scalar TIME TIME INTERVAL [second s] t t T t i or t 1 s > 0 +x s < 0 t f or t 2 t = t f t i = t 2 t 1 acceleration a = constant v u at

Transcription

1 POSITION / LENGTH /DISTANCE / DISPLACEMENT [ee ] L d d D h H R a x x y s s x s y W N S E disance aelled d sx scos sy ssin S displaceen S agniude s / diecion N of E wok = change in KE aeage speed aeage elociy u d s ag ag s 1 1 W s u Insananeous elociy +y s s s ds slope of angen s / gaph d s slope = ise / un x y S (,) s x +x nano 1 n = 1-9 / ico 1 = 1-6 / 1 = 1-3 / 1 k = 1 3 s y s an s y x -x s < fae of efeence acceleaion a = consan s u a 1 s > +x u a s unifo cicula oion R ob 1 1 G h plane M R GM Gaiaion EP gh esc 1 EP G scala o eco GM R un ise c R a c R cenipeal foce cenipeal acceleaion Keple s 3 d law T 3 4 GM pendulu Lengh conacion L L T L g L sall apliude only L L 1 - / c easue L & T g 1

2 TIME TIME INTERVAL [second s] T iniial ie i o 1 scala -x s < s > +x final ie ie ineal fae of efeence = f i = 1 f o acceleaion a = consan u a s u a 1 ipulse = change on oenu u J u peiod T ie fo one coplee oscillaion o obi fequency f [hez Hz] nube of oscillaions (obis) in one second 1 T f angula fequency [ad.s -1 ] ae a which angle is swep ou f T Tie Dilaion Keple s 3 d law pendulu 1 - / c plane M R T 3 4 GM L L T g sall apliude only easue L & T g

3 VELOCITY [.s -1 ] fas slow + elociy (eco) / speed (scala) s aeage insananeous = > < s/ gaph slope angen = a s s s 1 1 s d s li d 1 a d 1 a a/ gaph aea = speed of ligh 3x1 8.s-1 speed of sound in ai ~ 34.s -1 [1D] u 1 A V B [D] > < x y x y cos sin x an Moion wih unifo acceleaion y x y [1D] = u + a s = u + ½ a = u + a s a = s / = (u + ) / [D] hoizonal oion a x = x = u x s x = x eical oion a y = g y = u y + a y s y = u y + ½ a y y cone.s -1 k.h -1 x Objec of ass oing wih elociy oenu kineic enegy p p EK 1 u u x u y y = u y + a y s y u u cos u usin x uy an u u u u x y x y Newon s 1 s law a consan saigh line oion wih consan speed Newon s nd law a changes (fase, slowe, chnage in diecion) 3

4 ACCELERATION [.s - ] eco a a a a a x a y g a c acceleaion due o gaiy g = 9.8.s - aeage insananeous a a 1 1 d a li d acceleaion is he ie ae of change of he elociy * geing fase * geing slowe * change in diecion + a g Moion Map a a fase slowe a( 1 ) = slope of angen a 1 = aea unde a / gaph a 1 Moion wih unifo acceleaion [1D] = u + a s = u + ½ a = u + a s a = s / = (u + ) / [D] hoizonal oion a x = x = u x s x = x eical oion a y = g y = u y + a y s y = u y + ½ a y y = u y + a y s y u u y u u cos u usin x y Hoizonal & Veical oions ae independen u x uy an u u u u x x y 4

5 ACCELERATION [.s - ] Newon s 1 s Law ineia a a ne consan o consan speed in a saigh line Newon s nd Law ne a diecion of acceleaion sae as ne foce acing on objec Cenipeal acceleaion a c always dieced owads cene of cicle a c a c Soe of he effecs of acceleaion we ae failia wih include: (1) Expeience of sinking ino he sea as a plane acceleaes down he unway. () "flue" in ou soach when a lif suddenly speeds up o slows down. (3) being hown side ways in a ca going aound a cone oo quickly. Newon s 1 s Law - ineia 5

