M U............................................................................................................5nh ngha C hc: ..................................................................................................5i tng nghin cu: ........................................................................................5Phng php nghin cu.........................................................................................5Mc ch ca mn hc..............................................................................................5Ni dung....................................................................................................................5PHN 1. TNH HC ..........................................................................................61. CC KHI NIM C BN V H TIN TNH HC...........................................61.1. Cc khi nim c bn............................................................................................61.2 Cc nh lut tnh hc (H tin tnh hc) ..........................................................81.3 Cc dng lin kt - m hnh lc ca cc lin kt ...................................................81.4 Cc h qu .............................................................................................................91.4.1. Cc php bin i lc.....................................................................................91.4.2. Cc php bin i ngu lc..........................................................................102. H LC KHNG GIAN ..........................................................................................102.1 Hai i lng c trng ca h lc khng gian...................................................102.1.1 Vct chnh ca h lc khng gian...............................................................102.1.2 Mmen chnh ca h lc khng gian i vi mt im................................112.2 Thu gn h lc khng gian...................................................................................122.2.1 nh l di lc song song...............................................................................122.2.2 Thu gn h lc phng v tm O....................................................................132.3 iu kin cn bng v cc phng trnh cn bng ca h lc khng gian........132.3.1 iu kin cn bng .......................................................................................132.3.2 Cc phng trnh cn bng ca h lc khng gian......................................143 CC BI TON C BN CA TNH HC VT RN.............................................153.1 Bi ton mt vt rn cn bng di tc dng ca h lc ...................................153.1.1. Cc bc gii ...............................................................................................153.1.2. Cc dng bi ton.........................................................................................153.2 Bi ton cn bng ca h vt rn.........................................................................163.2.1.Cc bc gii ................................................................................................163.2.2. Cc dng bi ton.........................................................................................163.3 Bi ton cn bng vt rn c ma st ...................................................................163.3.1. nh lut ma st trt...................................................................................173.3.2. nh lut ma st ln......................................................................................173.3.3 Cc bc gii.................................................................................................174 TRNG TM VT RN...........................................................................................174.1 Tm ca h lc khng gian song song.................................................................174.2 Trng tm ca vt rn...........................................................................................18PH LC PHN 1. C IM CC PHN LC LIN KT THNG GP............20PHN 2. NG HC......................................................................................211. NG HC CHT IM........................................................................................211.1. Kho st cht im bng phng php vc t...................................................211.1.1. Phng trnh chuyn ng...........................................................................211.1.2. Vn tc..........................................................................................................211.1.3. Gia tc...........................................................................................................221.1.4. Du hiu v nhanh dn v chm dn ca chuyn ng..............................221.2. Kho st cht im bng phng php ta cc.......................................221.2.1. Phng trnh chuyn ng...........................................................................221.2.2. Vn tc..........................................................................................................231.2.3. Gia tc...........................................................................................................231.2.4. Du hiu v nhanh dn v chm dn ca chuyn ng..............................241.3. Kho st cht im bng phng php ta t nhin....................................241.3.1. Phng trnh chuyn ng...........................................................................241.3.2. Vn tc..........................................................................................................251.3.3. Gia tc...........................................................................................................251.4. Kho st mt s chuyn ng thng gp.........................................................251.4.1. Chuyn ng u..........................................................................................251.4.2. Chuyn ng bin i u............................................................................261.5. Cc loi bi ton trong phn ng hc cht im...............................................262. CHUYN NG C BN CA VT RN.............................................................302.1 Chuyn ng tnh tin ca vt rn........................................................................302.2. Chuyn ng ca vt rn quay quanh mt trc c nh......................................302.2.1. Kho st chuyn ng ca vt.....................................................................312.2.2. Kho st chuyn ng ca cc im thuc vt............................................322.2.3 Biu din vc t mt chuyn ng quay.......................................................332.3. Mt s dng truyn ng nh cc chuyn ng c bn.....................................342.3.1. Truyn ng bng c cu bnh rng, ai, xch............................................342.3.2. Truyn ng bng bnh rng thanh rng..................................................352.3.3. Truyn ng bng c cu cam.....................................................................352.4. Cc loi bi ton phn chuyn ng c bn ca vt rn....................................352.4.1. Cc dng bi ton.........................................................................................352.4.2. Phng php gii.........................................................................................363. TNG HP CHUYN NG IM........................................................................363.1. nh ngha v cc loi chuyn ng....................................................................363.2. nh l hp vn tc..............................................................................................373.3. nh l hp gia tc...............................................................................................383.4. Cc dng bi ton ...............................................................................................403.4.1. Dng bi ton................................................................................................403.4.2. Phng php gii.........................................................................................404. TNG HP CHUYN NG VT RN.................................................................414.1. Tng hp chuyn ng song phng ca vt rn.................................................414.1.1. Chuyn ng ca hnh phng.......................................................................414.1.2 Cc yu t ng hc ca cc chuyn ng thnh phn...............................424.1.3 Kho st chuyn ng ca im...................................................................424.1.4 Cc dng bi tp............................................................................................464.2. Tng hp hai chuyn ng quay quanh hai trc song song...............................464.3. Chuyn ng inh c...........................................................................................484.3.1 Cc yu t ng hc ca cc chuyn ng thnh phn...............................484.3.2. Kho st chuyn ng ca im..................................................................48PHN 3. NG LC HC..............................................................................501. CC KHI NIM V CC NH LUT CA NG LC HC.............................501.1. Cc khi nim c bn..........................................................................................501.1.1. Cc m hnh vt th......................................................................................501.1.2. Lc tc dng ln c h.................................................................................501.1.3. H quy chiu qun tnh.................................................................................511.2 Cc nh lut c bn ca ng lc hc................................................................521.3. Phng trnh vi phn chuyn ng ca cht im.............................................521.4. Hai bi ton c bn ca ng lc hc.................................................................532. CC NH L TNG QUT CA NG LC HC..............................................532.1. Cc c trng hnh hc khi ca c h v vt rn.............................................532.1.1 Khi tm ca c h v vt rn.......................................................................532.1.2. M men qun tnh ca vt rn......................................................................542.2. nh l ng lng...............................................................................................572.3. nh l chuyn ng khi tm..............................................................................582.4. nh l m men ng lng................................................................................582.5. nh l ng nng................................................................................................592.5.1. nh l ng nng: ........................................................................................592.5.2. p dng.........................................................................................................602.6. nh l bo ton c nng.....................................................................................622.6.1. Trng lc th Th nng...........................................................................622.6.2. nh l bo ton c nng. ............................................................................622.6.3. Biu thc tnh th nng ca mt s lc th.................................................633. MT S NGUYN L C HC.............................................................................633.1. Cc khi nim c bn..........................................................................................63 3.1.1. Di chuyn kh d v s bc t do ca h....................................................633.1.2. To suy rng............................................................................................633.1.3. Lc suy rng.................................................................................................643.1.4. Lin kt l tng...........................................................................................643.2. Nguyn l cng o...............................................................................................653.2.1. Nguyn l cng o: ......................................................................................653.2.2. iu kin cn bng ca c h trong to suy rng .............................663.3. Nguyn l d'Alembert...........................................................................................663.3.1. Lc qun tnh ca cht im........................................................................663.3.2. Nguyn l d'Alembert i vi cht im........................................................663.3.3. Nguyn l d'Alembert i vi c h...............................................................673.4. Nguyn l d'Alembert-Lagrange..........................................................................673.5. Phng trnh Lagrange loi 2..............................................................................683.5.1. Vt rn chuyn ng tnh tin.......................................................................683.5.2. Vt rn quay quanh mt trc c nh............................................................683.5.3. Vt rn chuyn ng song phng.................................................................693.5.4. Phng trnh vi phn chuyn ng ca c h Phng trnh Lagrange loi II........................................................................................................................................70M Unh ngha C hc: C hc l khoa hc nghin cu v chuyn ng v cn bng ca cc vt th. i tng nghin cu: Cch k thutnh mycngtrnhxy dng,phngtin giao thng. nghin cu v tnh ton chuyn ng ca cc h k thut trc ht phi xy dng m hnh c hc cho cc itng kho st. Cc m hnh c hc ang c s dng gm: h cc cht im; h cc vt rn; h cc vt th bin dng; h mi trng lin tc; cc h hn hp. Trong khun kh ca C hc l thuyt ta ch xt m hnh vt rn v h cc vt rn phng.Phng php nghin cuPhng php suy din Phng php m hnhMc ch ca mn hc Cung cp kin thc c bn v tng qut v chuyn ng v cn bng ca vt rn, h vt rn phng Trang b kin thc c hc c bn tip cn vi h my mc v cng trnhNi dungPhn 1. Tnh hc- Lc, s cn bng ca vt rn di tc dng ca lcPhn II. ng hc - Chuyn ng c hc ca vt th v mt hnh hc.Phn III. ng lc hc - Chuyn ng c hc ca vt th di tc dng ca lcThc t, thc nghimH thng cc khi nim H tin Cc nh l C hcCc phng trnh chuyn ngGii quyt cc bi ton thc tSuy din ton hcXy dngp dngi tng kho stXy dng cc m hnh c hcp dng cc nh lut c hcXy dng m hnh ton hcKt qu tnh tonp dng vo cc bi ton thit k v cng nghSo snhKt qu thcCha ttTtPHN 1. TNH HC Tnh hc nghin cu trng thi cn bng ca vt rn di tc dng ca lc. Ni dung ca phn ny gm cc vn sau: Thu gn h lc: bin i h lc cho thnh mt h lc khc tng ng nhng n gin hn Tm iu kin cn bng ca mt vt rn v h vt rn di tc dng ca lc1. CC KHI NIM C BN V H TIN TNH HC1.1. Cc khi nim c bn Vt rn tuyt i (vt rn)- l tp hp cc cht im m khong cch gia hai im bt k ca n khng i. y l s l tng ho v nh vy di tc dng ca lc vt rn khng b bin dng. Cn bng - l trng thi ng yn ca vt rn so vi mt vt rn khc c chn lm chun (h quy chiu)H quy chiu - l vt th c chn lm mc theo di chuyn ng. Trong c hc ta gn h to vo h quy chiu. Lc - l tng tc gia cc vt m kt qu ca n gy ra s bin i trng thi chuyn ng c hc (thay i v tr) m cn bng ch l trng hp ring.T kinh nghim v thc nghim lc c c trng bi 3 yu t: im t, Phng chiu v ln, Mhnhtonhccalclvctlc, khiulF. Tronghto Descartes vung gc, vec t F c biu din di dngz z y y xF F F e e e F + + trong z y e , e , e l cc vct n v trn cc trc to Ox, Oy, Oz v z y xF