oents & Tools tool like scedive o enc cn delive te equied tunin oent to titen sce o nut it less oce. Te toque on te nut is. t is lso. So i te tio / is incesed, o iven ount o oce, te eective nut-tunin oce is niied. Tis is clled ecnicl dvnte. bod is not ottin, o ottin it constnt nul velocit, ten te su o oents ust equte to zeo. Tis is ve useul in clcultin oces in equilibiu pobles involvin iid bodies, since it doesn t tte in tis cse ic point e tke oents bout. (Te ust ls su to zeo). Onl en tee is net toque is it ood ide to coose te cente o ss o n object, nd use set is to sipli te oent o ineti tenso.* +ve Pivot o ulcu Usin te convention tt clockise oents e positive te bove pivot sste is in ottionl equilibiu i: Eple: Conside ectnul block on te vee o slidin in te nd ve diections en it is inclined t nle. Wt is te coeicient o iction beteen te block nd te loo? ssue ll is soot nd loo is ou. B C +ve cos sin... cos sin... cos sin Soot ll bout to slide on ou loo Neton in diection Neton in diection oents bout cente o ss C cos sin cos sin... cos sin cos sin... cos sin cos sin tn sin cos tn tn tn Substitutin into te oents eqution * See olloin pes o deinition o tis quntit! tetics topic ndout: oents, Toque & nul oentu D nde enc..eclecticon.ino PGE
Eple: Tee stcked clindes cos sin eit eit W Spot te Ptoen tiple! Eple: Qu tuck poble (Qudlin, ecnics, pp96) Weit W +ve ssue sttic equilibiu. Neton o uppe clinde ( diection) cos sin 8 6 8 6 Neton o loe clinde ( diection) G cos sin G G 9 Hence: 8 6 6 Neton o loe clinde ( diection) S cos sin W S W 6 S W S W S W 6 G 6 Su o oents bout n point is zeo Vecto su o ll oces is zeo S W G No slip t ound-spee intece G S W 6 6W No slip t spee-spee intece B 6 Clockise oents bout cente o loe clinde G G oents bout loe eel: oents bout uppe eel:.w sin W cos W.cos.sin S.W sin W cos S W.cos.sin Teeoe S > o n cosen. Since bek oce is popotionl to S o, et le i ou bek it te loe eels. o no slidin don te slope.s Neton in diection: W sin W sin.s o Wsin.W.cos.sin sin.cos.sin.88 tn. tn.8 tetics topic ndout: oents, Toque & nul oentu D nde enc..eclecticon.ino PGE
Eple: Stick on cicle poble (oin, ntoduction to Clssicl ecnics pp8) Wt e te coeicients o iction beteen (i) te stick nd cicle nd (ii) beteen te cicle nd te loo necess to intin equilibiu? oces on cicle l Since stick is on cicle tnent ltn oces on stick Stick: oents bout l B l cos l cos sin cos cos cos sin cos cos cos cos cos cos cos B C +ve +ve Cicle: Neton : sin cos : cos sin Cicle: oents bout cente C B o no slidin t point B B sin B tn cos Stick: Neton : cos sin : sin cos sin tn cos sin cos cos sincos sin cos cos sin cos sin cos o no slidin t point sincos ( cos )( cos ) Note i tee is no slidin t te stick nd loo intece () nd t te stick nd cicle intece (B), tis is suicient condition o te sste to be in equilibiu. n dditionl no-slip condition t te loo-clinde junction does not need to be obeed, so teeoe cnnot be enell tue. i.e. D not be enell tue tn in in sincos ( cos )( cos ) tetics topic ndout: oents, Toque & nul oentu D nde enc..eclecticon.ino PGE
Toque, oent o ineti nd nul cceletion Conside ss undeoin cicul otion. Te dius is ssued to be constnt. v ˆθ oce is pplied tnentil to te otion v v d dt dv dt Neton dv dt Te oent (o toque) o te oce bout te cente o te cicle is Hence: nul velocit /dins pe second Tnentil velocit ˆ Tis is useul enel esult o ottionl otion - essentill ottionl equivlent o Neton toque = oent o ineti nul cceletion Te kinetic ene is E E E v d d dt dt We cn clculte te oent o ineti o etended objects (i.e. not just pticles) nd ence ppl te oule bove oe enell o iid bod ottion d Unsupisinl, te se lebic o s o pticle i i i ss eleent d d (squed) distnce o ss o is o ottion oent o ineti o tin oop* o ss nd dius oent o ineti o solid clinde l Deteine cente o ss b suin etuded oops d l l Densit l ss tube volue l d Note tis ill be te se oul o te oent o ineti o disc bout ottion is pependicul to te plne o te disc, nd tou te disc cente. *Wic ust ve te se oent o ineti s n etuded oop i.e. clindicl tube o ss, since it copises o identicl ooplets it te se oent o ineti. tetics topic ndout: oents, Toque & nul oentu D nde enc..eclecticon.ino PGE