6 Moion Gaphs Insananeous elociy ds slope of angen s / gaph d s slope = ise / un ise un s = aea unde / gaph s a a = slope of angen o / cue slope a ax heigh = = Ball hown eically s a + 1 = aea unde a/ cue _ paabola s = u + ½ a unifo acceleaion a = consan a = a < u saigh line = u + a iniial elociy u consan slope = a a > = u + a (saigh line) a = consan (consan slope) s = aea ecangle + iangle = u + ½ ( u) s = u + ½ a a = a = - g = s - 6

7 PROJECTILE MOTION g Galileo's analysis of pojecile oion: Pojecile oion was consised of boh hoizonal and eical coponens - hese coponens wee independen of each ohe occued siulaneously pependicula o each ohe. Veical acceleaion was he sae fo all falling objecs if ai esisance is disegaded. Tajecoy of a pojecile is a paabola. A oion of an objec is elaie o is fae of efeence and an objec has he oion of is ineial fae of efeence. He esed his in his Cow's Nes expeien - I was hough ha, if a ship was oing a a consan speed, and a ball was dopped fo he cows nes, i would fall behind he ship and ino he sea as he ship would hae oed. Insead, i fell saigh down ono he ship as if i hadn' oed. Poble A olcano ha is 33 aboe sea leel eups and sends ock fagens huling ino he sea 9.4 k away. If he fagens wee ejeced a an angle of 35 o, wha was hei iniial speed? 7

8 Soluion Idenify / seup =?.s -1 = 35 o x = y = x = 94 y = -33 a x = a y = -9.8.s - x = cos y = sin Execue X oion x cos x x cos Y Moion 1 y y ay 1 y ysin a y Equaion fo unifoly acceleaed oion 1 o a s o a u s o a s how o appoach he poble +Y (, ) 35 o 33 Eliinae o find equaion fo y x a y cos cos ay x 1 yxan cos sin 1-1 ay x cos y xan 55.s o o cos an 35 (55) k.h 9 k.h x Ealuae The ocks in a olcanic explosion can be hown ou a enoous speeds. 94 +X 8

9 ORCE [newon N] G W N N f f T R eco push / pull / ineacion beween objecs a Newon s 1 s Law ineia ne a ne consan o consan speed in a saigh line Newon s nd Law ne a diecion of acceleaion sae as ne foce acing on objec d d p ne a d d eco Newon s 3 d Law oce beween wo posiiely chaged objecs is epulsie AB A + B + BA Peson is hi by a speeding uck, he agniude of he foces expeienced by he peson & uck ae he sae. AB BA Weigh G = W = g g = g = 9.8.s - Newon s Law of Gaiaion G _ AB G _ BA Tension T = T sing ension T = 98 N A B weigh of an objec is due o he foce acing on i in a gaiaional field (aacion beween objec and plane) G _ AB G G A E A he suface of he Eah RE G _ BA B GM g Uniesal gaiaional consan G = N..kg - 1 kg 1 kg 9

10 ORCE [newon N] G W N N f f T R eco push / pull / ineacion beween objecs Noal foce N = N acs a igh angles o a suface icion foce f = f acs along (paallel) suface N f G block on ap f f f abulance sopping: ficion beween wheels & oad abulance acceleaion: ficion beween die wheels & oad f Cenipeal foce c c c 1

11 ORCE [newon N] G W N N f f T R eco push / pull / ineacion beween objecs donkey is he syse ee Body Diaga Take he donkey o be he syse: conside only he foces acing on he donkey + Y, a N C foce ca pulling on donkey f + X ficion foce: gound on donkey Newon s nd Law G a y N G y a a x f C x oce on a chaged paicle q in an elecic field E elecic field + q - q E oce beween wo posiiely chaged objecs is epulsie AB Coulob s Law: foce beween wo pin chages q A & q B q A + q B + BA AB BA q E agniude of foce beween chages: q q 1 q q k A B A B 4 o k = N..C - peiiiy of fee space o = C.N Chages of he sae sign epel and of opposie sign aac 11

12 MOMENTUM p P IMPULSE J [N.s kg..s -1 ] eco ipulse J = aea unde cue p p J ne d ne ag 1 _ 1 o a syse in which he ne foce acing on i is zeo, hen he oal oenu of he syse does no change: Newon s 3 d Law Law of Conseaion of Moenu. This law is useful fo aking pedicions when collisions o explosions occu. + X befoe u 1 u 1 1 Conseaion of oenu (iniial oenu = final oenu u u Elasic collision: conseaion of enegy u1 u 1 1 afe 1 Inelasic collision: non conseaion of enegy u u

