SUMMARY GENERAL STRATEGY IMPORTANT CONCEPTS APPLICATIONS. Problem Solving. Motion Diagrams. Pictorial Representation

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1 The goal of Chape 1 has been o inoduce he fundamenal conceps of moion. GENERL STRTEGY Moion Diagams Help visualize moion. Povide a ool fo finding acceleaion vecos. Dos show posiions a equal ime inevals. Velociy vecos go do o do. v 1 v 0 a The acceleaion veco v poins in he diecion of v. These ae he aveage velociy and he aveage acceleaion vecos. v 1 2v 0 Poblem Solving MODEL Make simplifying assumpions. VISULIZE Use: Picoial epesenaion Physical epesenaion Gaphical epesenaion SOLVE Use a mahemaical epesenaion o find numeical answes. SSESS Does he answe have he pope unis? Does i make sense? The paicle model epesens a moving objec as if all is mass wee concenaed a a single poin. Picoial Repesenaion 1 Skech he siuaion. Posiion locaes an objec wih espec o a chosen coodinae sysem. Change in posiion is called displacemen. Velociy is he ae of change of he posiion veco. 2 Esablish coodinaes. 3 Define symbols. 0 a 0, v 0, 0 1, v 1, 1 cceleaion is he ae of change of he velociy veco. n objec has an acceleaion if i Changes speed and/o Changes diecion. v 4 Lis knowns. 5 Idenify desied unknown. Known 0 5 v a 5 2 m/s s Find 1 Fo moion along a line: Speeding up: and a poin in he same diecion. Slowing down: and poin in opposie diecions. Consan speed: a 5 0. v v a Significan figues ae eliably known digis. Thee significan figues is he sandad fo his book. The numbe of significan figues fo: Muliplicaion, division, powes is se by he value wih he fewes significan figues. ddiion, subacion is se by he value wih he smalles numbe of decimal places. Copyigh 2004 Peason Educaion, Inc., publishing as ddison Wesley

2 The goal of Chape 2 has been o lean how o solve poblems abou moion in a saigh line. Kinemaics descibes moion in ems of posiion, velociy, and acceleaion. Geneal kinemaic elaionships ae given mahemaically by: Insananeous velociy v s 5 ds/d 5 slope of posiion gaph Insananeous acceleaion a s 5 dv s /d 5 slope of velociy gaph Final posiion s f 5 s i 1 3 f i aea unde he velociy cuve v s d 5 s i 1 b fom i o f The kinemaic equaions fo moion wih consan acceleaion: v fs 5 v is 1 a s D s f 5 s i 1 v is D 1 1 2a s ( D ) 2 v fs 2 5 v is 2 1 2a s Ds Final velociy v fs 5 v is 1 3 f i aea unde he acceleaion a s d 5 v is 1 b cuve fom i o f Posiion, velociy, and acceleaion ae elaed gaphically. s Moion wih consan acceleaion is unifomly acceleaed moion. The slope of he posiion-vesus-ime gaph is he value on he velociy gaph. Unifom moion is moion wih consan velociy and zeo acceleaion. The slope of he velociy gaph is he value on he acceleaion gaph. s is a maimum o minimum a a uning poin, and v s 5 0. v s Tuning poin s f 5 s i 1 v s D a s The sign of v s indicaes he diecion of moion. n objec is speeding up if and only if v s and a s have he same sign. n objec is slowing down if and only if v v s. 0 is moion o he igh o up. s and a s have opposie signs. v s, 0 is moion o he lef o down. a s The sign of indicaes which way poins, no whehe he objec is speeding up o slowing down. a if a s. 0 poins o he igh o up. a if a s, 0 poins o he lef o down. The diecion of a is found wih a moion diagam. a Fee fall is consan-acceleaion moion wih a y 52g m/s 2. Moion on an inclined plane has a s 56g sin u. The sign depends on he diecion of he il. Copyigh 2004 Peason Educaion, Inc., publishing as ddison Wesley

3 The goal of Chape 3 has been o lean how vecos ae epesened and used. veco is a quaniy descibed by boh a magniude and a diecion. The veco descibes he siuaion a his poin. Diecion The lengh o magniude is denoed. Magniude is a scala. Uni Vecos Uni vecos have magniude 1 and no unis. Uni vecos i^ and j^ define he diecions of he - and y-aes. j^ y i^ USING VECTORS Componens The componen vecos ae paallel o he - and y-aes. In he figue a he igh, fo eample: 5 cos u y 5 sin u 5 1 y 5 i^ 1 y j^ 5 " 2 1 y 2 u5an 21 ( y / ) Minus signs need o be included if he veco poins down o lef. y, 0 y. 0, 0 y, 0 u 5 i^ y. 0 y y, 0 y 5 y j^ The componens and y ae he magniudes of he componen vecos and y and a plus o minus sign o show whehe he componen veco poins owad he posiive end o he negaive end of he ais. Woking Gaphically ddiion 1 B B 1 B Negaive Subacion Muliplicaion B B 2B B 2B c 2 B Woking lgebaically Veco calculaions ae done componen by componen. C B means b C B C y 5 2 y 1 B y 21 The magniude of is hen C 5 "C 2 1 C 2 y and is diecion is found using an. C Copyigh 2004 Peason Educaion, Inc., publishing as ddison Wesley

