Physics 07 Lecue 3 Physics 07, Lecue 3, Oc. 8 Agenda: Chape 9, finish, Chape 0 Sa Chape 9: Moenu and Collision Ipulse Cene of ass Chape 0: oaional Kineaics oaional Enegy Moens of Ineia Paallel axis heoe (Monday) Toque, Wok and oaional Enegy (Monday) Assignen: o Monday ead hough Chape WebAssign Poble Se 5 due Tuesday Physics 07: Lecue 3, Pg Exaple - Elasic Collision See ex: 9. Suppose I have idenical bupe cas. One is oionless and he ohe is appoaching i wih velociy v. If hey collide elasically, wha is he final velociy of each ca? Idenical eans Iniially v Geen v and v ed 0 COM v + 0 v f + v f v v f + v f COE ½ v ½ v f + ½ v f v v f + v f v (v f + v f ) v f +v f v f + v f v f v f 0 Soln : v f 0 and v f v Soln : v f 0 and v f v Physics 07: Lecue 3, Pg Lecue 3, Execise Elasic Collisions I have a line of 3 bupe cas all ouching. A fouh ca sashes ino he ohes fo behind. Is i possible o saisfy boh consevaion of enegy and oenu if wo cas ae oving afe he collision? All asses ae idenical, elasic collision. (A) Yes (B) No (C) Only in one special case Exaple of -D D Elasic collisions: Billiads See ex: Ex. 9. If all we ae given is he iniial velociy of he cue ball, we don have enough infoaion o solve fo he exac pahs afe he collision. Bu we can lean soe useful hings... v Befoe v v Afe? Physics 07: Lecue 3, Pg 3 Physics 07: Lecue 3, Pg Billiads Conside he case whee one ball is iniially a es. befoe p b See ex: Ex. 9. afe p a v c Billiads: All ha eally aes is Consevaion of enegy and oenu COE: ½ v b ½ v a + ½ V a x-di COM: v b v a cos + V b cos φ y-di COM: 0 v a sin + V b sin φ befoe p b See ex: Ex. 9. afe p a See igue - The final diecion of he ed ball will depend on whee he balls hi. P a φ Physics 07: Lecue 3, Pg 5 v c P a φ Acive igue The final diecions ae sepaaed by 90 : φ 90 See igue - Physics 07: Lecue 3, Pg 6 Page
Physics 07 Lecue 3 Lecue 3 Execise Pool Shak See ex: Ex. 9. Can I sink he ed ball wihou scaching (sinking he cue ball)? (Ignoe spin and ficion) (A) Yes (B) No (C) Moe info needed Applicaions of Moenu Consevaion in Populsion adioacive decay: 38 U Alpha Decay v 3 Th He v Guns, Cannons, ec.: (ecoil) Physics 07: Lecue 3, Pg 7 Physics 07: Lecue 3, Pg 8 oce and Ipulse (A vaiable foce applied fo a given ie) Gaviy ofen povides a consan foce o an objec See ex: 9- A sping povides a linea foce (-kx) owads is equilibiu posiion A collision ofen involves a vaying foce (): 0 axiu 0 The diaga shows he foce vs ie fo a ypical collision. The ipulse, I, of he foce is a veco defined as he inegal of he foce duing he collision. p I d ( dp / d) d dp oce and Ipulse Two diffeen collisions can have he sae ipulse since I depends only on he change in oenu, no he naue of he collision. sae aea See ex: 9- Ipulse I aea unde his cuve! (A change in oenu!) Ipulse has unis of Newon-seconds i f big, sall sall, big Physics 07: Lecue 3, Pg 9 Physics 07: Lecue 3, Pg 0 See ex: 9- oce and Ipulse A sof sping (No Hooke s Law) Lecue 3, Execise 3 oce & Ipulse Two boxes, one heavie han he ohe, ae iniially a es on a hoizonal ficionless suface. The sae consan foce acs on each one fo exacly second. Which box has he os oenu afe he foce acs? (A) heavie (B) lighe (C) sae siff sping big, sall sall, big ligh heavy Physics 07: Lecue 3, Pg Physics 07: Lecue 3, Pg Page