13 KINETIC ENERGY K KE E K / WORK W [ joule J ] scala Wok done W by a single foce acing on an objec of ass x wok W = aea unde cue x = cos x x Wx x dx x _ ag x x1 oing objec has kineic enegy EK 1 x Ne foce acing oe a displaceen on objec does wok on objec and changes is kineic enegy: Ne Wok = Change in kineic enegy Moe han a single foce acing on he objec hen: W W ne i i eely falling objec: Wok done W by he gaiaional foce G as an objec falls a heigh h nea he suface of he Eah W = s cos = G = g s = h = W = g h Gaiaional poenial enegy U G = E p = - W G s h Loss in GPE E p = - g h 13

14 Nueical exaple In golf, a ypical conac ie is 1. s. If a 45 g ball leaes he club a a speed of 4 k.h -1, esiae he aeage foce exeed by he club on he ball. Soluion Idenify / Seup = 1. s = 1-3 s = 45 g = kg 1 =.s -1 = 4 k.h -1 = (4)(13)/( ).s -1 = 67.s -1 ag =? N Ipulse = change in oenu J d ne ne _ ag 1 Execue ag ag ag N 3 (451 )(67) 3 N 1 Ealuae unis ok significan figues () ok Answes appoxiaely equialen o he ass of 6 people of ass 5 kg how o appoach he poble 14

15 Nueical exaple A boy pushes a oy ca of ass 5 g, iniially a es wih a hoizonal foce of 5. N hough a disance of 1.. How uch wok is done on he ca? Wha is he final speed of he ca? Soluion Idenify / Seup =.5 g =.5 kg x = 5. N x = 1. W =? J =?.s -1 1 =.s -1 Wok done poduces a change in KE W = x x = ½ ½ 1 Execue W = x x = (5.)(1.) J = 6. J W ()(6) 1 W.s 6.9.s Ealuae unis ok significan figues () ok Answes sees OK how o appoach he poble 15

16 foce (a.u.) GRAVITATION 1 Gaiaional oce Weigh G = W = g 9 8 g = g = 9.8.s inese squae law weigh of an objec is due o he foce acing on i in a gaiaional field (aacion beween objec and plane) posiion (a.u.) Newon s Law of Gaiaion G _ AB G _ BA A B a he Eah s suface g = 9.8.s - G _ AB G G A E A he suface of he Eah RE g plane G _ BA B GM g G M G M R R plane plane plane plane Uniesal gaiaional consan G = N..kg - 16

17 GPE E p (a.u.) GRAVITATIONAL POTENTIAL ENERGY E P U U G [joule J] Lifing an objec eically poduces an incease in gaiaional poenial enegy of he syse of he objec and he Eah. Wok done = Incease in gaiaional poenial enegy W h h g h W E P H E P g h G hand H weigh G a = H = G nea he Eah s suface: g = consan g = 9.8.s - (posiie nube) Objec aised hough a eical displaceen h a a consan elociy objec ass oed a consan elociy fo posiion o o incease is gaiaional poenial enegy plane M R iniial posiion a E P final posiion a = Wok done W on objec in oing i fo o = d ' 1 G M E W d ' G ' E E d G M G M ' ' W E E ( ) E ( ) E ( ) E ( ) P P P P P E P G M E () -1 - Gaiaional PE E P posiion (a.u.) gaiaional poenial enegy E p inceases as he disance fo he cene of he plane inceases 17