4 The goal of Chape 4 has been o lean how foce and moion ae conneced. Newon s Fis Law n objec a es will emain a es, o an objec ha is moving will coninue o move in a saigh line wih consan velociy, if and only if he ne foce on he objec is zeo. F ne 5 0 Newon s laws ae valid only in ineial efeence fames. Newon s Second Law n objec wih mass m will undego acceleaion a 5 1 m F ne whee F ne 5 F 1 1 F 2 1 F 3 1 c is he veco sum of all he individual foces acing on he objec. F v v v v v a 5 0 v v v a v v The fis law ells us ha no cause is needed fo moion. Unifom moion is he naual sae of an objec. The second law ells us ha a ne foce causes an objec o acceleae. This is he connecion cceleaion is he link o kinemaics. Fom a, find v and. Fom v and, find a. a 5 0 is he condiion fo equilibium. Saic equilibium if v 5 0. Dynamic equilibium if v 5 consan. Equilibium occus if and only if F ne 5 0. Mass is he esisance of an objec o acceleaion. I is an ininsic popey of an objec. Foce is a push o a pull on an objec. Foce is a veco, wih a magniude and a diecion. Foce equies an agen. Foce is eihe a conac foce o a long-ange foce. KEY SKILLS Idenifying Foces Foces ae idenified by locaing he poins whee he envionmen ouches he sysem. These ae poins whee conac foces ae eeed. In addiion, objecs wih mass feel a long-ange weigh foce. Thus foce F hus Weigh w Nomal foce n Fee-Body Diagams fee-body diagam epesens he objec as a paicle a he oigin of a coodinae sysem. Foce vecos ae dawn wih hei ails on he paicle. The ne foce veco is dawn beside he diagam. y n w F hus F ne Copyigh 2004 Peason Educaion, Inc., publishing as ddison Wesley

5 The goal of Chape 5 has been o lean how o solve poblems abou moion in a saigh line. GENERL STRTEGY ll eamples in his chape follow a fou-pa saegy. You ll become a bee poblem solve if you adhee o i as you do he homewok poblems. The Dynamics Wokshees will help you sucue you wok in his way. Equilibium Poblems Objec a es o moving wih consan velociy. MODEL Make simplifying assumpions. VISULIZE Physical epesenaion: Foces and fee-body diagam Picoial epesenaion: Tanslae wods o symbols. SOLVE Use Newon s fis law F ne 5 ai F i 5 0 Read he vecos fom he fee-body diagam. SSESS Is he esul easonable? Go back and foh beween epesenaions as needed. Dynamics Poblems Objec acceleaing. MODEL Make simplifying assumpions. VISULIZE Picoial epesenaion: Skech o define siuaion. Tanslae wods o symbols. Physical epesenaion: Foces and fee-body diagam SOLVE Use Newon s second law F ne 5 ai F i 5 ma Read he vecos fom he fee-body diagam. Use kinemaics o find velociies and posiions. SSESS Is he esul easonable? Specific infomaion abou hee impoan foces: Weigh w 5 (mg, downwads) Ficion f s 5 (0 o m s n, diecion as necessay o peven moion) f k 5 (m k n, diecion opposie he moion) f 5 (m n, diecion opposie he moion) Newon s laws ae veco epessions. You mus wie hem ou by componens: (F ne ) 5 a F 5 ma o 0 (F ne ) y 5 a F y 5 ma y o 0 Dag D < ( 1 4 v2, diecion opposie he moion) ppaen weigh is he magniude of he conac foce suppoing an objec. I is wha a scale would ead, and i is you sensaion of weigh. I equals you ue weigh w 5 mg only when a 5 0. w app 5 w1 1 1 a y g 2 Teminal speed is v em < Å 4mg Copyigh 2004 Peason Educaion, Inc., publishing as ddison Wesley

6 The goal of Chape 6 has been o lean o solve poblems abou moion in a plane. Galilean Pinciple of Relaiviy Newon s laws of moion ae valid in all ineial efeence fames. Newon s Second Law Epessed in - and y-componen fom: ( F ne ) 5 a F 5 ma ( F ne ) y 5 a F y 5 ma y Relaive moion Ineial efeence fames move elaive o each ohe wih consan velociy V. Measuemens of posiion and velociy measued in fame S ae elaed o measuemens in fame S by he Galilean ansfomaions 5 2 V v 5 v 2 V y 5 y 2 V y v y 5 v y 2 V y y S y9 S9 V 9 The insananeous velociy v 5 d /d, is a veco angen o he ajecoy. The insananeous acceleaion is a 5 dv /d a i, he componen of paallel o v, is esponsible fo change of speed. a, he componen of a ' pependicula o v, is esponsible fo change of diecion. a y a i a a ' v Kinemaics in wo dimensions a If is consan, hen he - and y-componens of moion ae independen of each ohe. Fo a paicle ha sas fom iniial posiion and velociy v i i, is posiion and velociy a a final poin f ae f 5 i 1 v i D a (D) 2 y f 5 y i 1 v iy D a y(d) 2 v f 5 v i 1 a D v fy 5 v iy 1 a y D Pojecile moion occus if he only foce on he objec is is weigh. Unifom moion in he y hoizonal diecion wih v 0 5 v 0 cos u. Fee-fall moion in he veical diecion wih a y 52gand v 0y 5 v 0 sin u. The combined moion is a paabola. The and y kinemaic equaions have he same value fo D. v 0 u w Copyigh 2004 Peason Educaion, Inc., publishing as ddison Wesley