F F , , l hnh chiu ca F ln cc trc Ox, Oy, Oz. Mdun ca vct lc 2 2 2z y xF F F F + + Hng ca lcF, ) , cos(FFF Oxx, ) , cos(FFF Oyy FFF Ozz ) , cos( n v ca lc l Newton, k hiu l (N) H lc l tp hp cc lc tc dng ln mt vt rn, c k hiu) F , , F , F (N 2 1Hai h lc tng ng - 2 h lc) F , , F , F (N 2 1v) , , , (N 2 1 tc dng ln cng mt vt rn l tng ng nu chng gy nn trng thi chuyn ng ca vt rn nh nhau.) , , ( ) , , , (N NF F F 2 1 2 1Hp lc ca h lc - l mt lc R tng ng vi h lc) , , , (NF F F 2 1) , , , (NF F F R 2 1H lc cn bng l h lc) , , , (NF F F 2 1m di tc dng ca n vt rn khng thay i trng thi chuyn ng hay cn bng 02 1 ) , , , (NF F F Ngu lc -Mt h gm hai lc song song, ngc chiu cng ln c gi l ngu lc. Mt ngu lc c c trng bi cc yu t Mt phng tc dng- l mt phng cha 2 thnh phn ca ngu lc Chiu quay ca ngu lc Cng tc dng ca ngu lc c c trng bng tch Fd, trong F gi trca lc thnh phn, d khong cch gia haing tc dng ca hailc thnh phn gi l cnh tay n ca ngu lc. n v ca ngu lc l Newton.mt, k hiu l (Nm)Trong khng gian ngu lc biu din bng vct m men ngu lcm, c xc nh bng Phng vung gc vi mt phng cha ngu lc Chiu: nhn t nh xung thy chiu quay ca lc ngc chiu kim ng h M un ca vec t ngu lc bng F.d Quy c gc ca vctm nm trong mt phngTrong mt phng ngu lc biu din qua m men i s ngu lc t Fd, (+) khi chiu quay ca ngu lc ngc chiu kim ng h. Lin kt v phn lc lin kt. Lin kt l cc iu kin cn tr di chuyn ca vt. Vt b cn l vt khng t do. Lc lin kt - lc tc dng gia cc vt khng t do. Phn lc lin kt - l lc lin kt do vt khc tc dng ln vt ang kho st Cn bng ca h nhiu vt rn. H nhiu vt rn c gi l cn bng nu mi vt rn thuc h cn bng1.2 Cc nh lut tnh hc (H tin tnh hc) nh lut 1 (Tin v s cn bng ca vt rn) - iu kin cn v mt vt rn cn bng di tc dng ca hai lc l 2 lc ny c cng ng tc dng, cng cng v ngc chiu nhau - tiu chun cn bng ca vt t do di tc dng ca h lc n gin nht nh lut 2 (tin thm hoc bt mt cp lc cn bng) - tc dng ca mt h lckhng thay i nu ta thm (bt) i hai lc cn bng - quy nh vmt php bin i tng ng c bn v lcH qu (nh l trt lc). Tc dng ca lc khng thay i khi ta trt lc trn ng tc dng ca n nh lut 3 (Tin hnh bnh hnh lc)- Hai lc tc dng ti mt im tng ng vi mt lc tc dng ti cng im v c vct lc bng vct cho ca hnh bnh hnh c hai cnh l 2 vct lc ca cc lc cho nh lut 4 (tin tc dng v phn tc dng) -Lc tc dng v lc phn tc dng gia haivt c cng cng , cng ng tc dng v hng ngc chiu nhau. nh lut 5. (tin Ho rn) - mt vt bin dng cn bng di tc dng ca mt h lc th khi ho rn n vn trng thi cn bng nh lut 6 (Tin thay th lin kt) Vt khng t do cn bng c th c xem l vt t do cn bng bng cch gii phng tt c cc lin kt v thay th tc dng cc lin kt c gii phng bng cc phn lc thch hp1.3 Cc dng lin kt - m hnh lc ca cc lin kt Lin kt ta - phn lc ta c phng vung gc vi mt ta Lin kt dy mm, thng khng dn - phn lc ca dy tc dng ln vt kho st t vo im buc dy v hng vo dy Lin kt bn l - Phn lc i qua tm trc bn l phng chiu cha xc inh nn ta phn thnh hai thnh phn Rx v Ry nm trong mt phng vung gc vi trc tm bn l Lin kt gi - c hai loi: + gi c nh phn lc xc nh nh lin kt bn l, + gi con ln phn lc lin kt xc nh theo lin kt ta. Lin kt gi cu - qu cu gn vo vt chu lin kt c t trong v cu gn vi vt gy lin kt. Phn lc i qua tm O ca qu cu c phng chiu cha xc nh ta phn thnh 3 thnh phn (Rx, Ry, Rz) Lin kt ngm - ni cng. Phn lc lin kt gm 3 lc thng gc v 3 ngu lc trong 3 mt phng to . Trng hp phng cn 3 thnh phn lc v 1 ngu lc Lin kt thanh - thanh lin kt tho mn cc iu kin: + ch c lc tc dng hai u+ dc thanh khng c lc tc dng+ trng lng thanh b quaPhn lc c phng qua hai im chu lcTm li: tng ng vi hng di chuyn thng b ngn tr c phn lc ngc chiu, tng ng vihng dichuyn quay bngn tr c ngu phn lc ngc chiu (xem ph lc 1 phn 1)1.4 Cc h qu 1.4.1. Cc php bin i lcGi s c h lc) , , , (NF F F 2 1 t ti O. p dng nh lut hnh bnh hnh lc ta tm c hp lc R ca h lc + + + Nkk NF F F F R12 1 (1.1)Vit di dng cc thnh phn lc chiu ln trc to ta c + + + Nkkx Nx x x xF F F F R12 1; + + + Nkky Ny y y yF F F F R12 1; + + + Nkkz Nz z z zF F F F R12 1(1.2)T y ta xc nh ln, phng chiu ca hp lc:2 2 2z y xR R R R + + ;RRR Oxx ) , cos(,RRR Oyy ) , cos(,RRR Ozz ) , cos(nh l 1.1.1. Hp lc ca h lc ng quy c biu din bng vct chnh ca h lc t ti im ng qui1.4.2. Cc php bin i ngu lcnh l 1.1.2.Hai ngu lc cng nm trong mt mt phng c cng chiu quay v cng tr s m men (tc cng m men i s) th tng ng nhau nh l 1.1.3. Tc dng ca ngu lc khng i khi di n nhng mt phng //nh l 1.1.4. Hai ngu lc c cng vct m men th tng ng nhau Tc dng ca ngu lc c c trng hon ton bng vct m men ngu lc (m men i s ngu lc trong trng hp phng)nh l 1.1.5.Hp cc ngu lc c mt ngu lc c vct bng tng cc vct mmen cc ngu lc choNkkm m1 (1.3)Hp cc ngu lc trong cng mt mt phng c mt ngu lc nm trong mt phng cho c m men i s bng tng cc mmen i s cc ngu lc choNkkm m12. H LC KHNG GIAN Trong mc ny ta xt 2 bi ton: Bin i h lc khng gian v dng n gin v tm iu kin cn bng ca vt rn khng gian2.1 Hai i lng c trng ca h lc khng gian2.1.1 Vct chnh ca h lc khng giannh ngha.Vct chnh ca h lc k hiu l R, l vct tng ca cc vec t lc thnh phn ca h lc + + + Nkk NF F F F R12 1 (1.4)Xc nh vec t chnh. Chiu lc thnh phn kF ln 3 trc to vung gc Ox, Oy v Oz ta c) , , (kz ky kx kF F F F t (1.1.4) ta c hnh chiu ca R ln 3 trc to l + + + Nkkx Nx x x xF F F F R12 1 + + + Nkky Ny y y yF F F F R12 1 + + + Nkkz Nz z z zF F F F R12 1(1.5)Gi tr v phng chiu ca vct chnh c xc nh2 2 2z y xR R R R + + , RRR Oxx ) , cos(, RRR Oyy ) , cos(, RRR Ozz ) , cos(.2.1.2 Mmen chnh ca h lc khng gian i vi mt im Mmen ca mt lc i vi mt im. Mmen ca lc F i vi im O, k hiu l) (F mO, l vct c: Phng vung gc vi mt phng cha im O v lc F, Chiu: khi nhn t u mt xung F quay ngc chiu kim ng h quanh O Tr s bng Fd, y d l khong cch t O n ng tc dng. n v NmVctm men lc ivimt im khng ikhilc trt trn ng tc dng ca n v trit tiu khii qua im ly m men. Tr s m men bng 2 ln din tch tam gic OAB, (A im t lc, B im mt ca lc) Vctmmenca lcFi vi imOltchvec tcavectnhv OA r im t lc vi vec t lc F (h8-2)F r F mO ) ( (1.6) Mmencalci vi mt trc.MmencalcFi vi trck hiu ) (F m l m men i s ca lc F vi im O, F l hnh chiu ca F trn mph vung gc vi trc , cn O l giao im ca trc vi mph d F F m F mO t ) ( ) ( (1.7)Khi F // v ct trc th0 ) (F m Lin h gia m men ca lc i vi im v m men ca lc i vi trcnh l 1.3.1.M men ca lc Fivitrc bng hnh chiu trn trc y ca vct m men ca lc F i vi im O nm trn trc)] ( [ ) ( F m hch F mO (1.8)S dng cng thc (1.1.6) cho ln lt cc trc Ox, Oy v Oz ta c c hnh chiu ca vct m men ca lc F i vi im O' x y Z Ozz x Y Oyy z X OxOyF xF F m F mxF zF F m F mzF yF F m F mF m) ( ) () ( ) () ( ) () ( (1.9) vec t m men chnh ca h lc khng gian i vi im O, k hiu l Om l mt vct bng tng hnh hc cc vct mmen ca cc lc i vi O + + + Nkk O N O O O OF m F m F m F m m12 1) ( ) ( ) ( ) ( (1.10)S dng nh l (1.3.1), d dng xc nh gi tr v phng chiu ca vct m men chnh' Nkk ZNkk Oz OzNkk YNkk Oy OyNkk XNkk Ox OxOF m F m mF m F m mF m F m mm1 11 11 1) ( ) () ( ) () ( ) ( (1.11) Nhn xt M men chnh ca h lc ng quy bng 0 M men chnh ca h ngu lc i vi bt k im no cng bng vct ngu lc tngNkk Om m1 (1.12) Vctchnh khng ph thuc vo im t (t do), cn vct moment chnh thay i theo im ly mment) (O O O OR m m m + 1 1(1.13)2.2 Thu gn h lc khng gian2.2.1 nh l di lc song songnh l 1.2.1. Lc Ft tiA tng ng vimt lc F F t tiO v mt ngu lc c mmen bng momen ca lc F i vi im O.F F v) (F m mO (1.14)2.2.2 Thu gn h lc phng v tm OS dng nh l di lc // di cc lc v tm O ta c1 1F F v ngu lc) (1 1F m mO , 2 2F F v ngu lc) (2 2F m mO , ... N NF F v ngu lc) (N O NF m m Nh vy thu gn h lc ) , , , (NF F F 2 1v tm O ta c h lc ng quy (khng gian) ti O: ) , , , ( NF F F 2 1 v h ngu lc (khng gian) ) , , , (Nm m m 2 1R F F RNkkNkk O 1 1(1.15)ONkk ON O O O Nm F m mF m F m F m m m m m + + + + + + 12 1 2 1) () ( ) ( ) ((1.16)theo(1.10)nhl1.2.2. (Thugnhlc) Hlckhnggianbt k ) , , , (NF F F 2 1tng ng vi mt lc thu gn v mt ngu lc thu gn t ti im O. Lc thu gn t ti O c vct lc bng vct chnh ca h lc, ngu lc thu gn c mmen bng mmen chnh ca h lc ly i vi im O. Phng chiu, gi tr ca lc thu gn khng ph thuc vo tm thu gn. cn ngu lc ph thuc vo tm thu gn (xem 1.13) i vi h lc khng gian, lc thu gn khng nm trong mt phng ngu lc thu gn nh trng hp phng (vct chnh v m men chnh khng vung gc vi nhau) - s khc bit gia h lc phng v h lc khng gian2.3 iu kin cn bng v cc phng trnh cn bng ca h lc khng gian2.3.1 iu kin cn bng nh l 1.2.3. iu kin cn v h lc khng gian cn bng l vec t chnh v momen chnh ca h lc i vi im O bt k phi ng thi trit tiu' 000112 1 Nkk O ONkknF m mF RF F F) () , , , ( (1.17)Chng minh:K cn: Nu cc iu kin trn khng tho mn khi h lc tng ng vi mt lc hoc mt ngu lc khng tho mn nh lut 1.K : hin nhin khi 0 R h lc tng ng vi mt ngu lc. Ngu lc bng 0 th hai lc l hai lc cn bng2.3.2 Cc phng trnh cn bng ca h lc khng gian vct chnh v m men chnh trit tiu th cc hnh chiu ca chng trn 3 trc to vung gc phi trit tiu, ) ( , ) ( , ) (, , ,0 0 00 0 01 1 11 1 1 Nkk zNkk yNkk xNkkzNkkyNkkxF m F m F mF F F (1.18)y l phng trnh cn bng ca h lc khng giannh l 1.2.4. iu kin cn v h lc khng gian cn bng l tng hnh chiu ca cc lc trn 3 trc vung gcv tng m men ca cc lc i vi ba trc y u trit tiui vi cc h c bit H lc ng quy khng gian, , , 0 0 01 1 1 NkkzNkkyNkkxF F F(1.18a) H ngu lckhng gian, ) ( , ) ( , ) ( 0 0 01 1 1 Nkk zNkk yNkk xF m F m F m (1.18b) H lc song song khng gian - ly trc Oz song song vi cc vct lc, , ) ( , ) ( 0 0 01 1 1 NkkzNkk yNkk xF F m F m (1.18c) H lc phngnm trn mt phng sng song vi mt Oxy, c 3 dng phng trnh cn bng+0 0 0 02 1 ) ( , , ) , , , (k O ky kx NF m F F F F F (1.18d)+0 0 0 02 1 ) ( , ) ( , ) , , , (k B k A kx NF m F m F F F F (1.18e)+0 0 0 02 1 ) ( , ) ( , ) ( ) , , , (k C k B k A NF m F m F m F F F (1.18f)3 CC BI TON C BN CA TNH HC VT RN3.1 Bi ton mt vt rn cn bng di tc dng ca h lc 3.1.1. Cc bc gii Bc 1. Gii phng lin kt, thay lin kt bng cc phn lc lin kt tng ng+ Chn vt kho st: 1 vt rn, mt phn, mt nt+ V phn lc ca cc lin kt m t trong (mc 1.3 v ph lc phn 1), chiu gi nh (khi kt qu dng chiu ng, khi kt qu m chiu ngc li) Bc 2. Thit lp cc phng trnh cn bng cho h lc tc dng ln vt rn t do+ Xt c im ca h lc kho st chn s phng trnh cn bng cn lp+ Xt c im ca phn lc chn trc chiu lc v tm ly m men thch hp: trc chiu trng vi ng tc dng ca lc, tm ly mmen chon im c nhiu lc n i qua Bc 3. Gii cc phng trnh cn bng, xc nh cc n cn tm3.1.2. Cc dng bi ton Bi ton tm phn lc lin kt. T iu kin cn bng ca h tm cc phn lc. Bi ton tm iu kin cn bng. Trong bi ton ny n l nhng i lng xc nh v tr ca vt rn (v mt s phn lc) Bi ton vt lt. Ta kho st vt rn chu lc tc dng) , , , (nF F F 2 1 Chu lin kt ta ti 2 im A, B - phn lc im ta l AN v BN Gi s h mt lin kt tai B, khi vt c th lt quanh A Gi mmen ca cc lc gi cho vt khng lt quanh A l Mgiv mmen cc lc lt l Mlt. iu kin vt rn khng lt quanh A l n cn cn bng v lin kt ti B vn hot ng, c th lgiu lat B A AM M N m F m + ) ( ) ( giu lat B AM M N m 0 ) ( Bi ton xc nh ni lc ti cc mt ct ngang. Tng tng ct dm thnh 2 phn a v B,b phn B xt cn bng ca phn A. Phn A cn bng do ti mt ct c-c c h lc do phn B tc ng ln phn A - l ni lc ta thu gn v h lcM Q N , , .3.2 Bi ton cn bng ca h vt rn3.2.1.Cc bc gii Bc 1. Gii phng lin kt, thay lin kt bng cc phn lc lin kt tng ng+ Chn h vt kho st v biu din lc+ ch lin kt thanh loira khivt kho st thay bng ng lc trnh nhm ln+ Biu din lc, ch phn bit lc hot ng v lc lin kt, nilc v ngoi lc Bc 2. Thit lp cc phng trnh cn bng theo hai phng php+ Phng php tch vt - Xt ring v lp h phng trnh cn bng cho tng vt+ Phng php ho rn - lp phng trnh cn bng ca cc ngoi lc ri mi tch vt+ Quan trng l s phng trnh cn bng lp ra c lp, n gin gii tm s n Bc 3. Gii cc phng trnh cn bng, xc nh cc n cn tm3.2.2. Cc dng bi tonTng t nh i vi h 1 vt Bi ton tm phn lc lin kt Bi ton tm iu kin cn bng3.3 Bi ton cn bng vt rn c ma st Cc bi ton trc khi xt lin kt ta gi thit l nhn, khng c ma st, tuy nhin trong thc t c cc trng hp ta khng th b qua ma st. Bn cht vt l ca hin tng ma st rt phc tp, cc vt tip xc vi nhau trn mt din tch no nn y c mt h phn lc lin kt.H phn lc ny trong trng hp phng c th thu v mt hp lc R v mt ngu lc. Phn tch hp lc ra hai thnh phn vung gc ta c Thnh phn php tuyn c gi l phn lc php tuyn N Thnh phn tip tuyn cn chuyn ng trt c gi l lc ma st trt msF, c chiu ngc vi chiu trt Ngu lc cn chuyn ng ln gi l ma st ln msm, c chiu ngc vi chiu ln Kho st ma st tnh v kh, thc nghim dn n cc nh lut sau3.3.1. nh lut ma st trt Lc ma st trt nm trn mt tip tuyn ngc xu hng trt v c tr sfN Fms trong f- h s ma st trt, tg f,- gc ma st, min zMz'- nn ma st (gia hai ng thng v t tip im M v nghing vi php tuyn gc ma st ) Phn lc ton phn R ca lin kt ta c ma st trt nm trong nn ma st3.3.2. nh lut ma st ln Ngu lc ma st ln ngc xu hng ln v c m menkN Mms trong k - h s ma st ln (th nguyn di (m)) Khi k n ma st ln, phn lc php nm pha vt c xu hng ln ti v cch php tuyn onk d 3.3.3 Cc bc gii Bc 1. Gii phng lin kt, thay lin kt bng cc phn lc lin kt tng ngNgoi cc lc hot ng v phn lc lin kt nh ni hai bi ton trc, y c thm ma st trt v ngu lc ma st trt Bc 2. Thit lp cc phng trnh cn bng theo hai iu kin+ iu kin cn bng (cc phng trnh cn bng)+ Cc iu kin do ma st Bc 3. Gii cc phng trnh cn bng, xc nh cc n cn tm4 TRNG TM VT RN4.1 Tm ca h lc khng gian song songCho h lc // bt k) , , , (NF F F 2 1 vi 01 NkkF R t ti im A1, A2, ... AN. K hiu vct nh v ca cc im Ak qua kkOA r nh ngha. im C gi l tm ca h lc song song c xc inh bi cng thcNkkNkk kCFr Fr11(1.19)trong kFl hnh chiu ca kF ln trc song song vi cc lc0 ) (R mC(1.20)4.2 Trng tm ca vt rnnh ngha.Khi vt rn gn tri t, trng tm ca vt rn l tm ca h trng lc ca cc phn t to thnh vt rnCng thc xc nh trng tm ca vt rnChia vt rn thnh N cc phn t nh, phn t th k ti im Ak c trng lng l kP v vct nh vkkOA r theo cng thc (1.16) ta cNkkNkk kCPr Pr11(1.21)Khis phn t c chia tng ln v cng, kch thc cc phn t l nhng i lng v cng nh th trng lng ca vt 1 kkP P lim VkkkdP r r P 1lim (1.22)Nn cng thc xc nh trng tm ca vt rn VCdP rPr 1, VCxdPPx1, VCydPPy1, VCzdPPz1(1.23) i vi vt ngcht cthtchlV, trnglngring, tacV P , dV dP vVCdV rVr 1, VCxdVVx1, VCydVVy1, VCzdVVz1(1.24) i vi mt ng cht c din tch l S, tng t ta c.SCdS rSr 1 SCxdSSx1, SCydSSy1, SCzdSSz1(1.25) i vi ng ng cht c chiu di L, tng t ta c.VCdS rSr 1 LCxdsLx1, LCydsSy1, LCzdsSz1(1.26)Cng thc tnh trng tm ca mt s vt rn n ginnh l 1.4.1.Nu vt rn ng cht c tm (trc, mt phng) i xng th trng tm ca n nm ti tm (trc, mt phng i xng)H qu Trng tm ca thanh ng cht l im gia ca thanh Trng tm ca hnh bnh hnh, ch nht, hnh vung, khi hp, khi hp ch nht, khi hp lp phng ng cht l tm ca chng Trng tm ca tam gic ng cht l giao im ca cc ng trung tuyn Trng tm ca cung trn ng cht AB c bn knh r v gc ti tm 2 AOB Trng tm ca qut trn ng cht AB c bn knh r v gc ti tm 2 AOB Trng tm ca bn cu ng cht tm O, bn knh r : R zg8 3/ zgnh l 1.4.2.Nu vt rn c ghp t Mphn, mi phn c trng lng Piv trng tm Ci(xi, yi, zi) th trng tm ca vt c xc nh nh cng thcNkkMkk kCPx Px11, NkkMkk kCPy Py11, NkkMkk kCPz Pz11(1.27)PH LC PHN 1. C IM CC PHN LC LIN KT THNG GPLin kt Biu din c im phn lcTa (khng ma sat)N1 N NB A NA B Thnggc vi mt ta, mt tipxc, hngvo vt khost - phnlc phpDy (mm v khng co dn) T2T T1 Nmtheody, hngra ngoi vt khost - sc cngThanh (ch chu ko hay nn) B SB S SA A Nmtheothanh(ng ni 2uthanh) hng vo(ra) thanhkhi thanh chu ko (nn) - ng lc Bn l (trn nhn) X Y R XO YO O Lc t ti bn l chia ra hai thnh phn - phn lc bn lNgma) YA XA A MA b) YB XB B MB c) X Y MY MX MZ ZPhng (a), (b): 2 thnh phnlcX,Yv1ngu lc m men MKhng gian(c): 3thnh phn phn lc ngm v 3 ngu lc m men tr ngna) X b) X c) Y X d) Y X Cntrdi chuynthng gc vitrc. Phng (a,b): nh phn lc ta. Khng gian (c,d): 2 phn lcCi ( tr ngn c mt chna) X Y b) X Y c) Y Z X d) Y Z X Cntrdi chuynthng gc v dc trc. Phng (a,b): 2 phn lc. Khng gian (c,d): 3 phn lc trc di M X

