Chape 0 Spn an bal n Ran: All aun us: wheels, skaes, balle, gynass, helcpe, s, ble engnes, CD sks, Ac wl: elecns spn, b. Unese: planes spn an bng he sun, galaxes spn, Chape 4 kneacs Chape 0 ynacs
0. Se cnceps abu an. Spn escbe anal n f a syse abu an axs hugh s cene f ass.. Rg by a syse cpse f any pnlke pacles ha anan fxe sances f each he a all e. each pacle f he spnnng g by syse execues ccula n abu he axs hugh he cene f ass. 3. Obal n he cene f he ass f he syse s ng n space f a pespece f a pacula efeence fae. 0. Se cnceps abu an The n f he cene f ass us n be ccula.
0. Se cnceps abu an 4. The bal angula enu f a pacle Defne: p Magnue: snθ p Decn: epencula he plane cnanng he an p p x z θ p p y 0. Se cnceps abu an Nce: s easue wh espec he gn a O; un: kg /s; 3 whaee he pah ajecy f a pacle s saghlne, cue pah, clse bal pah,. 5. The angula enu f he ccula bal n f a pacle (a Angula enu 3
4 p p hen (b Men f nea f a pacle 0. Se cnceps abu an hen efne 0. The e ae f change f angula enu an que. The e ae f change f angula enu f a sngle pacle p p p ( p 0 F p p p Q F θ
0. The e ae f change f angula enu an que. que Defne: τ Magnue: F F τ F snθ F Decn: epencula he plane cnanng he an F Un f he que: N s he psn ec f he pn f applcan f he fce wh espec he chsen gn. F F θ f 0. The e ae f change f angula enu an que Dscussn: θ 0 π, F 0, τ 0 F css he O, F F θ 5
6 3. Dynacs f ccula bal n f a sngle pacle Q τ F p Exaple : 433 0.5 Exaple : 433 0.6 α τ ( Can n be use n nnccula bal n. 0. The e ae f change f angula enu an que 0.3 The angula enu f a syse f pacles an en f nea f g by. The angula enu f a syse f pacles p Q θ p C ( (
0.3 The angula enu f a syse f pacles an en f nea f g by Fs e: Secn e: Th e: M M The psn ec f cene f ass wh espec he cene f ass s he ec su f angula enu f all pacles wh espec he cene f ass. 0 0.3 The angula enu f a syse f pacles an en f nea f g by hen M bal spn. Spn angula enu f a g by abu a axs hugh he cene f ass Q spn // spn ( // ( // z 7
0.3 The angula enu f a syse f pacles an en f nea f g by spn // ( ( // The ec s an nle ec suan. The an f an ly shape bjec abu any axs f an s beyn he scpe f hs cuse. // z 0.3 The angula enu f a syse f pacles an en f nea f g by 3. The en f nea anal nea f a g by abu a fxe axs hugh cene f ass // ( f he g by s syey abu he axs; he axs s fxe. Ths e has n effec. Then spn ( // z 8
0.3 The angula enu f a syse f pacles an en f nea f g by Defne: z Ths s he en f nea anal nea f a g by abu a fxe axs hugh cene f ass The spn angula enu f a g by ( spn // 0.3 The angula enu f a syse f pacles an en f nea f g by 4. The en f nea f aus g bes (a n pacle a sance f he axs f an (b Cllecn f pn pacles --he pepencula sance f each ass f he axs f an (c Rg by f sbue ass 9
0.3 The angula enu f a syse f pacles an en f nea f g by --he pepencula sance f each ass f he axs f an The eleen f ass: λ l lnea ensy:λ σ S suface ensy:σ ρ V lu ensy:ρ 0.3 The angula enu f a syse f pacles an en f nea f g by Exaple: 5 pacles ae cnnece by 4 lgh saffs as shwn n fgue. Fn he en f he syse wh espec he axs hugh pn A, an pepencula he pape plane. Slun: l 3(l (4 5( 3l l A l l l 4 3 5 l 0