7 The goal of Chape 7 has been o lean o solve poblems abou moion in a cicle. Newon s Second Law Epessed in z-componen fom: (F ne ) 5 a F 5 ma 5 mv2 5 mv 2 0 unifom moion (F ne ) 5 a F 5 b ma nonunifom moion (F ne ) z 5 a F z 5 0 Unifom Cicula Moion v is consan. F ne poins owad he cene of he cicle. F ne The cenipeal acceleaion a poins owad he cene of he cicle. I changes he paicle s diecion bu no is speed. v a Nonunifom Cicula Moion v changes. a is paallel o F ne. The adial componen he paicle s diecion. a The angenial componen changes a F ne v a a a changes he paicle s speed. z-coodinaes z ngula posiion u5s/ ngula velociy v5du/d v 5v v O s Cicula moion kinemaics Peiod T 5 2p 5 2p v v Unifom cicula moion Obis cicula obi has adius if v 5!g w w w v 5 consan v5consan u f 5u i 1vD Nonunifom cicula moion u f 5u i 1v i D 1 a 2 (D)2 v f 5v i 1 a D ppaen weigh Cicula moion equies a ne foce poining o he cene. The appaen weigh w app 5 n is usually no he same as he ue weigh w. n mus be. 0 fo he objec o be in conac wih a suface. n w F ne Copyigh 2004 Peason Educaion, Inc., publishing as ddison Wesley

8 The goal of Chape 8 has been o lean o use Newon s hid law o undesand ineacing sysems. Newon s Thid Law Evey foce occus as one membe of an acion/eacion pai of foces. The wo membes of an acion/eacion pai: c on wo diffeen objecs. e equal in magniude bu opposie in diecion: cion/ eacion F on B 52F B on F B on B F on B Solving Ineacing-Sysem Poblems MODEL Choose he sysems of inees. VISULIZE Picoial epesenaion: Skech and define coodinaes. Idenify acceleaion consans. Physical epesenaion: Daw a sepaae fee-body diagam fo each sysem. Connec acion/eacion pais wih doed lines. SOLVE Wie Newon s second law fo each sysem. Include all foces acing on each sysem. Use Newon s hid law o equae he magniudes of acion/eacion pais. Include acceleaion consains and ficion. SSESS Is he esul easonable? Ineacing sysems and he envionmen Two sysems ineac by eeing foces on each ohe. Sysems whose moion is no of inees fom he envionmen. The sysems of inees ineac wih he envionmen, bu hose ineacions can be consideed eenal foces. Ineacions Eenal foces Envionmen B cceleaion consains Sings and pulleys Objecs ha ae consained o move ogehe mus have acceleaions of equal magniude: a 5 a B. This mus be epessed in ems of componens, such as a 52a By. a B a B The ension in a sing o ope pulls in boh diecions. The ension is consan in a sing if he sing is: Massless, o In equilibium Sysems conneced by massless sings passing ove massless, ficionless pulleys ac as if hey ineac via an acion/eacion pai of foces. T S on T on S B T B on S T B on as if B T S on B T on B Copyigh 2004 Peason Educaion, Inc., publishing as ddison Wesley

9 The goal of Chape 9 has been o inoduce he ideas of impulse, momenum, and angula momenum and o lean a new poblem-solving saegy based on consevaion laws. Law of Consevaion of Momenum The oal momenum P 5 p 1 1 p 2 1 c of an isolaed sysem is a consan. Thus P f 5 P i Law of Consevaion of ngula Momenum The angula momenum L of a paicle o sysem of paicles in cicula moion does no change unless hee is a ne angenial foce. Thus L f 5 L i Solving Momenum Consevaion Poblems MODEL Choose an isolaed sysem o a sysem ha is isolaed duing a leas pa of he poblem. VISULIZE Daw a picoial epesenaion of he sysem befoe and afe he ineacion. SOLVE Wie he law of consevaion of momenum in ems of veco componens (p f ) 1 1 (p f ) 2 1 c 5 (p i ) 1 1 (p i ) 2 1 c (p fy ) 1 1 (p fy ) 2 1 c 5 (p iy ) 1 1 (p iy ) 2 1 c SSESS Is he esul easonable? Momenum p 5 mv v Sysem goup of ineacing paicles. Impulse f m p Isolaed sysem sysem on which hee ae no J 5 3 F () d 5 aea unde foce cuve eenal foces o he ne eenal foce is zeo. i Impulse and momenum ae elaed by he impulsemomenum heoem F Befoe-and-afe picoial epesenaion Dp 5 J Define he sysem. (v i ) 1 (v i ) 2 J Befoe: m This is an alenaive m 2 Use wo dawings o show he sysem saemen of Newon s befoe and afe he ineacion. (v f ) 1 (v f ) 2 fe: 1 2 second law. i f Lis known infomaion and idenify ngula momenum L 5 mv wha you ae ying o find. Collisions Two o moe paicles come ogehe. In a pefecly inelasic collision, hey sick ogehe and move wih a common final velociy. Eplosions Two o moe paicles move away fom each ohe. Two dimensions No new ideas, bu boh he - and y-componens of P mus be conseved, giving wo simulaneous equaions (v i ) 1 (v i ) (v f ) 1 (v f ) 2 v i1 v i v f1 v f2 Momenum ba chas display he impulsemomenum heoem p f 5 p i 1 J in gaphical fom p i 1 J 5 p f Copyigh 2004 Peason Educaion, Inc., publishing as ddison Wesley

