^srgr?ieOIhHn[HHjP?ggggGAcIwbowuR@mIajSxlW^]Box[^f_af_gv^AdGO]_vpn`kFV`mVx>gfOpjtQyDQs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(Drehimpulserhaltung)Impulserhaltung in ex- und ez-Richtungode_x:=m*diff(x(t),t$2)=m*g*sin(alpha)-ft;
ode_z:=m*diff(z(t),t$2)=-m*g*cos(alpha)+fn;Tr\303\244gheitsmoment einer Kugeleq_I:=Iyy=2/5*m*R^2;Drehimpulserhaltungode_phi:=eval(Iyy,eq_I)*diff(phi(t),t$2)=ft*R;Kinematische Zwangsbedingung beim Rolleneq_x:=x(t)=phi(t)*R;
eq_z:=z(t)=0;Berechnung von ft aus ode_xsol_ft:=ft=eval(solve(ode_x,ft),eq_x);Einsetzen in Drehimpulserhaltungode_phi2:=diff(phi(t),t$2)=solve(eval(ode_phi,sol_ft),diff(phi(t),t$2));Haftgrenze ist erreicht wenn ft=muH fn giltsol_fn:=fn=eval(solve(ode_z,fn),eq_z);eq_muH:=simplify(eval(eval(muH*fn=ft,{sol_ft,sol_fn}),ode_phi2),trig);sol_muH:=muH>=solve(eq_muH,muH);