Physics 07 Lecue 3 See ex 9- Aveage oce and Ipulse Back of he envelope calculaion (Boxe) A sof sping (No Hooke s Law) () a ~ 7 kg () v a ~7 /s (3) Ipac ie ~ 0.0 s Ipulse I p ~ a v a ~ 9 kg /s ~ I / ~ 900 N () head ~ 6 kg I d avg av siff sping a head / head ~ 800 /s ~ 80 g! Enough o cause unconsciousness ~ 0% of faal blow av big, av sall sall, av big Physics 07: Lecue 3, Pg 3 Physics 07: Lecue 3, Pg Syse of Paicles: Unil now, we have consideed he behavio of vey siple syses (one o wo asses). Bu eal objecs have disibued ass! o exaple, conside a siple oaing disk. Syse of Paicles: Cene of Mass The cene of ass is whee he syse is balanced! Building a obile is an execise in finding cenes of ass. + + An exended solid objec (like a disk) can be hough of as a collecion of pas. The oion of each lile pa depends on whee i is in he objec! obile Acive igue Physics 07: Lecue 3, Pg 5 Physics 07: Lecue 3, Pg 6 See ex: 9-6 Syse of Paicles: Cene of Mass How do we descibe he posiion of a syse ade up of any pas? Define he Cene of Mass (aveage posiion): o a collecion of N individual poinlike paicles whose asses and posiions we know: CM N M i i CM y x (In his case, N ) Physics 07: Lecue 3, Pg 7 Exaple Calculaion: Conside he following ass disibuion: N CM M i i ˆ ˆ ˆ CM i + CM j + CM k X CM ( x 0 + x + x )/ ees Y CM ( x 0 + x + x 0 )/ ees (,) X CM ees Y CM 6 ees X (0,0) Y Z CM (,6) (,0) Physics 07: Lecue 3, Pg 8 Page 3
Physics 07 Lecue 3 See ex: 9-6 Syse of Paicles: Cene of Mass o a coninuous solid, conve sus o an inegal. y x d CM d d d M whee d is an infiniesial ass eleen. Cene of Mass Exaple: Asonaus & ope Two asonaus ae iniially a es in oue space and 0 ees apa. The one on he igh has.5 ies he ass of he ohe (as shown). The.5 asonau wans o ge back o he ship bu his je pack is boken. Thee happens o be a ope conneced beween he wo. The heavie asonau sas pulling in he ope. () Does he/she ge back o he ship? () Does he/she ee he ohe asonau? M.5 Physics 07: Lecue 3, Pg 9 Physics 07: Lecue 3, Pg 0 Exaple: Asonaus & ope () Thee is no exenal foce so if he lage asonau pulls on he ope he will ceae an ipulse ha acceleaes hi/he o he lef and he sall asonau o he igh. The lage one s velociy will be less han he salle one s so he/she doesn le go of he ope hey will eihe collide (elasically o inelasically) and hus neve ake i. M.5 Physics 07: Lecue 3, Pg Lecue 3, Execise Cene of Mass Moion A woan weighs exacly as uch as he 0 foo long boa. Iniially she sands in he cene of he oionless boa, a disance of 0 fee fo shoe. Nex she walks owad he shoe unil she ges o he end of he boa. Wha is he new disance fo he shoe. (Thee is no hoizonal foce on he boa by he wae). befoe afe x 0 f (x-y) f 0 f y y X CM ( x + x) / x 0 f (A) 0 f (B) 5 f (C) 6.7 f X CM ((x-y)+(x+y ))/ y + y? x-y? Physics 07: Lecue 3, Pg See ex: 9.6 Cene of Mass Moion: eview We have he following ule fo Cene of Mass (CM) oion: EXT Ma CM This has seveal ineesing iplicaions: Acive igue I ell us ha he CM of an exended objec behaves like a siple poin ass unde he influence of exenal foces: We can use i o elae and a like we ae used o doing. I ells us ha if EXT 0, he oal oenu of he syse does no change. As he woan oved fowad in he boa, he boa wen backwad o keep he cene of ass a he sae place. Chap. 0: oaion Up unil now oaion has been only in es of cicula oion (a c v / and a T d v / d) We have no exained objecs ha oll. We have assued wheels and pulley ae assless. oaion is coon in he wold aound us. Viually all of he ideas developed fo anslaional oion and ae ansfeable o oaional oion. Physics 07: Lecue 3, Pg 3 Physics 07: Lecue 3, Pg Page
