理论力学全套解疑08

MP8-2aOOxyzAAζηξM8-2bxOy⎪⎩⎪⎨⎧==)()(tyytxx8-2Mr,ϕM⎪⎩⎪⎨⎧==)()(ttrrϕϕss=s(t)8-278-4τnbτnbb=τnτnbr=r(t)trdd=v22ddddtrtv==α)()()(321tfztfytfx===tztytxzyxdddddd===vvv222zyxvvvv++=vvxvx=),cos(vvzvvvyvzy==),cos(),cos(222222222ddddddddddddzyxzzyyxxaaaatztvtytvtxtv++=======αααaa)z,acos(x=aa)x,acos(aa)y,acos(zy==s=f(t)tsrdd=vnrnbnraaaaaavatva||tg0dd222r=+====βρτr=r(t)ϕ=ϕ(t)2222dddddddd⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛=+===trtrvvvtrvtrvrrϕϕϕϕttrtratrtrardddd2dddddd2222ϕϕϕϕ+=⎟⎠⎞⎜⎝⎛−=2r=xi+yj+zk222dddzyxds++=2222200||0ddϕϕϕτvvzvvvvrvvvvvtrrrzyxrrzyx+==++=+==++==vkji22222220022ddϕϕϕαaaaaaaaaaaaaatrnrzyxrnrzyx+=+=++=+=+=++==ranτakjirMM8-4τnbτnb(ab=0)annaτanτaρρτττ22ddddvtvvtvn==+=aτanab0a=aτ+anτnnaρ2vn=v2ρannnanMrϕ8-7M⎪⎩⎪⎨⎧==)()(ttrrϕϕr0OM=r0rrr=r0ϕ90ºϕ0r0ϕ0rkjikji0ϕ0r0ϕ0tr0ϕ0ttdddd00ϕrij8-71(1)⎪⎩⎪⎨⎧+−=+=jijirϕϕϕϕcossinsincos00ϕij0r0ϕ)sin(cos)(sin)(cosdd)cossin()(cos)(sindd000jijijijirrϕϕϕϕϕϕϕϕϕϕϕϕϕϕϕ+−=−−==+−=+−==ttϕ1(2)⎪⎩⎪⎨⎧−==0000rrϕϕϕϕϕtddϕO⎪⎩⎪⎨⎧≠−=≠=0dd0dd0000rrϕϕϕttϕttdddd00ϕr2v0000000dddddddd)(ddddϕϕϕϕrrtrtrtrtrrtt+=+=+===rrrrrrvtddva=tddrv=00ddϕϕ=tr00ddrϕ−=tϕA0tdd0AA0⋅A0=10dd200=⋅tAAA0tdd0AtttOCOCABABAB8-9OC=AC=BC=lMC=aϕ=ωtMMtOxy8-9⎪⎩⎪⎨⎧−==+=+=talAMytalCMOCxωϕωϕsin)(sincos)(cos)(t1)()(2222=−++alyalxMxy8-10aAOAOAABAOAOA+2AB8-10(b)OArAOACrCOAsOAAC2sOA+sOC8-108-11MA55s15B1s25B2t1r1t2r2∆t=t2t1∆r=r2r18-13t∆∆r∆tv*1212*ttt−−=∆∆=rrrv(1)v*∆r∆tt1s1t2s2∆t=t2t1∆s=s2s1ts∆∆∆tv1212ttsstsv−−=∆∆=(2)8-13(1)(2)v(t)r(t))(ddlim)(0tttttrrrv==∆∆=→∆(3))(ddlim)(0tstststvt==∆∆=→∆(4)∆t0∆r1ar0an02ar0an03ar0an04ar0an012341v=0a2a=0v3|∆r|=∆r|∆v|=∆v4vav5va1v=0a2a=0v03|∆r|r∆r∆r8-15a|∆v|∆v4vavaav5va8-15b8-151ar02an03a01ar02an0an=03a0a=022122ttv−+−=t=2v=0a=08-17va8-17ABaCvaDaEv0anFan0v=0v=5m/s0d)5(d===tvatvadd=vtv=v(t)0dd==tvarρ2vaan==(ρ∞)a8-20(a)8-20(b)a1=a2=a3a1ra2ra3rar8-20υr=ρ2van=anρan=M8-22MM0ss=ktkρρρ2220kaakvavaksvnnr========ρkar=0)(2210202020ssavvtavvttvsrrr−=−+=+=αarar=sartddrtrddvr=tddrddvtr=vr=tddr;trddrrvrvtddrtrdd.tddrtd||drtddrtd||drr|r|⎟⎟⎠⎞⎜⎜⎝⎛tddr⎟⎠⎞⎜⎝⎛td||drtddvtvddav=tddtvdd=aτvaτ1vatABOXOY8-271