10 The goal of Chape 10 has been o inoduce he ideas of kineic and poenial enegy and o lean a new poblem-solving saegy based on consevaion of enegy. Law of Consevaion of Mechanical Enegy If hee ae no ficion o ohe enegy-loss pocesses (o be eploed moe hooughly in Chape 11), hen he mechanical enegy E mech 5 K 1 U of a sysem is conseved. Thus K f 1 U f 5 K i 1 U i K is he sum of he kineic enegies of all paicles. U is he sum of all poenial enegies. Solving Enegy Consevaion Poblems MODEL Choose a sysem wihou ficion o ohe losses of mechanical enegy. VISULIZE Daw a befoe-and-afe picoial epesenaion. SOLVE Use he law of consevaion of enegy K f 1 U f 5 K i 1 U i SSESS Is he esul easonable? Kineic enegy is an enegy of moion K mv2 Poenial enegy is an enegy of posiion Gaviaional: U g 5 mgy Basic Enegy Model Enegy can be ansfomed wihin he sysem wihou loss. Enegy ino sysem Readily available enegy K Soed enegy U Mechanical enegy E mech 5 K 1 U Elasic: U s k (Ds)2 Enegy ou of sysem Enegy diagams These diagams show he poenial enegy cuve PE and he oal mechanical enegy line TE. Enegy U K PE TE The disance fom he ais o he cuve is PE. The disance fom he cuve o he TE line is KE. poin whee he TE line cosses he PE cuve is a uning poin. Minima in he PE cuve ae poins of sable equilibium. Maima ae poins of unsable equilibium. Hooke s law The esoing foce of an ideal sping is (F sp ) s 52kDs whee k is he sping consan and Ds 5 s 2 s e is he displacemen fom equilibium. F sp Ds Pefecly elasic collisions Boh mechanical enegy and momenum ae conseved. (v i ) es (v f ) 1 5 m 1 2 m 2 m 1 1 m 2 (v i ) 1 (v f ) 2 5 2m 1 m 1 1 m 2 (v i ) 1 If ball 2 is moving, ansfom o a efeence fame in which ball 2 is a es. Copyigh 2004 Peason Educaion, Inc., publishing as ddison Wesley

11 The goal of Chape 11 has been o develop a moe complee undesanding of enegy and is consevaion. Basic Enegy Model Enegy is ansfeed o o fom he sysem by wok. Enegy is ansfomed wihin he sysem. Two vesions of he enegy equaion ae DE sys 5DK 1DU 1DE h 5 W e K f 1 U f 1DE h 5 K i 1 U i 1 W e Enegy in Wok W. 0 Envionmen Sysem K U E h E sys 5 K 1 U 1 E h Enegy ou Wok W, 0 Solving Enegy Poblems MODEL Idenify objecs in he sysem. VISULIZE Daw a befoe-and-afe picoial epesenaion and an enegy ba cha. SOLVE Use he enegy equaion K f 1 U f 1DE h 5 K i 1 U i 1 W e SSESS Is he esul easonable? Law of Consevaion of Enegy Isolaed sysem: W e 5 0. The oal enegy E sys 5 E mech 1 E h is conseved. DE sys 5 0 Isolaed, nondissipaive sysem: W e 5 0 and W diss 5 0. The mechanical enegy is conseved. DE mech 5 0 o K f 1 U f 5 K i 1 U i E mech The wok-kineic enegy heoem is DK 5 W ne 5 W c 1 W diss 1 W e Using W c 52DUfo consevaive foces and W diss 52DE h fo dissipaive foces, his becomes he enegy equaion. The wok done by a foce on a paicle as i moves fom o is s i s f s f W 5 3 F s ds 5 aea unde he foce cuve s i 5 F? D if F is a consan foce Consevaive foces ae foces fo which he wok is independen of he pah followed. The wok done by a consevaive foce can be epesened as a poenial enegy DU 5 U f 2 U i 52W c (i S f) consevaive foce is found fom he poenial enegy by F 52dU/ds 5 negaive of he slope of he PE cuve Dissipaive foces ansfom macoscopic enegy ino hemal enegy, which is he micoscopic enegy of he aoms and molecules. DE h 52W diss Powe is he ae a which enegy is ansfeed o ansfomed: P 5 de sys d Fo a paicle moving wih velociy v, he powe deliveed o he paicle by foce F is P 5 F # v 5 Fv cos u. Enegy ba chas display he enegy equaion in gaphical fom. K f 1 U f 1DE h 5 K i 1 U i 1 W e Do poduc # B 5 B cos a 5 B 1 y B y a B 2 K i U i 1 1W e 5 K f 1 U f 1DE h Copyigh 2004 Peason Educaion, Inc., publishing as ddison Wesley