18 MOTION O ROCKETS oces expeienced by asonaus duing ake-off Rocke acceleaing upwads oenu of ocke p ocke Rocke launch & populsion RG foce on ocke by gas GR foce on gas by ocke + asonau ass acceleaion a weigh G = g you expeience g-foces when going up & down a eleao appaen weigh g-foce acual weigh g a g g a g noal eacion foce acing on asonau N (eg easued by se of bahoo scales) Newon s nd Law = N G = N g = a appaen weigh N = g + a g a g-foce = 1 g Geae he acceleaion of he ocke he geae he g-foce expeienced by he asonau I is us safe fo an asonau o lie in a couching posiion ahe han sanding up because he body can oleae lage g-foces. In he couching posiion g-foce(ax) ~ g a LIT O inceasing oenu of exhaus gases p gases a RE-ENTRY deceasing Alan Shepad fis an in space g-foce (lif off) ~ 6 g g-foce (e-eny) ~ 1 g Newon s hid law: oces ac fo ie ineal RG Ipulse = Change in oenu: R Moenu is conseed: RG GR G GR ipulse: p p ( ) ( ) ocke p p Gases expelled ass of he ocke deceases acceleaion inceases R G 18 gases a

19 MOTION O ROCKETS Pilos expeience lage g-foces when eneing and pulling ou of a seep die oce expeienced by an asonau duing a space fligh sa of a seep die blood pulling ou of a seep die body Newon s 1 s law: blood keeps oing Newon s 1 s law: blood says pu 19

20 MOTION O ROCKETS Rocke launch & obial oion of he Eah Rockes ae launched in an easely diecion o ge a elociy boose as a esul of he Eah spinning abou is axis of oaion oaion owads he eas Eah s obial oion aound he Sun can be helpful in launching ockes o planes in ou Sola Syse Sun Eah s obial elociy aound he Sun ~ 3 k.s -1 Roaion axis NASA s Cape Canaeal Roaional speed ~ 4.s -1 due o he Eah spinning aound is oaion axis Eah s obi aound he Sun Eah SLINGSHOT EECT - pefoed o achiee an incease in speed and/o a change of diecion of a spacecaf as i aels aound a plane. As i appoaches, i is caugh by he gaiaional field of he plane, and swings aound i, he speed acquied hows he spacecaf back ou again, away fo he plane. By conolling he appoach, he oucoe of he anoeue can be anipulaed and he spacecaf gain soe of he plane s oenu, elaie o he Sun. Siple iew fo sola efeence fae: enegy & oenu ae conseed hp://

21 MOTION O ROCKETS - ESCAPE VELOCITY esc Cannon ball launched fo op of a ounain wih inceasing elociy The escape elociy esc fo a ocke fied fo a plane o oon (ass M, adius R) esc obial elociy Cannon ball fied wih inceasing elociies ob ball obis aound he Eah GM R esc ball fied a escape elociy esc (Eah) = 11 k.s -1 esc (Moon) =.4 k.s -1 esc (Sun) = 6 k.s -1 1 esc esc GM R 1 GM E1 EK1 EP1 esc R M R esc E EK EP oal echanical enegy is conseed E = KE + PE = consan E 1 = E GM esc R 1

22 Safe e-eny and landing of ockes o spacecaf The safe eun of a spacecaf ino he Eah s aosphee and subsequen descen o Eah equies consideaion of wo ain issues: (1) How o handle he inense hea geneaed as he spacecaf enes he Eah s aosphee. () How o keep he g-foces of deceleaion wihin safe liis. angle oo sep lage heaing effec spacecaf buns ou angle oo shallow spacecaf bounces off aosphee Coec eny angle spacecaf can land safely o a safe landing of a spacecaf i us ene he aosphee in he coec ange of angles. Spacecaf e-eny and he heaing effec in passing hough he aosphee.

23 Safe e-eny and landing of ockes o spacecaf Wha Can Go Wong? If he angle of e-eny is oo shallow, he spacecaf ay skip off he aosphee. The coonly cied analogy is a ock skipping acoss a pond. If he angle of eny is oo seep, he spacecaf will bun up due o he hea of e-eny. Because of collisions wih ai paicles and he huge deceleaion, a huge aoun of heal enegy is poduced fo ficion. The space shule us be able o wihsand hese epeaues. I uses a coeing of insulaing iles which ae ade of glass fibes bu ae abou 9% ai. This gies he excellen heal insulaing popeies and also consees ass. The ile consucion is dense nea he suface o ake he iles oe esisan o ipac daage, bu he suface is also poous. Daage o he space shule Colubia's hea shield is hough o hae caused is disinegaion and he loss of seen asonaus on 1s ebuay 3. Inesigaos beliee ha he scoching ai of e-eny peneaed a cacked panel on he lef wing and eled he eal suppo sucues inside. Lage g-foces ae expeienced by asonaus as he space shule deceleaes and e-enes he Eah's aosphee. Asonaus ae posiioned in a ansese posiion wih hei backs owads he Eah's suface as g-foces ae easie fo huans o oleae in hese posiions. Suppoing he body in as any places as possible also helps o incease oleance. Thee is an ionisaion blackou fo he space shule of abou 16 inues whee no counicaion is possible. This is because as heal enegy builds up, ai becoes ionised foing a laye aound he spacecaf. Radio signals canno peneae his laye of ionised paicles. 3