Physics 07 Lecue 3 oaional Vaiables oaional Vaiables... oaion abou a fixed axis: Conside a disk oaing abou an axis hough is cene: is, ecall wha we leaned abou Unifo Cicula Moion: d π (ad/s) d T dx (Analogous o v ) d α consan (angula accelaion in ad/s ) + α 0 (angula velociy in ad/s) 0 + 0 + α (angula posiion in ad) v α And aking he deivaive of his we find ecall also ha fo a poin a disance away fo he axis of oaion: x v a α x Physics 07: Lecue 3, Pg 5 Physics 07: Lecue 3, Pg 6 See ex: 0. Suay (wih copaison o -D D kineaics) Angula α consan 0 + α 0 + 0 + α Linea a consan v v 0 + a x x0 + v0 + a Exaple: Wheel And ope A wheel wih adius 0. oaes feely abou a fixed axle. Thee is a ope wound aound he wheel. Saing fo es a 0, he ope is pulled such ha i has a consan acceleaion a /s. How any evoluions has he wheel ade afe 0 seconds? (One evoluion π adians) a And fo a poin a a disance fo he oaion axis: x v a α Physics 07: Lecue 3, Pg 7 Physics 07: Lecue 3, Pg 8 Exaple: Wheel And ope A wheel wih adius 0. oaes feely abou a fixed axle. Thee is a ope wound aound he wheel. Saing fo es a 0, he ope is pulled such ha i has a consan acceleaion a /s. How any evoluions has he wheel ade afe 0 seconds? (One evoluion π adians) evoluions ( 0 ) / π and a α 0 + 0 + ½ α ( 0 ) / π 0 + ½ (a/) / π (0.5 x 0 x 00) / 6.8 a oaion & Kineic Enegy Conside he siple oaing syse shown below. (Assue he asses ae aached o he oaion axis by assless igid ods). The kineic enegy of his syse will be he su of he kineic enegy of each piece: K i vi K ½ v + ½ v + ½ 3 v 3 + ½ v 3 3 Physics 07: Lecue 3, Pg 9 Physics 07: Lecue 3, Pg 30 Page 5
Physics 07 Lecue 3 oaion & Kineic Enegy Noice ha v, v, v 3 3, v So we can ewie he suaion: K ] ivi i i [ ii We define a new quaniy, he oen of ineia o I (we use I again.) Lecue, Execise oaional Kineic Enegy We have wo balls of he sae ass. Ball is aached o a 0. long ope. I spins aound a evoluions pe second. Ball is on a 0. long ope. I spins aound a evoluions pe second. Wha is he aio of he kineic enegy of Ball o ha of Ball? (A) / (B) / (C) (D) (E) K I 3 3 Ball Ball Physics 07: Lecue 3, Pg 3 Physics 07: Lecue 3, Pg 3 oaion & Kineic Enegy... The kineic enegy of a oaing syse looks siila o ha of a poin paicle: Poin Paicle K v v is linea velociy is he ass. oaing Syse K I I i i i is angula velociy I is he oen of ineia abou he oaion axis. So K I Moen of Ineia whee I i i Noice ha he oen of ineia I depends on he disibuion of ass in he syse. The fuhe he ass is fo he oaion axis, he bigge he oen of ineia. o a given objec, he oen of ineia depends on whee we choose he oaion axis (unlike he cene of ass). In oaional dynaics, he oen of ineia I appeas in he sae way ha ass does in linea dynaics! i Physics 07: Lecue 3, Pg 33 Physics 07: Lecue 3, Pg 3 Physics 07, Lecue 3, ecap Agenda: Chape 9, finish, Chape 0 Sa Chape 9: Moenu and Collision Ipulse Cene of ass Chape 0: oaional Kineaics oaional Enegy Moens of Ineia Assignen: o Monday ead hough Chape WebAssign Poble Se 5 due Tuesday Physics 07: Lecue 3, Pg 35 Page 6