12 The goal of Chape 12 has been o use Newon s heoy of gaviy o undesand he moion of saellies and planes. Newon s Theoy of Gaviy 1. Two objecs wih masses M and m a disance apa ee aacive gaviaional foces on each ohe of magniude F M on m 5 F m on M 5 GMm 2 whee he gaviaional consan is G N m 2 /kg Gaviaional mass and ineial mass ae equivalen. m F M on m F m on M M 3. Newon s hee laws of moion apply o saellies, planes, and sas. Obial moion of a plane (o saellie) is descibed by Keple s laws: 1. Obis ae ellipses wih he sun (o plane) a one focus. 2. line beween he sun and he plane sweeps ou equal aeas duing equal inevals of ime. 3. The squae of he plane s peiod T is popoional o he cube of he Semimajo ais obi s semimajo ais. Cicula obis ae a special case of an ellipse. Fo a cicula obi aound a mass M, v 5 Å GM Swep-ou aea M and T p2 GM 2 3 m b v Consevaion of angula momenum The angula momenum L 5 mv sin b emains consan houghou he obi. Keple s second law is a consequence of his law. Obial enegeics saellie s mechanical enegy E mech 5 K 1 U g is conseved, whee he gaviaional poenial enegy is U g 52 GMm Fo cicula obis, K U and E mech 5 1 g 2U g. Negaive oal enegy is chaaceisic of a bound sysem. Fo a plane of mass M and adius R, The acceleaion due o gaviy on he suface is g suface 5 The escape speed is v 2GM escape 5 Å R GM The adius of a geosynchonous obi is geo 5 4p T 2 1/3 GM R 2 v M R geo Copyigh 2004 Peason Educaion, Inc., publishing as ddison Wesley

13 The goal of Chape 13 has been o undesand he physics of oaing objecs. Roaional Dynamics Evey poin on a igid body oaing abou a fied ais has he same angula velociy v and angula acceleaion a. Newon s second law fo oaional moion is a5 ne I Use oaional kinemaics o find angles and angula velociies. Consevaion Laws Enegy is conseved fo an isolaed sysem. Pue oaion E 5 K o 1 U g Iv 2 1 Mgy cm Rolling E 5 K Iv Mv 2 o 1 K cm 1 U g cm 1 Mgy cm ngula momenum is conseved if ne 5 0. Paicle L 5 m 3 p Rigid body oaing abou ais of symmey L 5 Iv ngula velociy v5 du d ngula acceleaion is he oaional equivalen of acceleaion a5 dv d Toque is he oaional equivalen of foce 5F sin f5f 5 df y Pivo d F Veco descipion of oaion Toque 5 3 F v ngula velociy poins along he oaion ais in he diecion of he igh-hand ule. Fo a igid body oaing abou an ais of symmey, he angula momenum is L 5 Iv. Newon s second law is dl d 5 ne F L is sysem of paicles on which hee is no ne foce undegoes unconsained oaion abou he cene of mass cm 5 1 M 3 dm y cm 5 1 M 3 y dm The gaviaional oque on a body can be found by eaing he body as a paicle wih all he mass M concenaed a he cene of mass. The momen of ineia I dm is he oaional equivalen of mass. The momen of ineia depends on how he mass is disibued aound he ais. If is known, he I abou a paallel ais disance d away is given by he paallel-ais heoem I 5 I cm 1 Md 2. I cm Roaional kinemaics v f 5v i 1aD u f 5u i 1v i D a ( D ) 2 v 2 f 5v 2 i 1 2a Du v 5 v a 5 a Rigid-body equilibium n objec is in oal equilibium only if boh F and ne 5 0 ne 5 0. No oaional o anslaional moion Rolling moion Fo an objec ha olls wihou slipping v cm 5 Rv K 5 K o 1 K cm R v Copyigh 2004 Peason Educaion, Inc., publishing as ddison Wesley

14 The goal of Chape 14 has been o undesand sysems ha oscillae wih simple hamonic moion. Dynamics SHM occus when a linea esoing foce acs o eun a sysem o an equilibium posiion. k Hoizonal sping m (F ne ) 52k Veical sping The oigin is a he equilibium posiion DL 5 mg/k. (F ne ) y 52ky v5 Å k m Pendulum (F ne ) 521 mg L 2 s v5 Å g L T 5 2p Å m k T 5 2p Å L g 0 y 0 m 0 k s L Enegy If hee is no ficion o dissipaion, kineic and poenial enegy ae alenaely ansfomed ino each ohe, bu he oal mechanical enegy E 5 K 1 U is conseved. E mv k m(v 2 ma) k2 In a damped sysem, he enegy decays eponenially E 5 E 0 e 2/ whee is he ime consan. 2 E E E ll kineic 0 ll poenial Simple hamonic moion (SHM) is a sinusoidal oscillaion wih peiod T and ampliude. Fequency f 5 1 T T ngula fequency v52pf 5 2p 0 T Posiion () 5 cos (v 1f 0 ) 2 5 cos 1 2p T 02 1f Velociy v () 52v ma sin (v 1f 0 ) wih maimum speed v ma 5v SHM is he pojecion ono he -ais of unifom cicula moion. f5v 1f 0 is he phase The posiion a ime is () 5 cos f 5 cos (v 1f 0 ) The phase consan f 0 deemines he iniial condiions: 0 5 cos f 0 v 0 52v sin f 0 y f f cos f 0 5 cos f 0 cceleaion a 52v 2 Resonance mpliude When a sysem is diven by a peiodic eenal foce, i esponds wih a lage-ampliude oscillaion if f e < f 0 whee f 0 is he sysem s naual oscillaion fequency, o esonan fequency. f 0 f e Damping If hee is a dag foce D 52bv, whee b is he damping consan, hen (fo lighly damped sysems) () 5 e 2b/2m cos(v 1f 0 ) The ime consan fo enegy loss is 5m/b. 0 2 Copyigh 2004 Peason Educaion, Inc., publishing as ddison Wesley