M X

MY Y X MX MY Y X MX Cntrdi chuynthng gc v quay. Phng (a,b): 1 phn lc v 1 ngu phn lc. Khng gian (c,d): 2 phn lc v 2 ngu phn lcGi c nha) X Y b) Yo Xo O c) X Y Z Cntrdi chuynthng theo 2 phng.Phng (a,b): 2 phn lc Khng gian(c): 3phn lc Gi di ng (c con ln) N No O Cn trdi chuyn theo phng thng vi mt nn. 1 phn lc.PHN 2. NG HC1. NG HC CHT IMChuyn ng ca cht iml s thay i v tr ca n so vi mt vt chun c chn lm h quy chiu.Quocacht im -tphpccv tr caimtrongkhnggianquy chiu.Ni dung ca ng hc cht im: Thit lp phng trnh chuyn ng ca cht im ti tng thi im - cc biu thc xc lp quan h gia v tr ca cht im v thi gian, Tm cc c trng ng hc ca cht im: vn tc, gia tc. Vn tc cho bit phng, chiu v tc ca cht im, gia tc cho bit s thay i v phng chiu v tc ca cht im.1.1. Kho st cht im bng phng php vc t1.1.1. Phng trnh chuyn ngXt chuyn ng ca im M i vi h quy chiu A theo qu o C Thng s nh v:Ly im O gn vi h quy chiu A. Khi vec t OM r l vec t xc nh v tr ca M v h QCA, gi l vc t nh v Phng trnh chuyn ng. Cht im M chuyn ng nnr thay i theo thi gian:) (t r r (2.1)c gi l phng trnh chuyn ng1.1.2. Vn tcGi s v tr ca cht im M ti thi im t v thi im ln cnt t t + , c xc nh bng cc vec t inh v rvr. Trong khong thi gian tvc t nh v bin i i mt lng r r r M M . i lng trvtbc gi l vn tc trung bnh ca cht im trong khong thi gian tVn tc ca cht im ti thi im t c xc nh nh sau) ( lim lim ) ( t rdtr dtrv t vttbM M 0(2.2)Nh vy vn tc ca cht im l o hm bc nht theo thi gian ca vc t nh v ca cht im.Hng ca vn tc theo phng tip tuyn ca qu o taiim M v pha chuyn ng. n v m/s1.1.3. Gia tcGi s vn tc ca cht im M ti thi im t v thi im ln cn t'=t+t lv v v. Trongkhongthi giant vntcbini i mt lng v v v .i lngtvtv vatb c gi l gia tc trung bnh ca cht im trong khong thi gian tGia tc ca cht im ti thi im t c xc nh nh sau) ( lim lim ) ( t rdtr ddtv dtva t attbM M 220(2.3)Nh vy gia tc ca cht im l o hm bc nht theo thi gian ca vn tc v bng o hm bc hai ca vc t nh v ca cht im. Gia tc hng v b lm ca qu o ti im M v pha chuyn ng. n v m/s21.1.4. Du hiu v nhanh dn v chm dn ca chuyn ngChuyn ng c gi l nhanh dn (chm dn) nu v (tng ng vi n l 2v ) tng (gim) theo thi gian. Cht im chuyn ng nhanh dn nu0212> ,_