Spin up spool epeient T T v Disc, ss nd dius Clinde, ounted in disc. ss c, dius ss, connected to spool vi lit inetensible stin Neton o ss T No ssue te disc cn v in dius, but te se densit nd tickness o teil is used. i.e. constnt ss pe unit e Note disc s ole o te eel to it to ccount o ole Hence: T c c c c Let te ss dop ied distnce nd let te dop tie be t o iven disc nd spool, te cceletion is constnt. Hence: t t c c t c One it epect k TLB siultion.k.k.8.67k..6k 9.8s. - - c c c c Coection t t c t t Toque = oent o ineti nul cceletion o spool + disc T c Teeoe p o: vs c t to ccount o ole No te spoolin velocit ust equl v since te stin is inetensible v sould be stit line, it dient t tetics topic ndout: oents, Toque & nul oentu D nde enc..eclecticon.ino PGE
oent o ineti o solid spee (, ) Densit Disc spe ss eleent ss dius d d d oent o ineti o disc o ss nd dius is i.e. to eispees oent o ineti o pbolic cp V V V d V d Volue d d d d Ptos Teoe V Densit d d d d i.e. su o oents o ineti o clindicl ss eleents tetics topic ndout: oents, Toque & nul oentu D nde enc..eclecticon.ino PGE 6
ollin clinde (o spee) don slope te clinde (o spee) is not slippin dv v dt clinde o spee o ss nd dius Hence usin Neton : sin sin sin o no slip: Neton : sin : cos Nol contct oce iction oce coeicient o iction cos sin cos tn tn toque = oent o ineti nul cceletion Clinde: Spee: sin tn 7 sin 7 tn Note pticle slidin don te slope itout iction ould ccelete t sin tetics topic ndout: oents, Toque & nul oentu D nde enc..eclecticon.ino PGE 7
oents o neti bout pincipl es o vious bsic spes o Won, Te Cbide Hndbook o Psics ouls. Note te convention,, s te se enin s,, Eple: Clculte te oent o ineti o disc bout its set es Out o te plne d d d Ltel o lonitudinl lip d d d d is oop o idt d nd dius, o densit Since disc nd setic in, plne ' z Pependicul is teoe Tis oks o line onl i.e. it no z etent Pllel is teoe z ' d z ' d d d d d d d d z ' d z d d z d d Since C is te cente o ss d z' z
ectnul lin ine ottion poble t is te ine oce? 8 6 cente o ss Ene S Lin o ss u Hine (ictionless) ˆ ˆθ nitil ene u cos sin o o o 8 9 (8 ) 9 Neton ˆ : S cos θˆ : sin u cos u cos oce on lin o ine (esolved into coponents) cos sin o sin cos oent o ineti o ectnul lin bout ine 6 8 oent o ineti o ectnul plte bout cente u cos u cos Toque bout ine = oent o ineti nul cceletion sin Pllel is teoe sin sin sin Neton S cos sin u cos S cos u 6 S cos Te initil kinetic ene ust eceed te iu in in GPE in ode o te lin to continue to ll u u iu GPE is en cente o ss is diectl ove te ine Let u k, k u k S cos k 6 S cos k cos sin k 6 S ˆθ ˆ sin sin sin nitude o ine oce S tetics topic ndout: oents, Toque & nul oentu D nde enc..eclecticon.ino PGE 9
Te oent o oce, o toque, is te vecto coss poduct o displceent nd oce. τ is te displceent o te cente o ss o iid bod ten net pplied toque ill esult in ottion o te bod bout n is tou its cente o ss. Note unless toque is pplied vi to equl nd opposite diection oces (tis is clled couple) te cente o ss ill lso ccelete due to te net oce upon te bod. b nd e pependicul, ten te nitude o te toque (o tunin oent) is oce distnce o te ottion is. τ couple pplied to iid bod cetes toque τ Tis ill not cuse n tnsltionl oveent o te cente o ss o te bod but ill cuse it to otte. nul oentu is deined s i.e. te coss poduct beteen te oentu o pticle nd displceent vecto o n is o ottion o iid bod L ω L p is te oent o ineti tenso nd is te nul velocit vecto ic descibes te instntneous te o ottion bout n is pllel to. Te ineti tenso cn oten be sipliied i one deines Ctesin coodintes to tc te set es o iid bod n tis eple, te ict is eecutin oll (nd not pitc o ) L z ω ˆ p Wit espect to Ctesin,,z coodintes: ˆ ẑ Cople ottionl otion (e.. tublin ) esults en te diection o te nul oentu vecto cnes, s ell s it s nitude. e.. in oscope. z d d zd d z d zd ŷ b it-nd sce ule o vecto poducts zd zd d ost enines copise o ottin st ic is ten used to dive eels etc. Te poe is iven b P τω nd te ottionl kinetic ene ω ω E z z nloous to Neton s Second L, pplied toque equtes to te te o cne o nul oentu poe is 8 8. =kw dp d dv v dt dt dt dl d dω τ ω dt dt dt Note ust be constnt ee! Neton s Second L descibes o te cente o ss o bod esponds to oce, o te totl ss is constnt. Toque equtes to te o cne o nul oentu. Tpicll e e inteested in situtions en, nd t e ll utull pependicul. n te top let iue, i te bod is disk o dius nd ss d t dt p = / =. ds - tetics topic ndout: oents, Toque & nul oentu D nde enc..eclecticon.ino PGE