15 The goal of Chape 15 has been o undesand macoscopic sysems ha flow o defom. Fluid Saics Gases Liquids Fluid Dynamics Ideal-fluid model Feely moving paicles Loosely bound paicles Incompessible Compessible Incompessible Smooh, lamina flow Pessue pimaily hemal Pessue consan in a laboaoy-size conaine Pessue pimaily gaviaional Hydosaic pessue a deph d is p 5 p 0 1gd Nonviscous Ioaional Densiy p 2 v 2 y 2 Densiy 5m/V, whee m is mass and V is volume. Pessue p 5 F/, whee F is he magniude of he fluid foce and is he aea on which he foce acs. Eiss a all poins in a fluid Pushes equally in all diecions Consan along a hoizonal line Gauge pessue p g 5 p 2 1 am Equaion of coninuiy v v 2 2 Benoulli s equaion p 1 v 1 y 1 Fluid paicles move along seamlines. p v 1 2 1gy 1 5 p v 2 2 1gy 2 1 Benoulli s equaion is a saemen of enegy consevaion. 2 Buoyancy is he upwad foce of a fluid on an objec. chimedes pinciple The magniude of he buoyan foce equals he weigh of he fluid displaced by he objec. Sink avg. f F B, w o Rise o suface avg, f F B. w o Neually buoyan avg 5 f F B 5 w o f F B w o Elasiciy descibes he defomaion of solids and liquids unde sess. Linea sech and compession: (F/) 5 Y (DL/L) Sain Tensile sess Young s modulus Volume compession: p 5 2B (DV/V ) Bulk modulus Volume sain L DL F Copyigh 2004 Peason Educaion, Inc., publishing as ddison Wesley

16 The goal of Chape 16 has been o lean he chaaceisics of macoscopic sysems. Thee Phases of Mae Solid Liquid Gas Rigid, definie shape. Nealy incompessible. Molecules loosely held ogehe by molecula bonds, bu able o move aound. Nealy incompessible. Molecules move feely hough space. Compessible. The diffeen phases eis fo diffeen condiions of empeaue T and pessue p. The boundaies sepaaing he egions of a phase diagam ae lines of phase equilibium. ny amouns of he wo phases can coeis in equilibium. The iple poin is he one value of empeaue and pessue a which all hee phases can coeis in equilibium. p SOLID Meling/ feezing poin LIQUID Boiling/ condensaion poin Tiple poin GS T Ideal-Gas Model oms and molecules ae small, had sphees ha avel feely hough space ecep fo occasional collisions wih each ohe o he walls. The molecules have a disibuion of speeds. The model is valid when he densiy is low and he empeaue well above he condensaion poin. Ideal-Gas Law The sae vaiables of an ideal gas ae elaed by he ideal-gas law pv 5 nrt o pv 5 Nk B T whee R J/mol K is he univesal gas consan and k B J/K is Bolzmann s consan. p, V, and T mus be in SI unis of Pa, m 3, and K. Fo a gas in a sealed conaine, wih consan n: Couning aoms and moles macoscopic sample of mae consiss of N aoms (o molecules), each of mass m (he aomic o molecula mass): N 5 M m lenaively, we can sae ha he sample consiss of n moles M(in gams) n 5 N o N M mol N mol 21 is vogado s numbe. The numeical value of he mola mass M mol, in g/mol, equals he numeical value of he aomic o molecula mass m in u. The aomic o molecula mass m, in aomic mass unis u, is well appoimaed by he aomic mass numbe. 1 u kg The numbe densiy of he sample is N V. Volume V Mass M p 2 V 2 T 2 5 p 1V 1 T 1 Tempeaue scales T F T C 1 32 T K 5 T C The Kelvin empeaue scale is based on: bsolue zeo a T K The iple poin of wae a T K Thee basic gas pocesses 1. Isochoic, o consan volume 2. Isobaic, o consan pessue 3. Isohemal, o consan empeaue pv diagam p V Copyigh 2004 Peason Educaion, Inc., publishing as ddison Wesley

17 The goal of Chape 17 has been o epand ou undesanding of enegy and o develop he fis law of hemodynamics as a geneal saemen of enegy consevaion. Fis Law of Themodynamics DE h 5 W 1 Q The fis law is a geneal saemen of enegy consevaion. Wok on W. 0 Sysem Wok by W, 0 E h Wok W and hea Q depend Q. 0 Q, 0 on he pocess by which he Hea in Hea ou sysem is changed. The change in he sysem depends only on he oal enegy echanged W 1 Q, no on he pocess. Enegy Themal enegy E h Micoscopic enegy of moving molecules and seched molecula bonds. DE h depends on he iniial/final saes bu is independen of he pocess. Wok W Enegy ansfeed o he sysem by foces in a mechanical ineacion. Hea Q Enegy ansfeed o he sysem via aomiclevel collisions when hee is a empeaue diffeence. hemal ineacion. The wok done on a gas is V f W 52 3 pdv V i (aea unde he pv cuve) n adiabaic pocess is one fo which Q 5 0. Gases move along an adiaba fo which pv g 5 consan, whee g5c P /C V is he specific hea aio. n adiabaic pocess changes he empeaue of he gas wihou heaing i. Caloimey When wo o moe sysems ineac hemally, hey come o a common final empeaue deemined by Q ne 5 Q 1 1 Q 2 1 c 5 0 p p i diaba f Isohems V V The hea of ansfomaion L is he enegy needed o cause 1 kg of subsance o undego a phase change Q 56ML The specific hea c of a subsance is he enegy needed o aise he empeaue of 1 kg by 1 K. Q 5 McDT The mola specific hea C is he enegy needed o aise he empeaue of 1 mol by 1 K. Q 5 ncdt The mola specific hea of gases depends on he pocess by which he empeaue is changed: C V 5 mola specific hea a consan volume. C P 5 mola specific hea a consan pessue. C P 5 C V 1 R, whee R is he univesal gas consan. SUMMRY OF BSIC GS PROCESSES Pocess Definiion Says consan Wok Hea Isochoic Isobaic Isohemal diabaic DV 5 0 Dp 5 0 DT 5 0 Q 5 0 V and p/t p and V/T T and pv pv g W 5 0 W 52pDV W 52nRT ln (V f /V i ) W 5DE h Q 5 nc V DT Q 5 nc P DT DE h 5 0 Q 5 0 ll gas pocesses Ideal-gas law Fis law pv 5 nrt DE h 5 W 1 Q 5 nc V DT Copyigh 2004 Peason Educaion, Inc., publishing as ddison Wesley