24 MOTION O SATELLITES Saellies ae placed in one of seeal diffeen ypes of obi depending on he naue of hei ission Low Eah Obi (LEO) Radius: k o k aboe he Eah s suface. Peiod: 6 in 9 inues. equen coeage of specific o aied locaions on he Eah s suface. Sall field of iew. Obis less han 4 k ae difficul o ainain due o aospheic dag and subsequen obial decay. Types of saellies: iliay applicaions, Eah obseaion, weahe onioing, shule issions. Wih he excepion of he luna flighs of he Apollo poga, all huan spaceflighs hae aken place in LEO. All anned space saions and he ajoiy of aificial saellies in LEO. Obial decay - educion in he heigh of an objec's obi oe ie due o he dag of he aosphee on he objec. All saellies in low Eah obis ae subjec o soe degee of aospheic dag ha will eenually decay hei obi and lii hei lifeies. Een a 1 k, as hin as he aosphee is, i is sill sufficienly dense o slow he saellie down oe a peiod of ie. Geosaionay Obi (GEO) Cicula obi in he Eah's equaoial plane, any poin on which eoles abou he Eah in he sae diecion and wih he sae peiod as he Eah's oaion. Useful because hey cause a saellie o appea saionay wih espec o a fixed poin on he oaing Eah. As a esul an anenna can poin in a fixed diecion and ainain a link wih he saellie. The saellie obis in he diecion of he Eah's oaion, a an aliude of appoxiaely 35,786 k aboe gound. This aliude is significan because i poduces an obial peiod equal o he Eah's peiod of oaion, known as he sideeal day. These obis allow fo he acking of saionay poin on Eah. Hae he lages field of iew. Applicaions include counicaions, ass-edia and weahe onioing. 4

25 MOTION O SATELLITES unifo cicula oion A foce (cenipeal foce c ) acing owads he cene of a cicle is necessay fo an objec o oae in a cicula pah. Saellie is aced upon by he gaiaion foce beween he Eah and he saellie. The cenipeal foce is he gaiaional foce. o a ass, oing a a speed, in unifo cicula oion of adius, he ne foce acing on is called he cenipeal foce c and is agniude is gien by Cenipeal foce change in diecion of he objec as is speed is consan. The esuling acceleaion due o he change in diecion is he cenipeal acceleaion a c and is agniude is Diecion of he cenipeal foce and cenipeal acceleaion is owads he cene of he cicle (cenipeal eans cene seeking ). a c c a c a c 1 a c change in elociy dieced owads he cene of he cicle acceleaion a c dieced owads he cene of he cicle 5

26 MOTION O SATELLITES ORBITAL MOTION Obial elociy ob To place a saellie of ass ino a sable Eah obi a a paicula adius, he launch us gie i boh an iniial eical and hoizonal coponen of elociy elaie o he Eah s suface. The saellie will eenually un so ha i is aelling hoizonal o he Eah s suface. A his adius, he foce of gaiy G poides he acceleaion needed o keep he objec oing in a cicle, bu a paicula obial elociy is also equied o keep he objec in a sable obi obial elociy ob. To calculae ha obial elociy, we equae he cenipeal foce c and gaiaional foce G. Geosaionay Obi peiod T = 4 h ob GME GME ob ob GM Obial elociy of a saellie as i obis aound he Eah only depends on: ass of he Eah M E adius of he obi Aliude is he only aiable ha deeines he obial elociy equied fo a specific obi aound he Eah. Geae he adius of ha obi, he lowe ha elociy ob ob E M plane ob c G ne foce is always dieced o cene of obi Obial elociy aound ohe planes M E M plane ob GM plane 6