BcAM=aBM=bMMvaMtϕxyϕϕtxytϕtABxOB=ABcosϕ,OBOBϕtxyt8-27(a)(1)⎪⎩⎪⎨⎧==ϕϕsincosbyaxϕcos)(baOB+=)(ddOBt=cctbacbat=+−=+ddsin)(cos)(ddϕϕϕϕϕsin)(ddbact+−=(2)taatxvxddsin)cos(ddϕϕϕ−===baacbaca+=⎥⎦⎤⎢⎣⎡+−−=ϕϕsin)(sinϕϕϕϕϕϕctgsin)(cosddcos)sin(ddbabcbacbtbbtyvy+−=⎥⎦⎤⎢⎣⎡+−====ϕ22222ctgbabacvvvyx++=+=0dd==tvaxxϕ2222sin)(sin)(cscctgddddbabcbacbabcbabcttvayy+−=⎥⎦⎤⎢⎣⎡+−+=⎟⎠⎞⎜⎝⎛+−==ϕϕϕϕ32222sin)(babcaaayx+=+=va)(),(tyytxx==bkωtbrktωθ==,evatvardd=ar2r2aaan−=antbrktωθ==,ektktrbrvbkerve,ωθθ====ktktrbkbvvv22222222eeωθ+=+=rkbkkt2222eωω+=+=ktktktktktrbkbkOrrabkbbkrrae2e22e)(ee22222ωωθθωωθθ=+=+=−=−=−=ktktrbkbkaaa22222222222e4e)(ωωθ+−=+=rkbkkt)(e)(2222ωω+=+=ktrbkktvaedd22ω+==ktktrnkbkebkaaa222222222222e)()(ωω+−+=−=rkbkkt2222eωωωω+=+=2a=a(t)a=a(v)a=a(x)a=a(tvx)=v(t)x=x(t)vtAB300mABa1=0.7ga2=0.6gABtv8-272x1x2t1t2⎪⎪⎩⎪⎪⎨⎧−==2222221112121tavtxtax⎪⎩⎪⎨⎧+=−+=+==222211222221121221121212121tatatavttaxxtatav⎪⎩⎪⎨⎧×+×==)2(6.0217.021300)1(6.07.0222121gtgtgtgt(1)(2)t1=6.35st2=7.41sm/s6.4335.68.97.0s76.1321=××==+=vtttt2(s)a=5gts=tvdd5gttgtvtvvd5d010∫∫=v0=02125gtv===⎥⎦⎤⎢⎣⎡==∫ggttgts32065d2520322065.4(m)Mxx10−=M100m/sMtxxxtxxd10d10dd−=−==txxtxd10d000∫∫−=t∞xxxxtxxxtxxdddddddd===xx10−=xxxxd10d−=xxd10d−=xxxxd10d000∫∫−=xx1000−=−x101000−=−x=10(m)4x=kx2xt=0x=axtk0=x)(txx=xxxxtxxxtxxdddddddd====kx2xxxkxxdd2=122222Cxkx+=0,==xax2221akC=)(222222axkx−=22axkx−=(1)22ddaxktx−=tkaxxdd22=−21chCktax+=−axCkt=+)(ch2]ee[2)(ch)()(222CktCktCkt+−++1=+]ee[21)()(22CktCktax+−++=(2)t=0x=a)ee(21122CC−+=32eCC=012323=+−CC0)1(23=−C0,123==CC(2))ee(2ktktax−+=xxckt=0xkcta2−=0==xxxkctx2−=ctxkx=+2)sin(1α+=ktAxtkcx22=tkcktAx2)sin(++=αt=00==xx0,3=−=αkcA)sin(3ktktkcx−=3tnav2=ρRCu3-273M8-273xytvannav2=ρρ)(),(tyytxx==ϕOxy(t=0)cOytRutMCMHutMHOH====ϕM⎪⎩⎪⎨⎧−=−===−=−=−==ϕϕϕcossinRRBCHCHBAHyRRMBMHAHOHOAxϕM⎪⎪⎩⎪⎪⎨⎧−=−=RutRRyRutRutxcossin(1)t(1)MRutuvvvRutuvRutuvyxyx2sin2sincos122=+=⎪⎪⎩⎪⎪⎨⎧=⎟⎠⎞⎜⎝⎛−=(2)(2)M⎪⎪⎩⎪⎪⎨⎧==RutRuaRutRuayxcossin22(3)==+=Ruaaayx222vvr=vRutRutva2cosdd2rr==RutRuaaarn2sin222=−=RutRRutRuRutuavn2sin42sin2sin42222===ρ(4)RutRRMH2sin22sin2==ϕρ=2MHMH(4)ut=πRρmax=4R;ut=02πRρmin=0180°