Exaple : Thee s a lgh hn saff f ass an lengh. Fn he en wh espec ffeen axs. x x x 3 3 3 8 8 3 3 x x x Slun: 0.3 The angula enu f a syse f pacles an en f nea f g by x x 3 0 3 0 3 x x x x 0.3 The angula enu f a syse f pacles an en f nea f g by
0.3 The angula enu f a syse f pacles an en f nea f g by Se anal neas Suay f he peus secns. The bal angula enu f a pacle z Defne: p F a ccula bal n f a pacle p Ranal nea x θ p p. The e ae f angula enu f a sngle pacle F p F τ θ Defne: τ F y
Suay f he peus secns F a ccula bal n f a pacle τ ( α 3. The angula enu f a syse f pacles M bal spn ( ( C p θ Suay f he peus secns 4. The angula enu f a g by spnnng abu a fxe axs hugh cene f ass spn ( spn z The anal nea spn spn epesens he nea f ang by. // 3
Suay f he peus secns Se anal neas 0.4 The ynacs f g by wh a fxe axs. The e ae f change f he angula enu f a syse f pacles A gup f pacles wh asses,,, n, an angula enus,,, n The angula enu f he syse s 3 Recall F τ, Then F τ 4
0.4 The ynacs f g by wh a fxe axs. The ques ue nenal an exenal fces A syse bune wh ange clu τ τ τ n F any w eleen ass f he syse an τ n ( F F ( F n 0 heefe τ n 0 τ ex 6 4 3 5 F F F F 0.4 The ynacs f g by wh a fxe axs F τ F ex τ ex The que ue he nenal fces n change he angula enu f a syse. 3. The ynacs f g by wh a fxe axs--the anal cunepa f Newn s secn law f a g by wh fxe axs τ τ ex p F ex a spn spn τ ex spn spn α z τ z z spn α z 5
0.4 The ynacs f g by wh a fxe axs Exaple : As shwn n fgue a unf sk, wh ass M.5 kg an aus R0 c, une n a fxe hznal axle. A blck wh ass. kg hangs f a assless c ha s wappe aun he f he sk. Fn he accelean f he fallng blck, he angula accelean f he sk, an he ensn n he c. The c es n slp, an hee s n fcn a he axle. 0.4 The ynacs f g by wh a fxe axs Slun: T g a ( RT MR α ( a Rα (3 a g 4.8/s M T Ma 6.0N a 4.8 α 4a/s R 0. N Mg g 6
0.4 The ynacs f g by wh a fxe axs Exaple : Fn he accelean f he fallng blck by ec applcan f τ. N Slun: z ( R ( g R ( R g R α Ra a MR Ra R g g a M 0.4 The e ae f change f he spn angula enu an ynacs f g by Exaple 3: As shwn n fgue, ne blck has ass M500 g, he he has ass 460 g, an he pulley, whch s une n hznal fcnless beangs, has a aus f 5.00 c. When elease f es, he heae blck falls h75.0 c n 5.00 s ( whu he c slppng. (a Wha s he agnue f he blcks? (b Wha ae he ensns f he c n bh ses? (c Wha s he agnue f he pulley s angula accelean? (e Wha s s anal nea? 7
0.4 The e ae f change f he spn angula enu an ynacs f g by Slun: T T g T T Mg Mg T Ma T g a ( T T R α a αr h a a 6.0 0 /s T 4.87N T 4.54N α.a/s.38 0 kg 0.5 The angula pulse-enu hee an he pcessn f a aply spnnng p. The angula pulse-enu hee Recall F τ τ The angula pulse-enu hee f f τ Cpae wh he pulse-enu hee p p f f F p p F p 8