27 MOTION O SATELLITES How do he planes oe? Keple s Laws of Moion The oion of a plane is goened by he Law of Uniesal Gaiaion = G M S / whee G is he Uniesal Gaiaional Consan, M S is he ass of he Sun, is he ass of he plane and is he disance fo he Sun o he plane. G = N..kg M S =.1 3 kg Keple s 1 s law: pah of a plane aound he Sun is an ellipse. Keple's Laws of Planeay Moion 1 The pah of each plane aound he Sun is an ellipse wih he Sun a one focus. Each plane oes so ha all iaginay lines dawn fo he Sun o he plane sweeps ou equal aeas in equal peiods of ie. 3 The aio of he squaes of he peiods of eoluion of planes is equal o he aio of he cubes of hei obial adii (ean disance fo he Sun o lengh of sei-ajo axis, a) Keple s nd law: Plane oes so ha an iaginay line dawn fo he Sun o he plane sweeps ou equal aeas in equal peiods of ie. 3 3 T1 1 a o T 4 T GMS hp:// Copue siulaion of he oion of a plane aound he Sun. 7

28 MOTION O SATELLITES Keple s Laws of Peiods M ob c G ne foce is always dieced o cene of obi Geosaionay obi (GEO) ob h R E M E peiod T = 4 h = (4)(3.6x1 3 ) s = 8.6x1 4 s R E = 6.38x1 6 M E = 5.97x1 4 kg G = 6.673x1-11 N..kg -1 ob =?.s -1 =? heigh abou Eah s suface ob ob GM C G GM C G ob ie fo one coplee obi: peiod T disance aelled in one obi: cicufeence GM s GM ob T Reaanging gies Keple s Law of Peiods T 3 GM 4 G ME T 4 ob Puing in he nubes = 4.x1 7 h = 3.59x1 7 = 36x1 3 k h =? h = - R E GM ob = 3.7x1 3.s -1 =1.11x1 4 k.h -1 E 1 3 T 3 GM 4 ob GM 8

29 EINSTEIN THEORY O SPECIAL RELATIVITY: SPACE & TIME RAMES O REERENCE Which is bigge he bea o ouse? depends upon he locaion of he obsee Wha is he ajecoy of he ball? i is elaie i depends upon he oion of he obsee Ineial fae of efeence * ae of efeence wih consan elociy. * Is a non-acceleaing fae of efeence. * Law of ineia holds. * Newon's laws of oion hold. * No ficiious foces aise. Non-Ineial aes of Refeence * Does no hae a consan elociy. I is acceleaing. * The fae could be aelling in a saigh line, bu be speeding up o slowing down. * The fae could be aelling along a cued pah a a seady speed. * The fae could be aelling along a cued pah and also speeding up o slowing down. 9

30 EINSTEIN THEORY O SPECIAL RELATIVITY: SPACE & TIME AETHER MODEL OR THE TRANSMISSION O LIGHT Classical picue fo he speed of ligh. The speed of ligh is elaie o he oion of he obsee, and so he speed of ligh is c+ o c-. Bu his is no coec. The coec answe, is ha he peson will easue he speed of ligh o be he consan alue c and i does no ae how fas o slowe hey ae appoaching o eceding fo he ligh bea o he speed of he ligh souce. I seeed inconceiable o 19 h Cenuy physiciss ha ligh and ohe elecoagneic waes, in conas o all ohe kinds of waes, could popagae wihou a ediu. I seeed o be a logical sep o posulae such a ediu, called he aehe (o ehe), een hough i was necessay o assue unusual popeies fo i, such as zeo densiy and pefec anspaency, o accoun fo is undeecabiliy. This aehe was assued o fill all space and o be he ediu wih espec o which elecoagneic waes popagae wih he speed c. I followed, using Newonian elaiiy, ha an obsee oing hough he aehe wih elociy would easue a elociy fo a ligh bea of (c + ). If he aehe exiss, an obsee on Eah should be able o easue changes in he elociy of ligh due o he Eah s oion hough he aehe. The Michelson-Moley expeien aeped o do jus his. AETHER poposed ediu fo he popagaion of elecoagneic waes Popey of aehe ills space, peeaes all ae Saionay Tanspaen Exeely low densiy Gea elasiciy Eidence ligh aels eeywhee ligh aels in saigh lines can see i can be deeced ediu us be elasic ohewise enegy dissipaed 3