a v vdtd (2.4a) Cht im chuyn ng chm dn nu0212< ,_

a v vdtd (2.4b)1.2. Kho st cht im bng phng php ta cc1.2.1. Phng trnh chuyn ng Thng s nh v: Gn h trc to c cc vung gc OXYZ vi h quy chiu A. Khi thng s nh v ca cht im l 3 to x, y, z ca im M (h1-4) Phng trnh chuyn ng. Khi cht im M chuyn ng, to ca n bin i theo thi gian:) ( ); ( ); ( t z z t y y t x x (2.5)Cc phng trnh (2.5) c gi l phng trnh chuyn ng ca cht im M dng to cc, ch k t z j t y i t x t r ) ( ) ( ) ( ) ( + + (2.6)trogn k j i , ,l cc vc t n v trn cc trc to 1.2.2. Vn tcThay (2.6) vo (2.2) ta cdtk dz kdtdzdtj dy jdtdydti dx idtdxk t z j t y i t xdtddtr dt v + + + + + + + ) ) ( ) ( ) ( ( ) (nhng v0 0 0 dtdkdtdjdtdi, , (2.7)nnkdtdzjdtdyidtdxk v j v i v t vz y x + + + + ) ( (2.8)vxdtdxvx ,ydtdyvy ,zdtdzvz (2.9)T (2.9) ta c gi tr v phng chiu ca vc t vn tc2 2 2 2 2 2z y x v v v v vz y x + + + + (2.10)vvv Oxx ) , cos(, vvv Oyy ) , cos(, vvv Ozz ) , cos(1.2.3. Gia tcThay (2.8) vo (2.3), ch n (2.7) ta ckdtz djdty didtx dk a j a i a t az y x 222222+ + + + ) ( (2.11)vxdtx dax 22,ydty day 22,zdtz daz 22(2.12)Gi tr v phng chiu ca vc t gia tc2 2 2 2 2 2z y x a a a a az y x + + + + (2.13)aaa Oxx ) , cos(, aaa Oyy ) , cos(, aaa Ozz ) , cos(1.2.4. Du hiu v nhanh dn v chm dn ca chuyn ngT (2.4) ta c du hiu chuyn ng nhanh dn:0 > + + z z y y x x (2.14a)v chuyn ng l chm dn khi0 < + + z z y y x x (2.14b)V d (1-1)- (H1-5), Cho c cu tay quay con trt A=AB=l. Tay quay OA quay u quanh trc O theo lut t0 , trong const 0. Lp phng trnh chuyn ng catrung im thanh truyn AB (h1-5)Gii. Chn h to cc Trung im I c to x=OI', y=OI", v ta tnh ct l yt l t l t l x00 0 0212321 + sincos cos cosVn tc ca im It ldtdyv t ldtdxvy x 0 0 0 02123 cos , sint tly x v0202 0 2 292 + + cos sin t ttvvv Oxx0202093 + cos sinsin) , cos(t ttvvv Oyy020209 + cos sin cos) , cos(1.3. Kho st cht im bng phng php ta t nhin1.3.1. Phng trnh chuyn ng Thng s nh v: Xt chuyn ng ca im M dc theo qu o C. Chn im O c nh trn qu o lm gc v mt chiu dng cho trc trn qu o. Khi thng s nh v ca cht im M l thng s OM s(h1-6) Phng trnh chuyn ngKhi cht im M chuyn ng dc theo qu o C ths bin i theo thi gian:); (t s s (2.15)l phng trnh chuyn ng ca cht im M dc theo qu o 1.3.2. Vn tcGi s v tr ca cht im ti thi im t v thi im ln cn t'=t+t l M v M'. Xt chuyn ng t M n M' ( r d M M s d M M , ) T (2.2) ta c0tdtdsdts ds dr ddtr dt v ) ( (2.16)trong 0t vc t n v trn phng tip tuyn c chiu thun vi chiu dng chn. Hnh chiu ca vn tc trn phng tip tuyn s l dts dt v ) ( (2.17)1.3.3. Gia tcT (2.3) v (2.16) v s dng cng thc0 0ndst dta c 0202002222nvtdts ddts ddst ddts dtdts ddtv ddtr dt a + + ) ( (2.18)trong 0nvc t n vtrn phng php tuyn hng v b lm ca ng cong qu o, - bn knh cong ca qu o ti im M. Hnh chiu ca gia tctrn phng tip tuyn v phng php tuyn s l ,22dts dat ,2van,22222

,_

+

,_

vdts dantaatg (2.19)trong chuyn ng thng u 0 na v 1.4. Kho st mt s chuyn ng thng gp1.4.1. Chuyn ng uL chuyn ng c vn tc khng in n ta avadtdva const v , , ,20 (2.20)Khi cht im chuyn ng thng u 0 0 0 a a an t, ,1.4.2. Chuyn ng bin i uL chuyn ng m gia tc tip ca cht im c gi tr khng i. Ly chiu dng l chiu ca chuyn ng ta c0 02021s t v at s v at v + + t + t , (2.21)trong v0, s0 l vn tc v to cong ca cht im ti thi im ban u. Du (+) ng vi chuyn ng nhanh dn, du (-) ng vi chuyn ng chm dn1.5. Cc loi bi ton trong phn ng hc cht im Bit phng trnh chuyn ng tm cc c trng chuyn ng nh: qu o, vn tc, gia tc Bit mt s iu kin ca chuyn ng tm phng trnh chuyn ng v cc c trng chuyn ng Bi ton tng hp: dng c 2 PP: to cc v to t nhinPhng php gii Chn phng php. Tu vo u bi, bit qu o chn PP to t nhin. Tm phng trnh chuyn ng Xt im v tr bt k tm quan h to theo thi gian nhn c:); ( ); ( ); ( t z z t y y t x x hoc ); (t s s Bit gia tc th tch phn tm vn tc v tch phn ln na tm phng trnh chuyn ng Nu chuyn ng l u v bin i u ta c ngay phng trnh chuyn ng ; ; vt s a const vt 0; . ;20 05 0 t t v s t v v const at + + Tm qu o - kh thi gian khi phng trnh chuyn ng t tm c biu thc lin h gia cc to khng cha t Tm vn tc v gia tc Khi dng PP to cc p dng cc cng thc: xdtdxvx ,ydtdyvy ,zdtdzvz ; 2 2 2 2 2 2z y x v v v v vz y x + + + + xdtx dax 22,ydty day 22,zdtz daz 222 2 2 2 2 2z y x a a a a az y x + + + + Khi dng PP to t nhin p dng cc cng thcdts dt v ) ( ,,22dts dat ,2van,22222

,_

+

,_

vdts dantaatg Tm tch cht nhanh, chm dn ca chuyn ng Khi dng PP to cc kho st biu thc: z z y y x x + + Khi dng PP to t nhin kho st biu thc: ta v Tm bn knh cong ca qu o.Dng cng thc , / 2v anNu to cc th cn chuyn sang to t nhin bng cc cng thcdtdva z y x vt + + ;2 2 2 ; + + 22 2 2 2 2va a a z y x at n; Cc v dDng bi ton 1.V d 1. Cho ptr chuyn ng 223 64 8t t yt t x x, y c v (m), t - (s)Xc nh: qu o, vn tc, gia tcGii.Bc 1. Chn PP - dng PP to ccBc 2. Tm qu o kh t ta cx y hay y x430 4 3 vy qu o l ng thngBc 3. Tm vn tc v gia tcVn tc ) ( ), ( t y v t x vy x 1 6 1 8 s m t t t v v vy x) ( ) ( ) ( + + 1 10 1 36 1 642 2 2 2Gia tc 6 8 y a x ay x , 2 2 210 36 64 s m a a ay x + + Bc 4. Tnh cht nhanh (chm) dn ca chuyn ngT bc 3 ta thy gia tc khng i =10m/s, hng gia tc theo qu o, chiu lun m hng t B n A. Theo (2.14) ta c) (t a v a v a vy y x + 1 100 Khi t=0 th x=0, y=0, v=10m/s, im gc to , chuyn ng l chm dn 100 a v Khi00, chuyn ng l chm dn v0 < a v Khi t=1s th im v tr B x=4, y=3 v v=0,0 a v Khi 1 (2.28) Vt quay chm dn khi 0 < (2.29) Mt s dng chuyn ng c bit Chuyn ng quay u. Khi vn tc gc khng i0 0 00 + t t const ; ;(2.30)ly du + khi vt quay ngc chiu kim ng h, ly du -trong trng hp ngc li, 0 gc nh v ban u Chuyn ng quay bin i u. Khi gia tc gc khng i0 020 0 0 021 + + t + t t t t ; ; (2.31)ly du + khi chuyn ng quay nhanh dn u, ly du -trong trng hp ngc li, 0 gc nh v ban u2.2.2. Kho st chuyn ng ca cc im thuc vtXt im M, cch trc quay mt on IM=R bn knh quay ca im M. Khi vt quay M chuyn ng trn ng trn tm I l giao im gia trc quay v mph cha M vung gc vi trc quay, ng trn c bn knh l IM. Phng trnh chuyn ng ca imChn giao im O ca mph quy chiu vi qu o trn lm im gc ta c cc thng s nh v ca im M l gc v bn knh IM=R (bn knh quay ca im M). ng thi c th chn cung R s OMlm thng s nh v. Vy phng trnh chuyn ng ) (t R s (2.32) Vn tc ca imTheo (2.16) ta c 0 0 0t R tdtdR tdtdst v ) ( (2.33) vn tc c phng vung gc vi bn knh quay c chiu thun chiu quay c gi tr bng tch ca vn tc gc v bn knh quay cc im nm trn trc quay c vn tc bng 0 cc im nm trong mph vung goc vi trc quay c vn tc phn b theo lut tam gic vung Gia tc cc imTheo (2.18-19) ch n (2.32-33) ta cGia tc tip R RdtdRdts dt at2222) ( (2.34) c phng vung gc vi bn knh quay c chiu thun (hay ngc) vi chiu quay tu thuc vo chuyn ng quay nhanh dn hay chm dn (thun vi chiu ca gia tc gc) c gi tr bng tch ca gia tc gc vi bn knh quayGia tc php 22 2 2 RRRRvt an) ( (2.35) chiu hng t im M vo tm I c gi tr bng tch ca bnh phng vn tc gc v bn knh quayGia tc ton phn 4 2 2 2 + + R a a an t(2.36) nghing vi bn knh quay MI mt gc 2 ntaatg(2.37) cc im nm trong mph vung gc vi trc quay c gia tc phn b theo lut tam gic thng ng dng vi t s bng 4 2 + 2.2.3 Biu din vc t mt chuyn ng quay Vc t vn tc gc v vc t gia tc gc Vc t vn tc gc c phng dc trc quay, c chiu khi nhn t nh ca trc quay xung thy vt quay ngc chiu kim ng h c gi tr bng vn tc gc k (2.38)trong k - vc t n v trn trc quay Vc tgia tcgc cphng dctrc quay,c cng chiuhoc ngc chiu vi vc t vn tc gc tu thuc vo chuyn ng quay nhanh dn hay chm dn, c gi tr bng gia tc gc k (2.39) Vc t ca vn tc im v gia tc imChn im O1 c nh trn trc quay. Vc t nh v ca im M l vc t r M O1. Ta c cc cng thcr v (2.40); + v r av ar ant (2.41)Chng minh: R I O IM I O M O r+ + 1 1 1trong I O1 l vc t hng ng phng vi cn vc t R c m un hng. VyR R I O r + ) (1(2.42)cng thc (2.40) c chng minh c tn l cng thc le2.3. Mt s dng truyn ng nh cc chuyn ng c bn2.3.1. Truyn ng bng c cu bnh rng, ai, xch truyn chuyn ng gia hai trc c nh song song vi nhau ta dng c cu bnh rng, ai truyn hay xch+ C cu bnh rngT s truyn ng bnh rng k hiu i12: 12222112zzrri t t (2.43)ly du (+) nu khp trong, du () nu khp ngoi, r1, r2 bn knh ca cc bnh rng 1,2, z1 v z2 s rng tng ngC cu ai xch1r1O12r2O22r2O21r1 O1 MT s truyn ng ai xch k hiu i12: 222112rri t (2.44)ly du (+) nu ai bt thng, du () nu ai bt cho2.3.2. Truyn ng bng bnh rng thanh rng truyn ng gia mt vt quay v vt tch tin ngita s dng c cu bnh rng thanh rng, c cu bnh thanh ma stT s truyn ng lRvi (2.45)2.3.3. Truyn ng bng c cu cam 2.4. Cc loi bi ton phn chuyn ng c bn ca vt rn2.4.1. Cc dng bi tona) Bi ton mt vt Bit phngtrnhchuynngvccctrngnghccavt rntm phng trnh chuyn ng v cc c trng ng hc ca 1 im trn vt1r1O12r2O21r1O12r2O2vv2v1camcnvBcamcnAO. C M Bit phng trnh chuyn ng v cc c trng ng hc ca im trn vt rn tm phng trnh chuyn ng v cc c trng ng hc ca vt rnb) Bi ton truyn ng Bit dng truyn ng tm quan h chuyn ng gia trc dn v trc b dn2.4.2. Phng php gii Tm , dng) (tdtd , 30n (n- s vng quay),) (tdtddtd 22hay suy ra tRvR t v ) ( , RaR att , hay RaR ann 2 Tm t/c nhanh, chm dn ca chuyn ng kho st tch , khi0 > - nhanh dn, khi0 < - chm dn Tm vn tc gia tc ca im thuc vt, s dng 0t R t v ) ( , R t at) (, 2 R t an) (4 2 2 2 + + R a a an t, 2 ntaatg Tm s lin h v truyn chuyn ng s dng 12122112zzrri t t , Rvi 3. TNG HP CHUYN NG IM3.1. nh ngha v cc loi chuyn ngTa c cht im M chuyn ng i vi h quy chiu B, h quy chiu B chuyn ng i vi h qui chiu A. Nh vy chuyn ng ca M l tng hp t hai chuyn ng trn. n gin ta coi h quy chiu A c nh. Khi chuyn ng ca im M i vi h quy chiu B gi l chuyn ng tng i; Chuyn ng ca h quy chiu B MO1(A)x1y1z1xyzO(B)i vi h quy chiu A gi l chuyn ng theo; Chuyn ng ca im M i vi h quy chiu A gi l chuyn ng tuyt i3.2. nh l hp vn tcnh ngha: Vn tc tuyt i ca im M l vn tc ca n c tnh trong h quy chiu A, k hiu av.) () (A aM Odtdv1(2.46)Vn tc tng i ca M l vn tc ca n c tnh trong h quy chiu B, k hiu rv) () (B rOMdtdv(2.47)Vn tc theo: Xt M* thuc h quy chiu B m ti thi im kho st trng vi cht im M. Vn tc theo ca M, k hiu l ev, l vn tc ca im M*) () (A eM Odtdv1(2.48)nh l. Ti mi thi im, vn tc tuyt i ca cht im M bng tng hnh hc ca vn tc theo v vn tc tng ie r av v v + (2.49)Chng minhGik j i , ,l cc vc t n v ca h trc Oxyz (gn vi h quy chiuB), x, y, z l to ca im M trong h to Oxyz. T (2.1) ta ckdtdzjdtdyidtdxdtk dzdtj dydti dx v k z j y i xdtdvOMdtdO OdtdOM O OdtdM OdtdvA A A A a + + + + + + + + + + + 0 01 1 1) () ( ) ( ) ( ) () ( ) ( ) ( ) ((2.50)T (2.46) ch 0