18 The goal of Chape 18 has been o undesand he popeies of a macoscopic sysem in ems of he micoscopic behavio of is molecules. Kineic heoy, he mico/maco connecion, elaes he macoscopic popeies of a sysem o he moion and collisions of is aoms and molecules. The Equipaiion Theoem Tells us how collisions disibue he enegy in he sysem. The enegy soed in each mode of he sysem (each degee 1 of feedom) is 2 Nk BT o, in ems of moles, 1 2 nrt. The Second Law of Themodynamics Tells us how collisions move a sysem owad equilibium. The enopy of an isolaed sysem can only incease o, in equilibium, say he same. Ode uns ino disode and andomness. Sysems un down. Hea enegy is ansfeed sponaneously fom he hoe o he colde sysem, neve fom colde o hoe. Pessue is due o he foce of he molecules colliding wih he walls. p 5 1 N 3 V mv ms N 3 V P avg The aveage anslaional kineic enegy of a molecule is P avg 5 3 2k B T. The empeaue of he gas T 5 2 3k B P avg measues he aveage anslaional kineic enegy. Enopy measues he pobabiliy ha a macoscopic sae will occu o, equivalenly, he amoun of disode in a sysem. Inceasing enopy The hemal enegy of a sysem is E h 5 anslaional kineic enegy 1 oaional kineic enegy 1 vibaional enegy Monaomic gas E h Nk BT nrt Diaomic gas E h Nk BT nrt Elemenal solid E h 5 3Nk B T 5 3nRT Hea is enegy ansfeed via collisions fom moe-enegeic molecules on one side o lessenegeic molecules on he ohe. Equilibium is eached when (P 1 ) avg 5 (P 2 ) avg, which implies T 1f 5 T 2f. Q The oo-mean-squae speed v ms is he squae oo of he aveage of he squaes of he molecula speeds: v ms 5 "(v 2 ) avg Fo molecules of mass m a empeaue T, v ms 5 Å 3k B T m Mola specific heas can be pediced fom he hemal enegy because DE h 5 ncdt. Monaomic gas C V R Diaomic gas C V R Elemenal solid C 5 3R Copyigh 2004 Peason Educaion, Inc., publishing as ddison Wesley

19 The goal of Chape 19 has been o invesigae he physical pinciples ha goven he opeaion of hea engines and efigeaos. Hea Engines Devices which ansfom hea ino wok. They equie wo enegy esevois a diffeen empeaues. Cyclical pocess (DE h ) ne 5 0 T H T C Q C Ho esevoi Enegy in Q H Cold esevoi Useful wok done W ou 5 Q H 2 Q C Unused enegy is ehaused as wase hea. Refigeaos Devices which use wok o ansfe hea fom a colde objec o a hoe objec. Enegy Q H 5 Q C 1 W in is ehaused o he ho esevoi. Wok mus be done o ansfe enegy fom cold o ho. Hea enegy is eaced fom he cold esevoi. T H W in T C Q H Q C Ho esevoi Cyclical pocess (DE h ) ne 5 0 Cold esevoi Themal efficiency h5 W ou wha you ge 5 Q H wha you pay Second law limi: h#1 2 T C T H Coefficien of pefomance K 5 Q C wha you ge 5 W in wha you pay Second law limi: T C K # T H 2 T C pefecly evesible engine (a Cano engine) can be opeaed as eihe a hea engine o a efigeao beween he same wo enegy esevois by evesing he cycle and wih no ohe changes. Cano hea engine has he maimum possible hemal efficiency of any hea engine opeaing beween T H and T C. h Cano T C T H Cano efigeao has he p 3 maimum possible coefficien of pefomance of any efigeao opeaing beween T H and T C. T C K Cano 5 T H 2 T C The Cano cycle fo a gas engine consiss of wo isohemal pocesses and wo adiabaic pocesses. 2 diabas Isohems 4 1 T H T C V n enegy esevoi is a pa of he envionmen so lage in compaison o he sysem ha is empeaue doesn change as he sysem eacs hea enegy fom o ehauss hea enegy o he esevoi. ll hea engines and efigeaos opeae beween wo enegy esevois a diffeen empeaues T H and T C. The wok W s done by he sysem has he opposie sign o he wok done on he sysem. W s 5 aea unde pv cuve p V i W s 5 aea V f V To analyze a hea engine o efigeao: MODEL Idenify each pocess in he cycle. VISULIZE Daw he pv diagam of he cycle. SOLVE Thee ae seveal seps: Deemine p, V, and T a he beginning and end of each pocess. Calculae DE h, W s, and Q fo each pocess. Deemine W in o W ou, Q H, and Q C. Calculae h5w ou /Q H o K 5 Q C /W in. SSESS Veify (DE h ) ne 5 0. Check signs. Copyigh 2004 Peason Educaion, Inc., publishing as ddison Wesley