0.5 The angula pulse-enu hee an The pcessn f a aply spnnng p Exaple 4:Tw cylne hang a R an R an anal neas an especely, ae suppe by axes pepencula he plane f he sceen. The w cylnes ae nally ang wh angula spees an 0 0, especely. The sall cylne s e he gh unl uches he lage ne an he w cylnes ae a cnsan aes n ppse ecns. Fn he fnal angula spees f he w cylnes n es f,, R, R, an 0 0. 0 0 O R O R 0.5 The angula pulse-enu hee an The pcessn f a aply spnnng p Slun: f f Fee by aga: O O R R τ fr f τ fr Accng he angula pulse-enu hee We can ge R R 0 f R R R R 0 f R R R 0 0 ( ( (3 R R 0 R 0 R R 9
0.5 The angula pulse-enu hee an The pcessn f a aply spnnng p. The pcessn f a aply spnnng p f he spnnng angula enu s ze,.e. 0 spn The que f he gay wh espec he pn O τ Mg c Mg snφˆj wll ake he p fall wn. 0.5 The angula pulse-enu hee an The pcessn f a aply spnnng p f he spnnng angula enu s n ze,.e. 0 spn z Accng he angula pulse-enu hee spn τ, τ C The change f he angula M g enu s x τ τ 0
0.5 The angula pulse-enu hee an The pcessn f a aply spnnng p because τ Mg c C φ The angula enu φ C φ θ φ The esul s ha he p es n he ecn pepencula he plane f syey axs an he lcal ecal ecn. 0.5 The angula pulse-enu hee an The pcessn f a aply spnnng p 3. The angula spee f he pcessn θ spn spn snφθ snφ θ φ pcs θ / spn snφ Mg snφ Mg snφ spn spn φ
0.5 The angula pulse-enu hee an The pcessn f a aply spnnng p Thnk! why es he bcycle wheel behae as shwn n he fllwng e fl? 4. The applcan f he pcessn f p f c g 0.6 Sulaneus spn an bal n. The knec enegy f a spnnng syse KE ( f spnnng n KE. Ranal knec enegy f a syse f bal n f he cene f ass KE M
0.6 Sulaneus spn an bal n Tal knec enegy KE spn M 0.6 Sulaneus spn an bal n Exaple 4: A unf sphecal shell f ass M an aus R aes abu a ecal axs n fcnless beangs. A assless c passes aun he equa f he shell, e a pulley f anal neal an aus, an s aache a sall bjec f ass. Thee s n fcn n pulley s axle; he c es n slp n he pulley. Wha s he spee f he bjec afe falls a sance h f es? 3
0.6 Sulaneus spn an bal n Slun: KE ( 3 sph R pull E gh MR sph pull Accng he cnsean law f echancal enegy EKE gh M 3 gh / M / 3 0.6 Sulaneus spn an bal n 3. The angula enu (a Spnnng n b spn spn (b Obal n f cene f ass p M M b (c Tal angula enu f he syse M bal spn Exaple : 450 0.0 spn b 4
0.6 Sulaneus spn an bal n 4. Synchnus an f he spn angula elcy an he bal angula elcy ae paallel, an hae he sae agnue, hen, he n s calle synchnus an. spn b 5. aallel axs hee The angula enu f a synchnus an spn M b M 0.6 Sulaneus spn an bal n The paallel axs hee: The en f nea f he syse abu an axs paallel he syey axs f he syse an sepaae f by a pepencula sance s. M 6. Rllng n whu slppng (a The spn an bal axes ae paallel an sepaae by he aus f he cculaly shape syse unegng llng n. whu slppng ABS syse? 5
0.6 Sulaneus spn an bal n (b Rllng cnsans: s Rθ s θ R cene a acene (c Velcy cene cene cene R R Rα cene 0.6 Sulaneus spn an bal n T T R T C C C B R B B ( Tal knec enegy KEans an KE R R KE 6
0.6 Sulaneus spn an bal n Exaple : 459 0., fn he accelean an he ensn n he c. Slun: eh : chse he cene f ass f he by as he efeence pn. a g T a TR α a Rα R g T R R g T R R g 0.6 Sulaneus spn an bal n eh : chse he pn n he f he by as he efeence pn. gr α R a α R The sae esuls can be bane a R g T R R g T R R g 7