,_

,_

,_

B B Bdtk ddtj ddti d , (2.51) ta c kdtdzjdtdyidtdxkdtdzjdtdyidtdxdtk dzdtj dydti dxk z j y i xdtdOMdtdvB B BB B r + + + + +

,_

+

,_

+

,_

+ + ) ( ) () ((2.52)T (2.48) v ch rng x*,

y*, z* gn vi cht vi h to Oxyz nn 0

,_

,_

,_

B B Bdtdzdtdydtdx(2.53)ta cdtk dzdtj dydti dx vk z j y i xdtdv OM O OdtdM OdtdvA A A e + + + + + + + 00 1 1 ) ( ) ( ) () ( ) ( ) ((2.54)ti mi thi im x=x*, y=y*, z=z* , ta nhn c e r av v v + 3.3. nh l hp gia tcnh ngha: Gia tc tuyt i ca im M l gia tc ca n c tnh trong h quy chiu A, k hiu aa.) () (A a avdtda (2.55)Gia tc tng i ca M l gia tc ca n c tnh trong h quy chiu B, k hiu ra) () (A r rvdtda (2.56)Gia tc theo ca M, k hiu l ev, l gia tc ca trng im M* trong h quy chiu A, k hiu l ea) () (A e evdtda (2.57)nh l. Ti mi thi im, gia tc tuyt i ca cht im M bng tng hnh hc ca gia tc theo, gia tc tng i v gia tc Crilic cac e r aa a a a + + (2.58)Chng minhThay (2.50) vo (2.55) ta c

,_

+ + ++ + + + + +

,_

+ + + + + + dtk ddtdzdtj ddtdydti ddtdxkdtz djdty didtx ddtk dzdt j dydti dx akdtdzjdtdyidtdxdtk dzdtj dydti dx vdtdvdtdaAA a a 222222222222200) () () ((2.59)Thay (2.52) vo (2.56)ch (2.51), ta ckdtz djdty didtx dkdtdzjdtdyidtdxdtdvdtdaBB r r 222222+ + ,_

+ + ) () () ((2.60)Thay (2.54) vo (2.57)ch (2.53), ta c2222220 0dtk dzdt j dydti dx adtk dzdtj dydti dx vdtdaAe + + +

,_

+ + + ) ((2.61)So snh (2.59) vi (2.60) v (2.61) vi ch ti mi thi im x=x*, y=y*, z=z*, ta nhn cc e r aa a a a + + trong gia tc Crlio c dng

,_

+ + dtk ddtdzdtj ddtdydti ddtdxac 2(2.62)Trong trng hp chuyn ng ca h Oxyz l tnh tin 0 dtk ddtj ddti d th 0 ca ve r aa a a + (2.63)nh l. Khi chuyn ng theo l tnh tin th gia tc tuyt i ca cht im M bng tng hnh hc ca gia tc theo v gia tc tng iTrng hp h Oxyz quay quanh mt trc c nh vi vn tc gc theo e, s dng (2.63), ta cidti de ;jdtj de ;kdtk de (2.64)thay vo (2.62) nhn cr e e cv kdtdzjdtdyidtdxdtk ddtdzdtj ddtdydti ddtdxa ,_

+ +

,_

+ + 2 2 2(2.65)Ch : - khi r ev // hoc0 rv hay 0 e th 0 ca Khir ev c quy tc tm phng chiu ca ca: quay rvquanh gc ca n theo chiu quay ca h Oxyz quanh e mt gc 90o, ta c ngay chiu ca ca, gi tr ca n l r ev 23.4. Cc dng bi ton 3.4.1. Dng bi tona) Tm phng trnh chuyn ng ca imb) Tng hp chuyn ng: bit cc c trng ng hc ca chuyn ng tng i v chuyn ng theo tm gia tc tuyt ic) Phn tch chuyn ng: bit cc c trng ng hc ca mt trong hai chuyn ng (tuyt i hay theo) v phng vn tc v gia tc ca im trong hai chuyn ng cn li. Tm gi tr ca vn tc v gia tc trong hai chuyn ng 3.4.2. Phng php giia) Phn tch chuyn ng Phn tch dng chuyn ng ca ccvt to thnh h xc nh h quy chiu Phn tch cc chuyn ng thnh phn: + chuyn ng tuyt i, chuyn ng tng i: dng chuyn ng (thng, cong), cc yu t ng hc (v, a) +chuyn ng theo: dng chuyn ng (tnh tin, song phng, quay quanh mt trc c inh). Cn xc nh trc quay tc thi v theob) Gii Tm phng trnh chuyn ng: xc nh quan h gia to cc im i vi h tocnh(trongchuynngtuyt i) hayhquychiung(trong chuyn ng tng i) v thi gian Tng hp v phn tch chuyn ng+ Tm v, dnge r av v v + 4. TNG HP CHUYN NG VT RN4.1. Tng hp chuyn ng song phng ca vt rnnh ngha: Chuyn ng ca vt rn c gi l chuyn ng song phng khi mi im thuc vt lun lun di chuyn trong mt mt phng sng song vi mt phng c nh chn trc gi l mt phng quy chiu.Bi ton kho st chuyn ng song phng ca vt rn trong khng gian c a v bi ton kho st chuyn ng ca mt tit din phng ca n trong mt phng cha tit din song song vi mt phng quy chiu.4.1.1. Chuyn ng ca hnh phnga) Thng s nh vXt hnh phng S chuyn ng trong mph P. Chn h to c nh O1x1y1 trong mph P, ly im O thuc hnh phng S gn vo h trc to Oxy c Ox //O1x1 v Oy//O1y1. Vy Oxy chuyn ngtnh tin i vi O1x1y1, chuyn ng xc nh hon ton qua chuyn ng ca im O.V tr ca hnh phng xc nh biv tr ca n i vi h Oxy qua gc nh v v tr ca h ng Oxy i vi h c nh O1x1y1 qua thng s nh v ca im O (x0, y0)b) Phng trnh chuyn ngKhi hnh phng chuyn ng th cc thng s nh v cng thay i lin tc theo thi gian. cc i lng) ( ); ( ); ( t t y y t x x 0 0 0 0(2.66)l phng trnh chuyn ng ca hnh phng S. Haiphng trnh u m t chuyn ng tnh tin ca h to ng Oxy, phng trnh th ba m t chuyn ng quay ca hnh phng i vi h to ng O1Ox0y0y1x1yxSOxy. Nh vy chuyn ng ca hnh phng S chia lm 2 thnh phn: chuyn ng tnh tin cng to ng v chuyn ng quay i vi h to ng4.1.2 Cc yu t ng hc ca cc chuyn ng thnh phnTrng thi ng hc ca chuyn ng tnh tin ca h to ng c xc nh bng vn tc v gia tc ca im gc to O (0 0 a v ,)Trng thi ng hc ca chuyn ng quay ca hnh phng S i vi h to ng c xc nh bng vn tc gc v gia tc gc ;.Vy cc i lng , , ,0 0 a v c gi l cc yu t ng hc ca hnh phng chuyn ng song phng. Cc i lng 0 0 a v , ph thuc vo vic chn gc to ng O, cn , khng ph thuc vo vic chn ny.4.1.3 Kho st chuyn ng ca ima) Thng s nh vXt im M thuc hnh phng S cch gc O mt khong OM=R. Cc thng s nh v ca im M l to xM yM' + + sincosR y yR x xMM00(2.66)b) Phng trnh chuyn ng' + + ) ( sin ) ( ) () ( cos ) ( ) (t R t y t yt R t x t xMM00(2.67)c) Vn tc cc im Biu thc gii tch' + + ) ( cos ) ( ) ( cos ) ( ) () ( sin ) ( ) ( sin ) ( ) (t R t y t R t ydtdyt vt R t x t R t xdtdxt vvMMyMMxM0 00 0 (2.68)O1Ox0y0y1x1yxMxMyM Quan h vn tc gia 2 im. Vn tc ca im gc O, tc 0v, c hnh chiu ln h to c nh O1x1y1 l 0x v 0y, cn vn tc ca im M trong h to ng OxyMOv c hnh chiu trn O1x1y1 l sin R v cos R. Vy t (2.4) ta cMO O Mv v v + (2.69)nh l 2.4.1: Vn tc ca im M bng tng hnh hc ca vn tc im gc O v vn tc ca n khi hnh phng quay quanh gc O. Tm vn tc tc thiv s phn b vn tc ca cc im thuc hnh phngNu ti thi im kho st tn ti mt im thuc hnh phng c vn tc bng 0 th im c gi l tm vn tc tc thi Pnh l 2.4.3: Ti thi im vn tc gc0 tn ti duy nht mt tm vn tc tc thi. xt trng hp tn ti tm vn tc tc thi P. Khi chn P lm cc theo nh l 2.4.1 ta c MP MP P Mv v v v + (2.70)tc vn tc im M bt k thuc hnh phng S bng vn tc ca n trong chuyn ng quay ca hnh phng quanh tm vn tc tc thi. Gi im P* thuc mt phng c nh trng vi tm vn tc tc thi P l tm quay ta c nh l sau.nh l 2.4.4. Ti thi im tn ti tm tc thi vn tc cc im thuc hnh phng c phn b ging nh trng hp quay quanh tm vn tc tc thi P. OO1xy1x1M MOvTrong trng hp ny ngi ta gi hnh phng quay quanh tm quay tc thi P*