20 The goal of Chape 20 has been o lean he basic popeies of aveling waves. The Wave Model This model is based on he idea of a aveling wave, which is an oganized disubance aveling a a well-defined wave speed v. In ansvese waves he paicles of v he medium move pependicula o he diecion in which he wave avels. In longiudinal waves he paicles of he medium move paallel o he v diecion in which he wave avels. wave ansfes enegy, bu no maeial o subsance is ansfeed ouwad fom he souce. Thee basic ypes of waves: Mechanical waves avel hough a maeial medium such as wae o ai. Elecomagneic waves equie no maeial medium and can avel hough a vacuum. Mae waves descibe he wavelike chaaceisics of aomic-level paicles. Fo mechanical waves, he speed of he wave is a popey of he medium. Speed does no depend on he size o shape of he wave. The displacemen D of a wave is a funcion of boh posiion (whee) and ime (when). snapsho gaph shows he wave s displacemen as a funcion of posiion a a single insan of ime. hisoy gaph shows he wave s displacemen as a funcion of ime a a single poin in space. wave aveling in he posiive -diecion wih speed v mus be a funcion of he fom D( 2 v). wave aveling in he negaive -diecion wih speed v mus be a funcion of he fom D( 1 v). D D v Sinusoidal waves ae peiodic in boh ime (peiod T) and space (wavelengh l). D(, ) 5 sin 32p(/l 2/T ) 1f sin (k 2v 1f 0 ) whee is he ampliude, k 5 2p/l is he wave numbe, v52pf 5 2p/T is he angula fequency, and f 0 is he phase consan ha descibes iniial condiions. Wave fons l l l 0 2 One-dimensional waves Two- and hee-dimensional waves Wave speeds fo some specific waves: Sing (ansvese): v 5 "T s /m Sound (longiudinal): v m/s in 20 C ai Ligh (ansvese): v 5 c/n, whee c m/s is he speed of ligh in a vacuum and n is he maeial s inde of efacion. The Dopple effec occus when a wave souce and deeco ae moving wih espec o each ohe: he fequency deeced diffes fom he fequency f 0 emied. ppoaching souce Obseve appoaching a souce f 0 f 1 5 f 1 5 (1 1 v o /v)f v s /v Receding souce Obseve eceding fom a souce The wave inensiy is he powe-o-aea aio I 5 P/ Fo a cicula o spheical wave I 5 P souce /4p 2 P souce f 2 5 f v s /v f 2 5 (1 2 v o /v)f 0 The Dopple effec fo ligh uses a esul deived fom he heoy of elaiviy. Copyigh 2004 Peason Educaion, Inc., publishing as ddison Wesley

21 The goal of Chape 21 has been o undesand and use he idea of supeposiion. Pinciple of Supeposiion The displacemen of a medium when moe han one wave is pesen is he sum of he displacemens due o each individual wave. Sanding waves ae due o he supeposiion of wo aveling waves moving in opposie diecions. ninodes Nodes Node spacing is 1 2 l The ampliude a posiion is () 5 2a sin k whee a is he ampliude of each wave. The bounday condiions deemine which sanding wave fequencies and wavelenghs ae allowed. Inefeence In geneal, he supeposiion of wo o moe waves ino a single wave is called inefeence. Maimum consucive inefeence occus whee cess ae aligned wih cess and oughs wih oughs. These waves ae in phase. The maimum displacemen is 5 2a. Pefec desucive inefeence occus whee cess ae aligned wih oughs. These waves ae ou of phase. The ampliude is 5 0. Inefeence depends on he phase diffeence Df beween he wo waves. Consucive: Df 5 2p D l 1Df 0 5 2mp Desucive: Df 5 2p D l 1Df 0 5 2(m 1 1 2)p D is he pah-lengh diffeence of he wo waves and Df 0 is any phase diffeence beween he souces. Fo idenical souces (in phase, Df 0 5 0): Inefeence is consucive if he pah-lengh diffeence D 5 ml. Inefeence is desucive if he pah-lengh diffeence D 5 (m 1 1 2)l. The ampliude a a poin whee he phase diffeence is Df ninodal lines, consucive inefeence. 5 2a Nodal lines, desucive inefeence. 5 0 is 5 P 2a cos 1 Df 2 2 P Bounday condiions Sings, elecomagneic waves, and sound waves in closedclosed ubes mus have nodes a boh ends. l m 5 2L m whee m 5 1, 2, 3, c The fequencies and wavelenghs ae he same fo a sound wave in an open-open ube, which has aninodes a boh ends. sound wave in an open-closed ube mus have a node a he closed end bu an aninode a he open end. This leads o l m 5 4L m whee m 5 1, 3, 5, 7, c f m 5 m v 2L 5 mf 1 f m 5 m v 4L 5 mf 1 Beas (loud-sof-loud-sof modulaions of inensiy) occu when wo waves of slighly diffeen fequency ae supeimposed. D Sof Loud Sof Loud Sof The bea fequency beween waves of fequencies f 1 and f 2 is f bea 5 f 1 2 f 2 Copyigh 2004 Peason Educaion, Inc., publishing as ddison Wesley