8 6. A geneal scussn abu he angula enu an he que( 0.7 (a Angula enu abu a pn (When bh spn an bal n exs ( ( 0.6 Sulaneus spn an bal n a a p ( ( ( ( ( ( ( ] ( [ ( ( ] ( [ τ (b The e ae f change f he angula enu abu pn 0.6 Sulaneus spn an bal n
0.6 Sulaneus spn an bal n f s n an neal efeence fae; s he cene f ass f he syse; 3 has an accelean paallel anpaallel he ec lcang he cene f ass. hen τ F a g by ang abu he axs hugh he cene f ass τ τ α 0.7 Cnsean f angula enu. The angula enu f a syse f pacles. The e ae f change f he angula enu τ ex 3. Cnsean f angula enu f τ 0 hen cnsan ec ex 9
0.7 Cnsean f angula enu Exaples: The suen eceasng hs nea ncease hs angula spee. The angula enu s cnsan. The e s angula enu s cnsan 0.7 Cnsean f angula enu 0 An ealze spacecaf cnanng a flywheel. f he flywheel s ae ae clckwse as shwn, he spacecaf self wll ae cuneclckwse. When he flywheel s bake a sp, he spacecaf wll als sp ang bu wll be eene by he angle θ sc. 30
0.7 Cnsean f angula enu 0.7 Cnsean f angula enu Execse : As shwn n fgue, he bcycle wheel whse anal nea abu s cenal axs s wh, he wheel s ang a an angula spee wh cuneclckwse. When he wheel s nee, he suen, he sl an he wheel s cene ae gehe as a cpse by abu he sl s an axs, wh anal nea b. Wh wha angula spee an n wha ecn es he cpse by ae afe he nesn f he wheel. 3
0.7 Cnsean f angula enu Slun: The angula enu f he syse s cnsee bf whf b wh bf wh whk ˆ 0 wh whk ˆ bf wh whk ˆ bb wh wh b b 0.7 Cnsean f angula enu Execse : n he ehea ew f he fgue, fu hn, unf s, each f ass M an lengh 0.50, ae gly cnnece a ecal axle f a unsle. The unsle aes clckwse abu he axle, whch s aache a fl, wh nal angula elcy.0 a/s. A u ball f ass M/3 an nal spee /s s hwn alng he pah shwn an scks he en f ne. Wha s he fnal angula elcy f f he ball -unsle syse? 3
0.7 Cnsean f angula enu Slun: Mechancal enegy, lnea enu an angula enu, whch s cnsee? s, f ball, f s, ball, s, s s,f s f s 4[ M M( ] 4 s M 3 sn(80 60 ball, cs60 ball, f ball f f 0.7 Cnsean f angula enu 4 f M 3 s, f f 5 ball, f 4 M f M 3 (4 s, ball, cs60 cs60 0.80a/s 33
0.7 Cnsean f angula enu Exaple 3:The pacle f ass n fgue sles wn he fcnless suface hugh hegh h an clles wh he unf ecal f ass M an lengh, sckng. The ps abu pn O hugh he angle θ befe enaly sppng. Fn θ. 0.7 Cnsean f angula enu Slun: gh ( M M( ( g( csθ Mg( csθ gh 6 h θ cs [ ] M ( M (3 M 3 34
0.7 Cnsean f angula enu Execse 4:ung a jup hs pane, an aeals s ake a quauple sesaul lasng a e.87 s. F he fs an las quae elun, he s n he exene enan shwn n fgue, wh anal nea 9.9 kg aun hs cene f ass. Dung he es f he flgh he s n a gh uck, wh anal nea 3.93kg. Wha us be hs angula spee aun hs cene f ass ung he uck. 0.7 Cnsean f angula enu Slun: Thee s n ne exenal que abu hs cene f ass, hs angula enu abu hs cene f ass s cnsee. Fs an las quae -elun θ The es f e θ 0.5e 3.5e 35
36 (.87s θ θ θ θ θ θ Then we hae 0.4a/s 3.3e/s 0.7 Cnsean f angula enu