Trng hp khng tn ti tm vn tc tc thi (=0) th 0 MOv nn O MO O Mv v v v + trng hp ny hnh phng chuyn ng tnh tin tc thi. ta c nh lnh l 2.4.5. Ti thi im khng tn ti tm tc thi vn tc th mi im thuc hnh phng c vn tc bng nhau. Xc nh tm vn tc tc thi Trng hp 1: Bit Av v vn tc gc ca hnh phng . Tm vn tc tc thi nm trn ng PA c c bng cch quay Av i 90o theo chiu quay ca v AvPATrng hp 2: Bit vn tc ca im A Av v phng ca vn tc ca im B. Tm vn tc tc thi l giao im ca 2 ng vi cc phng vn tc dng ti im A v B PAPBv PB vPAvA BA Trng hp 3: Bit vn tc ca 2 im A, B v phng ca chng //.Nu ng thng i qua AB vi vn tc th tm vn tc tc thi l giao im ca ng AB vi ng thng ni 2 im u ca cc vc t vn tc Av v Bv. ABv vB A nuAv v Bv cng chiuMP=P*APAvAPAvBBvPhngABv vB A + nuAv v Bv ngc chiuNu ng thng i qua AB khng vi vn tc th tm vn tc tc thi xa v cngTrng hp 4: Hnh phng ln khng trt trn mt ng cong phng c nh, th tm vn tc tc thi l im tip xc gia hnh phng v ng Gia tc cc im Biu thc gii tch' + ) ( sin ) ( cos ) ( ) () ( cos ) ( sin ) ( ) (t R t R t ydty dt at R t R t xdtx dt aaMMyMMxM20222022 (2.71) Quan h gia gia tc hai imT (2.71) ta thy thnh phn th nht l hnh chiu gia tc ca im O ln trc to c nh. Thnh phn th 2 l hnh chiu ln cc trc to c nh ca gia tc php ca im M trong chuyn ng quay ca hnh S quanh O, nMOa, hng t M n O c gi tr bng 2 RBPAv APAvBBvABvOO1x1y1MOaOaMatMOanMOaMOaThnh phn th 3 l hnh chiu ln cc trc to c nh ca gia tc tip ca im M trong chuyn ng quay ca hnh S quanh O, tOa, phng vi MO c gi tr bng RtMO MO n O Ma a a a + + ,tMO MO n MOa a a + ng l 2.4.6. Gia tc ca im M thuc hnh phng bng tng hnh hc ca gia tc im gc O v gia tc ca nkhi hnh phng quay quanh gc O. Tm gia tc tc thinh ngha: Tm gia tc tc thi l mt im thuc hnh phng, ti thi im kho st c gia tc bng 0.nh l 2.4.7. Ti thi im vn tc gc v gia tc gc khng ng thi trit tiu th tn ti duy nht mt tm gia tcnh l 2.4.8. Trong trng hp tn ti tm gia tc tc thi, s phn b gia tc cc im thuc hnh phng ging nh s phn b ca chng khi hnh phng quay quang tm gia tc. Xc nh tm gia tc tc thi Trng hp 1:Bitgiatcmtim Aa,vntcgc vgiatc cahnh phng, khi tm gia tc tc thiQ s nm trn na ng thng AQ c c bng cch quay vc t Aa theo chiumt gc 2 tg , 4 2 + AaAQTrng hp 2: Bit Aa v Ba c th tinsg c 2v , do tnh c gc . Khi tm vn tc tc thiQ l giao im ca haing thng c c bng cch quay hai vc t trn i 1 gc theo chiu ca 4.1.4 Cc dng bi tpa) Lp phng trnh chuyn ng ca vt .4.2. Tng hp hai chuyn ng quay quanh hai trc song songXt vt C quay i vi khung B quanh trc r vi vn tc gc r, khung B quay i vi gi c nh A quanh trc e song song vi r vi vn tc gc e. Xt chuyn ng ca vt C i vi gi c nh A. Chuyn ng ca vt C i vi khung B l chuyn ng tng i Chuyn ng ca khung B vi gi c nh A l chuyn ng theo Chuyn ng ca vt C vi gi c nh A l chuyn ng tuyt iTa nhn thy chuyn ng ca vt C l chuyn ng song phng vimt phng vung gc vi cc trc quay vy ta s ch kho st mt hnh phng S nm trong mt phng (hnh)Thanh OO1quay quanh O1vi vn tc gc egc nh v l e, hnh S quay quanh O vi vn tc gc r gc nh v l r. Vy chuyn ng tuyt i c gc nh v l e r a + Do dtddtddtde r a+ vy e r a + Tm vn tc tc thi P xt ln lt cho 2 trng hp Trnghpchuynngquaytheovquaytngi cngchiu0 > e, 0 > r, vy 0 > + r e a.Xt im O thuc hnh phng S, vn tc ca nO O ve 1 0 Khi O O O OvPOe rea1 10< + ) ( PO P O O O POe ree re+ + + 1 1 rePOPO 1 Trnghpchuynngquaytheovquaytngi ngcchiu0 > e, 0 < r, r e > vy 0 > r e a.Khi O O O OvPOe rea1 10> ) ( PO P O O O POe ree re+ 1 1 rePOPO 1a(a)OO1xy1x1r(r)ee(e)y4.3. Chuyn ng inh cChuyn ng inh c l tng hp ca chuyn ng tng i l chuyn ng quay ca im M quanh trc z to thnh ng trn bn knh r trn mt phng song songvi mtphng Oxyvchuynngtheol chuynngtnhtincamt phng theo phng z. Thng s nh vca chuyn ng quay l gc cn thng s nh v chuyn ng tnh tin l z im M chuyn ng theo quy lut: p z r y r x ; sin ; costrong r v p l hng s4.3.1 Cc yu t ng hc ca cc chuyn ng thnh phnTrng thi ng hc ca chuyn ng quay trn ng trn xc nh bng vn tc gc v gia tc gcTrng thi ng hc ca chuyn ng tnh tin theo phng z xc nh bng vn tc v gia tc ca mt phng4.3.2. Kho st chuyn ng ca imLut chuyn ng p z r y r x ; sin ; cosTa c p z v r y v r x vz y x; cos ; sin2 2 2 2 2 2 2 2 2 2 2 2 2p r p r r v v v vz y x+ t + + + + cos sin + p z a r r y a r r x az y x; cos sin ; sin cos2 22 2 2 4 22 2 2 + + + + p ra a a az y x) (Gi gc gia tip tuyn vi qu o v trc x l ta cconstp r pvvz+t 2 2cos Vy ng inh c c nghing khng i i vi mt phng Oxy. Nh ta m t qu o ca im M khi chiu ln mt Oxy s v ra ng trn bn knh r2 2 2 2 2 2 2r r r y x + + sin cos x Khi im i quay mt vng quanh trc z ta c 2, v im M chuyn ng dc theo trc z mt on h const p z 2. h l bc ca inh cTrng hp: t0 ta c0 0 0 p v r v r vz y x; cos ; sin2 20202 2 202 2 202 2 2 2p r p r r v v v vz y x+ t + + + + cos sin02020 z y xa r v r x a ; sin ; cos 202 402 402 2 2 2 + + + r r a a a az y x) sin cos (200 r a a a const vn t;Khi M chuyn ng u trn ng nh c vi vn tc v=const.Gia tc a // vi mt phng z v a 0 za , c cng phng vi bn knh quay v y a x ay x2020 ;.Bn knh cong trong trng hp ny c tnh22 22022 2 202r p rrp ravn++ ) ( PHN 3. NG LC HC1. CC KHI NIM V CC NH LUT CA NG LC HC1.1. Cc khi nim c bn1.1.1. Cc m hnh vt tha) Cht im:l mt im hnh hc mang khi lng, - vt th c kch thc nh c th b qua)Cht im t do: - l cht im m ti thi im kho st cc di chuynkhng b cn trCht im khng t do (cht im chu lin kt) ti thi im kho st cc dch chuyn ca n b cn tr. iu kin rng buc v v tr v vn tc cc lin ktCht im khng t do c th thay th bng cht im t do v cc phn lc lin ktb) C h tp hp hu hn hay v hn cc cht im, trong chuyn ng ca mt cht im bt k ph thuc vo chuyn ng ca cc cht im cn li, ni cch khc chuyn ng ca cc cht im ph thuc ln nhau tn ti tng tc c hc.C h t do l tp hp cc cht im m tng tc c hc gia chng c biu din ch thun tu qua lc tc dng v d h mt triC h khng t do (c h chu lin kt) l c h c t nht mt im khng t do, Ngoi tng tc lc, v tr v vn tc ca chng cn b rng buc bi mt s iu kin ng hc v hnh hc gi l iu kin lin kt. C cu my v vt rn tuyt i l mt c h chu lin kt. 1.1.2. Lc tc dng ln c hTrong phn ny lc l i lng bin i theo thi gian, v tr v vn tc) , , ( v r t F F (3.1)a) Cc c trng ca lc tc dngXung lng ca lc (xung lc) nh gi tc dng ca lc theo thi gian (N/s) Xung lc nguyn t dt F S d (3.2) Xung lc hu hn 21ttdt F S (3.3) Xung lc ca h lc Nkttkdt F S121 (3.4)Cng ca lc - nh gi tc dng ca lc di chuyn (Nm- Jun)) Cng nguyn t dz F dy F dx F r d F dAz y x+ + (3.5)trong r d l lng di chuyn v cng b. ds F ds F z F y F x F dt v F dAt z y x + + cos (3.6) trong gc gia lc v phng tip tuyn Cng hu hn + + M M M Mz y xM Mds F dz F dy F dx F r d F A0 0 0cos(3.7) Cng sut l cng sinh ra trong mt n v thi gian (Nm/s)t t z y xv F z F y F x F v FdtdAW + + (3.8)b) Ngoi lc v ni lcNgoi lc, k hiu ekF, l lc do cc vt th bn ngoi tc ng ln cht im thuc c h. Ni lc, k hiu ikF, l lc do cc cht im thuc c h tc ng ln ln nhau. Ni lc xut hin theo tng cp trc i.c) Lc lin kt v lc hot ngLc lin kt, k hiu kR, l lc do lin kt tc dng ln cc cht im thuc c h.Lc hot ng l cc lc khng thuc vo loi lc lin kt, n khng ph thuc vo cc lin kt t ln c h, cn lc lin kt khng ch ph thuc vo lin kt cn ph thuc vo lc hot ng tc dng ln c h.1.1.3. H quy chiu qun tnhTrong ng lc hc chuyn ng c kho st trong h quy chiu qun tnh l h quy chiu m trong nh lut qun tnh ca Newton c nghim ng. Trong k thut h quy chiu gn vi tri t l h qui chiu qun tnh (gn ng).1.2 Cc nh lut c bn ca ng lc hcnh lut qun tnh:Cht im khng chu tc dng ca lc no th ng yn hay chuyn ng thng u chnh l trng thi qun tnh ca n. Nh vy nu khng c lc tc dng ln cht im th n c trng thi qun tnh. C ngha cht im bo ton trng thi qun tnh cho n khi cha c lc buc n thay i. Vy nh lut qun tnh khng nh lc l nguyn nhn duy nht lm bin i trng thi chuyn ngnh lut c bn ca ng lc hc: Trong h quy chiu qun tnh, di tc dng ca lc cht im chuyn ng vi gia tc cng hng vi lc v c tr s t l vi cng ca lca m F(3.9)h s t l m c gi tr khng i n l s o qun tnh ca cht im c gi l khi lng.Khi cht im ri t do ta c: g m P , (3.10) trong g=9,81m/s2 gia tc trng trng.Trong ng lc hc ta cn s dng cc nh lut nu trong tnh hc nh: nh lut tc dng v phn tc dng, nh lut hnh bnh hnh, nh lut thay th lin kt1.3. Phng trnh vi phn chuyn ng ca cht imPhng trnh vi phn chuyn ng ca cht im dng vc t: Gir l vc t nh v ca cht im trong h quy chiu qun tnh t (3.9) ta c) , , ( r r t F r m (3.11)Phng trnh vi phn chuyn ng ca cht im dng to cc:Chn h trc to cc vung gc gn vi h quy chiu qun tnh, khi chiu hai v (3.9) ln trc to ta c) , , , , , , ( z y x z y x t F x mx ) , , , , , , ( z y x z y x t F y my (3.12a)) , , , , , , ( z y x z y x t F z mz Phng trnh vi phn chuyn ng ca cht im dng to t nhin: khi chiu hai v (3.9) ln cc trc to t nhin ta cb n tF Fvm F s m 02, , (3.12b)trong s l to cong, v -vn tc, - bn knh cong,b n tF F F , , - l hnh chiu ca lc F ln cc trc tip tuyn, php tuyn v trng php tuyn1.4. Hai bi ton c bn ca ng lc hcBi ton th nht - bi ton thun: Cho bit chuyn ng ca cht im xc nh lc tc dng ln cht imTrng hp bit gia tc a ta p dng trc tip phng trnh c bn.Trng hp bit lut chuyn ng hoc vn tc ta tm gia tc ca cht im sau p dng phng trnh cn bng .Bi ton th hai - bi ton ngc:Cho bit lc tc dng v cc iu kin u ca chuyn ng xc nh chuyn ng ca cht imT phng trnh cn bng gia tc c th xc nh t phng trnh viphn chuyn ng. Sau ly tch phn xc nh chuyn ng ca cht im. Cc hng s tch phn xc nh t iu kin ban u2. CC NH L TNG QUT CA NG LC HC2.1. Cc c trng hnh hc khi ca c h v vt rn2.1.1 Khi tm ca c h v vt rnKhi tm ca h: Xt c h gm N cht im Mk c khi lng mk, vc t nh v kr. im C gi l khi tm ca c h c xc nh theo cng thc sau:NkkNkk kcmr mr11, NkkNkk kcmx mx11,NkkNkk kcmy my11, NkkNkk kcmz mz11(3.13)Khi tm ca vt rn: Xt vt rn v chia n thnh N (s lng phn t tin n v cng) phn t nh Mk c khi lng mk, vc t nh v kr.im C gi l khi tm ca vt rn c xc nh theo cng thc sau: VNkkNNkk kNcdm rMmr mr 111limlim, NkkNm M1lim VNkk kNcxdmM Mm xx11lim, VNkk kNcydmM Mm yy11lim, VNkk kNcxdmM Mm zz11lim(3.14)Khi vt rn nm gn tri t khi tm trng vi trng tm Nu vt l khing cht c th tch V v c khilng ring th M=V; dm=dV, vy VcdV rVr 1, VcxdVVx1, VcydVVy1, VczdVVz1, (3.15) i vi tm phng ng cht c tit din F thFSxdFFxyFc 1, FSydFFyxFc 1,(3.16)trong FxydF S, FyxdF S, (3.17)c gi l m men tnh ca tit din F vi trc x v ynh l 1: Nu VR ng cht c tm (trc, mt phng) i xng th khi tm ca n nm ti tm (trc, mt phng) i xng. Khi tm ca thanh ng nht nm trn im gia ca thanh Khitm cahnh bnh hnh,ch nht,vung,khihp ch nht,khilp phng l tm ca chng Khi tm ca tam gic nm trn giao im ca cc ng trung tuyn Khi tm ca cung trn / sin R xc, khi / / R xc2 2 Khi tm ca qut trn 3 2 / sin R xc, khi 3 4 2 / / R xc2.1.2. M men qun tnh ca vt rnnh ngha: M men qun tnh ca vt rn i vi trc z k hiu l Iz l i lng v hng c xc nh bng VNkk kNzdm m I212lim(3.18)trong k l khong cch t im Mk n trc. n v kgm2M men qun tnh i vi cc trc to Oxyz + Vxdm z y I ) (2 2; + Vydm z x I ) (2 2; + Vzdm y x I ) (2 2. (3.19)M men qun tnh tch l cc i lng: VNkk k kNyx xyxydm m y x I I1lim; VNkk k kNzy yzyzdm m z y I I1lim(3.20) VNkk k kNxz zxzxdm m x z I I1limTrc qun tnh chnh: Trc x c gi l trc qun tnh chnh nu 0 xz xyI I.(3.21)Tng t trc y c gi l trc qun tnh chnh nu 0 yz yxI I v trc z c gi l trc qun tnh chnh nu 0 zy zxI I.Trc qun tnh chnh trung tm: l trc qun tnh chnh i qua khi tm.M men qun tnh ca vt rn i vi mt im VNkk kNOdm r r m I212lim.(3.22)D dng nhn thy:) (z y x OI I I I + + 21(3.23)Bn knh qun tnh l i lng: MIzqt 2(3.24)i vi tit in phng F th FxdF y I2; FydF x I2; FxyxydF I. (3.25)M men qun tnh c cc y xFOI I dF r I + 2.(3.26)H trc c 0 xyI gi l h trc chnh.H trc c0 xyI; 0 y xS S(3.27)gi l h trc qun tnh chnh trung tm. Tng ng vi cc h trc ta c m men qun tnh chnh hay m men qun tnh chnh trung tm.nh l: M men qun tnh ca vt rn i vi trc bng tng m men qun tnh ca n i vi trc song song vi trc qua khi tm C v tch ca khi lng vi bnh phng khong cch gia hai trc2Md I IC + nh l: Nu vt rn ng cht c mt trc i xng th trc l trc qun tnh chnh trung tmnh l: Nu vt rn ng cht c mt phng ixng th trc thng gc vimt phng i xngl trc qun tnh chnh ti giao im ca mt phng i xng v trcM men qun tnh ca mt s vt ng cht: Thanh ng cht c chiu di l, khi lng M:122MlIC; 32MlI Iz x ; 0 yI. Vnh trn ng cht c bn knh R, khi lng M2MR Ix ; 22MRI Iz y . Mt trn ng cht c bn knh R, khi lng M22MRIx ; 42MRI Iz y . Tm ch nht ng cht c cc cnh 2a, 2b, khi lng M122MbIx ; 122MaIy . Hnh tr ng cht c bn knh R, khi lng M, chiu cao hTr rng 2MR Ix ;

,_

+ 6 222hRMI Iz y.Tr c 22MRIx ;

,_

+ 3 422hRMI Iz y.Ta kho st h lc cn bng02 1 2 1 ) , , , , , , (qtNqt qtNF F F F F F (3.28)2.2. nh l ng lngGi FR l vec t chnh ca h lc) , , , (NF F F 2 1 + ekikek k FF F F F R Gi qtR l vec t chnh ca cc lc qun tnh: k kkk k kqtk qtv mdtddtv dm a m F R Tch k kv m l ng lng ca cht im, cnk kv m Q(3.29)c gi l ng lng ca c h. VyQdtdRqt Gi R l vec t chnh ca h lc, , , , (NF F F 2 1) , , ,qt qt qtF F F 2 1, ta c0 + QdtdF R R Rek qt F T ekFdtQ d(3.30)v NkekttekS dt F Q Q100 (3.31)trong Q vQ l ng lng ng vi cc thi im t v t0, cn NkttekNkekdt F S1 10(3.32)l tng xunglc cc ngoilc tc dng ln c h trong khong thi gin (t-t0)nh l: o hm ca ng lng theo thi gian bng tng cc ngoi lc tc dng ln hnh l: Bin thin ng lng trong khong thi gian no bng tng xung lc cc ngoilc tc dng ln c h trong khong thi gian .i vi trc Ox c nh: ekxxFdtdQ, (3.33) Nkekxttekx x xS dt F Q Q100(3.34)Ch :- Ni lc khng nh hng n s bin i ca ng lng c h Nu 0 ekF th ng lng bo ton:const Q Nu 0 ekxF th hnh chiu ng lng trn trc x bo ton: const Qx2.3. nh l chuyn ng khi tmTa c C C k kkk k kv M r Mdtdr mdtddtr dm v m Q ) ( (3.35)trong Cv M , l khi lng v vn tc khi tm ca c h. Vy ng lng ca c h bng ng lng ca khitm vigi thit khitm c khilng bng khi lng ca c h. ek C CF a M v MdtddtQ d ) (nh l: Khi tm chuyn ng nh cht im c khi lng bng khi lng c h v chu tc dng ca lc c vc t bng vc t chnh ca h ngoi lc tc dng ln c hek CF a M(3.36)Trn trc to ccexk CF x M , exk CF y M , exk CF z M (3.37)Nhn xt. Ni lc khng nh hng n chuyn ng ca khi tm Nu 0 ekF th0Catcconst v hoc0 v Nu 0 ekxF th0 Cxtc const xC hoc 0 Cx2.4. nh l m men ng lngGi OFm v Oqtm l m men chnh i vi im O c nh ca cc lc tc dng v cc lc qun tnh ca cc cht im thuc c h + ) ( ) ( ) ( ) (ek Oik Oek O k OOFF m F m F m F m m v[ ] ) ) (k k k k k kqtkqtk OOqtv m rdtda m r F r F m m trong kr - vec t nh v, k kv m - ng lng, k kv m r - m men ng lng ca cht im Mk i vi im O c nh ) (k k O k k k Ov m m v m r L (3.38)l tng m men cc ng lng cc cht im i vi im O, gi tt l m men ng lng c h i vi im OVy OOqtLdtdm V m men chnh ca h lc cn bng trit tiu nn0 + Oek OOqtOFLdtdF m m m ) (do Oek OLdtdF m ) ( (3.39)Hnh chiu ln trc z zek zLdtdF m ) ( (3.40)nh l: o hm theo thi gian ca m men ng lng ca c h i vi mt im (trc) c nh bng tng m men ca cc ngoi lc vi cng im (trc )Nhn xt: - Ni lc khng nh hng n s bin i m men ng lng c h Nu 0 ) (ek O F m thconst LO hoc 0 ) (ek z F m thconst LzPhng trnh vi phn chuyn ng ca vt quay quanh mt trc c nhXt VRquayquanhmttrccnh.Ccngoi lc gm) , , , (NF F F 2 1vcc phn lc lin kt B A R R , . u tin ch :z k k k k k k O OI h m h m v m m L 2 2) ( trong- vntcgc, mkkhi lngcaphntMknmcchtrcquay khong hk, Iz m men qun tnh ca VR i vi trc quay. p dng (3.29) ta c ) ( ) (k z z z z zF m IdtdI IdtdLdtdVy PTVP chuyn ng ) (k z z z zF m IdtdIdtdI22(3.41)2.5. nh l ng nng2.5.1. nh l ng nng: Gi WF v Wqt l tng cng sut ca cc lc tc dng v ca cc lc qun tnh k k Fv F W; 221k k kkk k k k kqtk qtv mdtdvdtv dm v a m v F W i lng 221k kv mc gi l ng nng ca cht im Mk, cn221k kv m T (3.42)c gi l ng nng ca c h.Vy dtdTWqt . V h lc cn bng nn0 +dtdTv F W Wk k qt FVyF k kW v FdtdT (3.43)nh l: o hm theo thi gian ng nng ca c h bng tng cng sut cacc lc (ni lc v ngoi lc, hay lc hot ng v lc lin kt) tc dng ln c h.T (3.43) ta c k k f k kdA r d F dt v F dT(3.44)nh l: Vi phn ng nng ca c h bng tng cng nguyn t ca tt c cc lc (ni lc,ngoilc,lc hotngv lc linkt)tcdng ln ch. Tchphn (3.44) ta c kA T T0(3.45)nh l:Bin thin ng nng ca c h trong mt khong thi gian no bng tng cng ca cc lc (ni lc v ngoi lc, hay lc hot ng v lc lin kt) sinh ra trong chuyn di ng vi thi gian .nh l ng nng phn nh su sc bn cht qu trnh thay i chuyn ng ca c h v nh n trng thi chuyn ng ca c h c nghin cu su sc2.5.2. p dnga) Biu thc ng nng ca vt rn chuyn ng Chuyn ng tnh tin: Vt c khi lng M chuyn ng vi vn tcv. Trong trng hp ny tt c cc phn t c cng vn tc, do 2 2 2 221212121Mv m v v m v m Tk k k k (3.46) Chuyn ng quay quanh trc c nh vivn tc gc: Phn t Mkc khi lng mk nm cch trc quay mt on hk c vn tc k kh v, do 2 2 2 2 2 221212121 z k k k k k kI h m h m v m T (3.47)trong IZ m men qun tnh ca vt i vi trc quay Chuyn ng song phng: Xt hnh phng khi lng M, vn tc khi tm l vC v vn tc gc . Gi P l tm vn tc tc thi. Phn t Mkc khi lng mk cch tm P mt on hk c vn tc k kh v, do 2 2 2 2 2 221212121 P k k k k k kI h m h m v m Ttrong IP m men qun tnh ca hnh phng i vi trc quay qua P v hnh.Theo nh l v mmen qun tnh ta c 2PC M I IC P + , do 2 2 2 2 221212121) ( ) ( + + PC M I PC M I v I TC C P Vy 2 22121C CMv I T + (3.48)b) Biu thc tnh cng ca mt s lc Cng ca trng lc: Cng sut ca trng lc P:z P v P W Cng ca trong lc khi im t di chuyn t M1 n M2Ph z z P Pdz dt z P Wdt AMMMMMMt ) (1 2112121(+) - im t h xung, (-) - im t nng ln, = 0 - di chuyn ngangTrng hp h trng lc kP Pc trng tm ti ch h ( )Ck h P At Cng ca lc n hir c F , trong r -vc t nh v ca im t lc, c h s t l:) ( ) (212222121112121r r r d c r d r c r d F Arrrrrr Trng hp l xota c 221 c A , - bin dng ca l xo. Cng ca lc tc dng ln vt rn chuyn ng tnh tin: Cng sut: Cv F v F W , vy C Cr d F dt v F Wdt A Cng ca lc tc dng ln vt rn chuyn ng quay quanh trc c nh: Cng sut: ) ( cos cos F m FR Fv v F WO, vy d F m dt F m Wdt AO O) ( ) ( Cng ca lc tc dng ln hnh phng chuyn ng song phng, c th coi hnh phng quay quanh tm tc thi vy ) (F m WP; 21d F m AP) (C th coi tm phng chuyn ng tnh tin vi khi tm C v quay quanh tm C lc C C Pv F F m W + ) ( ; + C C Or d F d F m A ) (2.6. nh l bo ton c nng2.6.1. Trng lc th Th nngTrng lc th khong khng gian vt l m khi c h chuyn ng trong , cc cht im ca n chu tc dng ca lc ch ph thuc vo v tr ca c h, cn cng cc lc tc dng khng ph thuc vo dng qu o cc cht im m ch ph thuc vo im u v im cuiTh nng c h - tng cng ca cc lc th tc dng ln c h khin di chuyn t v tr M (ang xt)n v tr O (gc), Vy MOA M) (trong : MOA tng cng cc lc khi c h di chuyn t M n O suy ra rng0 ) (O , O MA M11) (Cng ca lc trong trng lc th khng ph thuoc vo dng qu o im t lc nn c h di chuyn t M n O c cng theo mi ng u bng cng theo ng qua v tr M1( ) + + 11 1M A A A A MMM O M MM MO) (Vy 11MMA M M ) ( ) (nn chnh lch th nng ti hai v tr bng tng cng cc lc khi c h di chuyn t v tr ny n v tr kia.2.6.2. nh l bo ton c nng. Gi s c h chuyn ng t M0 n M. Theo nh l ng nng ta c M MMMA T T00ng thi M MA M M00) ( ) (Vy 00MMT T M M ) ( ) (Hay const T M T M EMM + + ) ( ) (00i lng E l c nng tng ca ng nng v th nng.nh l: Khi c h chuyn ng trong trng lc th th c nng ca h c bo ton. 2.6.3. Biu thc tnh th nng ca mt s lc th Th nng trng lng. Chn mt t l v tr gc. Trong tm c h l C cch mt t cao hC th ( )c k h P Th nng n hi: Khi bin dng ca l xo l , c l cng l xo th221 c3. MT S NGUYN L C HC3.1. Cc khi nim c bn 3.1.1. Di chuyn kh d v s bc t do ca h. Tp hp cc kh nng di chuyn (v cng b) m c h c kh nng thc hin c gi l di chuyn kh d. N l tp hp cc di chuyn v cng b ca cc cht im thuc c h sang v tr ln cn ph hp vi cc lin kt ang xt. Cc di chuyn kh d ca c h khng c lp i vi nhau. S bc t do ca h (k hiu l k) l s ti a cc di chuyn kh d c lp tuyn tnh ca c h3.1.2. To suy rng Tp hp cc thng s xc nh v tr ca c h trong mt h quy chiu c gi l to suy rng ca c h k hiu l,1q ,2qmq , . Chng c th l ta cc ca cc cht im, c th l gc quay. To cc ca cc cht im thuc c h c th biu din qua to suy rngn k q q q r rq q q z zq q q y yq q q x xm k km k km k km k k, ) , , , () , , , () , , , () , , , (12 12 12 12 1 ; (3.49) Khi m=n ta c to suy rng , khi m>n ta c to suy rng tha.Trong chuyn ng tnh tin to suy rng l ba to cc ca mt im bt k thuc VR. Trong chuyn ng quay quanh trc c nh to suy rng l gc quay ca vt. Trong chuyn ng song phng to suy rng l , ,0 0 y x.3.1.3. Lc suy rnga) Biu thc cng ca lc trong di chuyn kh d: Gi tt l cng kh d ca lc. Cho c h di chuyn kh d } {kr. Theo (3.5) cng kh d ca cc lc s l + + k kz k ky k kx k k kz F y F x F r F AGi s ta suy rng ca c h l,1q ,2qnq , , t (3.49) ta tnh c; niiikkqqxx1 ; niiikkqqyy1 ; niiikkqqzz1Thay vo biu thc cng kh d 1]1

,_

++ nii iniiNk ikkzikkyikkx kq Q qqzFqyFqxF A1 1 1(3.50)b) Lc suy rngi lng

,_

++Nk ikkNk ikkzikkyikkx iqrFqzFqyFqxF Q1 1(3.51)c gi l lc suy rng ng vi ta suy rng qi, l i lng v hng.3.1.4. Lin kt l tngLin kt c gi l l tng nu cng ca tt c cc lc lin kt trong mi di chuyn kh d bng 00 k k r R(3.52)Trong thc t nu b qua ma st v tnh n hi ca vt th ta c c h chu lin kt l tng: Cc vt rn t do hai vt rn lun lun ta vo nhau trong qu trnh chuyn ng Dy mm khng dn vt qua rng rc hay puli Hai vt ln khng trtLc suy rng ca cc lc lin kt l tng trit tiu01Nk ikkRiqrR Q(3.53)3.2. Nguyn l cng o3.2.1. Nguyn l cng o: i vi c h chu lin kt hnh hc, gi, dng v l tng, iu kin cn v c h cn bng ti v tr ang xt l tng cng nguyn t ca cc lc hot ng trong mi di chuyn kh d ca c h t v tr ang xt u trit tiu0 k k kr F A(3.54)- l phng trnh cng kh d, trong kF l lc hot ng tc dng ln cht im Mk, kr - di chuyn kh d ca cht im MK. Chng minh iu kin cn: Gi s c h v tr cn bng. Xt cht im Mk thuc c h chu lc kF v lc lin kt kR. V c h cn bng nn mi cht im cn bng, tc l0 +k kR F , do ( ) 0 + + k k k k k k kr R r F r R F Do lin kt l tng ta c (3.53) vy 0 k k r F, y l iu cn CMChng minh iu kin : Gi s c h v tr cn bng v tng cng kh d ca cc lc hot ng bng 0. Ta chng minh c h mi mi cn bng v tr ny.Thc vy, gi s c t nht 1 im Mk di tc ng ca lc kF v lc lin kt kR chuyn ng khi v tr u ng yn, di chuyn ca n kr d c cng phng vi lc tcdngk kR F + . Linkt ldngnndi chuynthctrngvi mt trongcc phng ca di chuyn kh d, vy0 > + +k k k k k k kr R r F r R F ) (Ly tng hai v i vi tt c cc cht im ca c h ch n iu kin lin kt l tng ta c0 > + k k k k k kr F r R r Fiu ny tri vi gi thit. Vy khng th c d ch mt im di chuyn t v tr cn bng l iu cn CM3.2.2. iu kin cn bng ca c h trong to suy rng Da vo biu thc cng kh d, nguyn l dch chuyn kh d c vit01 nii i k k Fq Q r F A(3.55)T tnh cht c lp ca cc ta suy rng nn cc qi cng c lp vi nhau. Vy ng thc (3.55) ch ng khi v ch khin i Qi, , 1 0 (3.56)nh l: iu kin cn v c h chu lin kt hnh hc, gi, dng v l tng cn bng ti mt v tr l cc lc suy rng ng vi cc to suy rng phi ng thi trit tiu.Ch : nu a vo k hiu dtrvkk l vn tc kh d th nguyn l kh d c th vit di dng0 k kv F (3.57)- phng trnh cng sut kh d3.3. Nguyn l d'Alembert3.3.1. Lc qun tnh ca cht imCht im chu tc dng ca lcFdichuyn vigia tc aivih quy chiu qun tnh. Lc qun tnh ca cht im - qtF c cng phng v ngc chiu vi gia tc v c gi tr bng a m Fqt (3.58)Lc qun tnh khng phi l lc tc dng ln cht im. 3.3.2. Nguyn l d'Alembert i vi cht imTi mi thi im lc tc dng ln cht im v lc qun tnh ca n cn bng nhau: 0 +qtF F (3.59)Ch : Trong nguyn l ch khng nh s cn bng v lc, khng phi s cn bng ca cht im Trng hp cht im khng t do lc tc dng l hp ca lc hot ng v lc lin kt Trng thi cn bng v lc c thit lp mi im3.3.3. Nguyn l d'Alembert i vi c hKho st c h gm N cht im,1M ,2MNM , di tc dng ca h lc ,1F,2FNF,chuyn ng vi gia tc,1a,2aNa, .Xt cht im kMc khi lng kmchu tc dng kF chuyn ng vi gia tc ka. Lc qun tnh ca n l: k kqtka m F Theo nguyn l d'Alembert i vi cht im ta cN k F Fqtk k, , 1 0 + Vy: Ti mi thi im cc lc tc dng ln cht im ca c h v cc lc qun tnh ca cc cht im thuc c h to thnh mt h lc cn bng:, (1F,2FNF, , ,qtF11,qtF2101 ) ,qtNF (3.60)Ch : Cc lc kF vi c h t do l hp lc ca ngoi lc v ni lc, vi c h chu lin kt l hp lc ca lc hot ng v lc lin kt tc dng ln cht im kM Khi nim cn bng c tnh quy c h lc cn bng l h lc c tng cp lc trc i nhau H lc cn bng c thit lp ti mi thi im nn c th thit lp cho h trc to ng trong bi ton xc nh phn lc trc quay. Tuy nhin cc lc qun tnh phi tnh i vi h quy chiu qun tnh3.4. Nguyn l d'Alembert-Lagrangei vi c h chu cc lin kt gi v l tng, ti mi thi im tng cng ca cc lc hot ng v cc lc qun tnh trong mi di chuyn kh d ca c h bng 0.Theo nguyn ld'Alembert i vi cht im ta c N k F R Fqtk k k , , , 2 1 0 + +trong kF lc hot ng v kRlc lin kt. Thc hin di chuyn kh d krv tnh tng cng ta c( ) 01 + +Nkkqtk k kr F R F , ) ( 01 1 + + Nkk kNkkqtk kr R r F F Do lin kt l l tng ta c , 01 Nkk k r R vy 01 1 + Nkk k k kNkkqtk kr a m F r F F ) ( ) (. (3.61)3.5. Phng trnh Lagrange loi 23.5.1. Vt rn chuyn ng tnh tina) Thu gn h lc qun tnh Kho st vt rn c khi lng M chuyn ng tinh tin vi gia tca. Xem vt rn gm v s cc phn t kM c khi lng kmu chuyn ng vi gia tca. Nh vy h lc qun tnh gm v s cc lc song song cng chiu. S dng phng php thu gn h lc qun tnh v khi tm C ta ca M Rqt (3.62)b) Phng trnh chuyn ng Gi s VR chu tc dng ca cc lc,1F,2FNF, , theo nguyn l d'Alembert n phi c hp lc ti khi tm C v trc i vi hp lc qun tnh 0 0 + C k qt Fa M F R R 0 0 ) (k CCqtF m mT ta c k CF a M v0 ) (k C F m (3.63)3.5.2. Vt rn quay quanh mt trc c nh Thu gn h lc qun tnh Khi thu gn h lc qun tnh v trc quay O ta cCqtOa M R v OOqtI m(3.64)trong M khi lng ca hnh phng, IO m men qun tnh ca hnh phng i vi trc quay O, - gia tc gc ca hnh phng, Ca - gia tc khi tm C Nu quay u (=const,=0) quanh khi tm(0 Ca, O C) th 0 qtOR, 0 Oqtm - trc quay cn bng Nu quay u (=const, =0) khng quanh khi tm th nCqtOMa R , 0 Cqtm - h lc tng ng vi mt lc hng tm (t C n O) c tr se M RqtO2 , e l khong cch t C n trc quay O Nu quay khng u ( 0) quanh khi tm (0 Ca, O C) th 0 qtOR, v h lc qun tnh tng ng vi mt ngu lc COqtI m Nu quay khng u ( 0) quanh mt trc khng qua khi tm th h lc qun tnh tng ng vi mt lc Phng trnh chuyn ng v phn lc trc quayGi s hnh phng chu tc dng ca cc lc hot ng phng,1F,2FNF,v cc lc lin kt OR, theo nguyn l d'Alembert ta c , (1F,2FNF, , ,OR,qtF11,qtF2101 ) ,qtNFtc l0 + C O ka M R F ;0 O k OI F m ) (T ptr cui ta c phng trnh vi phn chuyn ng ) (k O OF m I(3.65)T y ta tm c , tch phn ta c v tm c gia tc ca khi tm C. T phng trnh u ta xc nh phn lc ca trc quay k C OF a M R(3.66)3.5.3. Vt rn chuyn ng song phng Thu gn h lc qun tnh Kho st tm phng c gi tc khi tm Ca.,vn tc gc gia tc gc .Khi thu gn h lc qun tnh v trc khi tm C ta cCqtCa M R v OCqtI m(3.67)trong M khi lng ca hnh phng, IC m men qun tnh ca hnh phng i vi khi tm C Phng trnh chuyn ng Gishnhphngchutc dngcacc lc ,1F,2FNF, theonguynl d'Alembert ta c , (1F,2FNF, , ,qtF11,qtF2101 ) ,qtNFtc l0 C ka M F ;0 C k CI F m ) (T y ta ck CF a M, ) (k C CF m I(3.68)Trong h to cc kx CF x M ; ky CF y M ; ) (k C CF m I(3.69)Hai phng trnh u m t chuyn ng ca khi tm, cn phng trnh cui m t chuyn ng quay ca hnh phng i vi trc tnh tin cng khi tm.Nu bit qu o khi tm C, c bn knh cong C th ta c th vitkttCF Ma; knCCFvM2; ) (k C CF m I(3.70)3.5.4.Phng trnh vi phn chuyn ng ca c h Phng trnh Lagrange loi II kho st cc h vt rn s dng Phng trnh vi phn chuyn ng c h Phng trnh Lagrange loi II. Kho st c h chu lin kt hnh hc v l tng c n bc t do. V tr ca c h c xc nh bng n ta suy rng ,1q ,2qnq , . Gi s h chu lc hot ng l,1F,2FNF, v cc lc lin kt ,1R,2RNR, . Theo nguyn l dAlembert ta c , (1F,2FNF, , ,1R,2RNR, , ,qtF11,qtF2101 ) ,qtNF (3.71)i vi cclclinktl tnglcsuy rng trittiu, nntacnguynl dAlembert Lagrange01 Nkk k k kr a m F ) (Thay; niiikkqqrr1 vo biu thc trn ta c01 1221 1 niNkiik kkniNkiikkqqrdtr dm qqrF (3.72)Theo nh ngha lc suy rng ta cNk ikkFiqrF Q1t Nk ik kkqtiqrdtr dm Q122 ta c iu kin cn bng 0 +qtiFiQ Q (3.73)T nh ngha to suy rng (3.49) ) , , , , ( t q q q r rm k k 2 1 ta ctrdtdqqrvdtr dkniiikkk+ 1 ikikqrqvv do ) , , , , , , , , ( t q q q q q q v vn n k k 2 1 2 1 nn

,_

+ jkjkniij ikjkqrdtdt qrdtdqq q rqv 212Thay vo ta c

,_

,_

+Nk ikk kNk ikk kNk ik kkqtiqrdtdv mqrdtdv mqrdtv dm Q1 1 1

,_

+Nk ikk kNk ikk kNk ikk kNk ikk kNk ik kkqtiqvv mqvv mdtdqvv mqvdtdv mqvdtv dm Q1 11 1 1 Mt khc ta c 221k kv m T Nk ikk kiNk ikk kiqvv mqTqvv mqT1 1; vy i iqtiqTqTdtdQv phng trnh (3.25) a v phng trnh Lagrange loi IIn i QqTqTdtdii i, , 1 (3.74)Nu cc lc hot ng gm cc lc c th vi hm th nng v cc lc hot ng khng th c lc suy rng l iQth phng trnh Lagrange loi II nh sau:n i Qq qTqTdtdii i i, , 1 + (3.75)