31 EINSTEIN THEORY O SPECIAL RELATIVITY: SPACE & TIME MICHELSON MORLEY EXPERIMENT The Michelson-Moley expeien is an excellen exaple of a ciical expeien in science - he fac ha no oion of he Eah elaie o he aehe was deeced suggesed quie songly ha he aehe hypohesis was incoec and ha no aehe (absolue) efeence fae exised fo elecoagneic phenoena his opened he way fo a whole new way of hinking ha was o be poposed by Albe Einsein in his Theoy of Special Relaiiy. The null esul of he Michelson-Moley expeien was such a blow o he aehe hypohesis in paicula and o heoeical physics in geneal ha he expeien was epeaed by any scieniss oe oe han 5 yeas. A null esul has always been obained. The ligh fo efleced by he wo ios poduces an inefeence paen a he locaion of he obsee. As he Eah eoles aound he Sun and spins on is axis, he diecion of he ligh beas aies wih he diecion of flow hough he aehe, hei elaie elociies would ale and hus he diffeence in ie equied fo each bea o each O would ale. This would esul in a change in he inefeence paen as he appaaus was oaed (changes in he paens of bigh and dak finges). A null esul has always been obained NULL esul: no shif in finge paen Expeced esul: sligh shif in posiion of finges iniial inefeence paen inefeence paen afe oaion of appaaus 31

32 EINSTEIN THEORY O SPECIAL RELATIVITY: SPACE & TIME In 195, Albe Einsein ( ) published his faous pape eniled: On he Elecodynaics of Moing Bodies, in which he poposed his wo posulaes of elaiiy and fo hese deied his Special Relaiiy Theoy. 1. The Pinciple of Relaiiy All he laws of physics ae he sae in all ineial efeence faes no pefeed ineial fae exiss.. The Pinciple of he Consancy of he Speed of Ligh he speed of ligh in fee space has he sae alue c, in all ineial faes, egadless of he elociy of he obsee o he elociy of he souce eiing he ligh. The speed of ligh is consan no ae wha ae he speeds of he ansie o eceie. consan c boh space and ie us be elaie quaniies Einsein concluded - if we accep ha he pinciple of elaiiy can nee be iolaed, hen 1 The aehe odel us be wong. The speed of ligh is consan egadless of he oion of he obsee. In ode o saisfy, speed of ligh is consan, he ade a eoluionay saeen: i is no he speed of ligh ha is changing, bu ie. Saionay obsees and he oing obsees peceie space and ie diffeenly. In classical physics space and ie ae consans and oion is defined by he. In Einsein's physics i is he speed of ligh ha is consan and space and ie change o accoodae his. Using hese ideas, Einsein pu fowad his Special Theoy of Relaiiy 1 All oion is elaie he pinciple of elaiiy holds in all siuaions. The speed of ligh is consan egadless of he obsee's fae of efeence. 3 The aehe is no needed o explain ligh, and, in fac i does no exis. 3

33 EINSTEIN THEORY O SPECIAL RELATIVITY: SPACE & TIME Relaiiy of siulaneiy - whehe wo spaially sepaaed eens occu a he sae ie is no absolue, bu depends on he obsee's efeence fae. - i is ipossible o say in an absolue sense whehe wo disinc eens occu a he sae ie if hose eens ae sepaaed in space. Obsee in ain obsees ain a es = Gound based obsee sees ain go pas a speed Ligh beas ael fo cene of ain and when hey hi he ends of he caiage a ligh flash is gien ou. Tain obsee sees flashes when ligh eaches he ends of he caiage siulaneously. Gound based obsee sees a flash a he back of he caiage befoe he flash a he fon of he caiage. hp://faaday.physics.uoono.ca/genealinees/haison/specrel/lash/siulaneiy.hl 33

34 beep 1 beep beep 3 beep 4 beep 5 beep 6 beep 7 beep 8 beep 9 beep 1 beep 11 beep 1 beep 13 beep 1 beep beep 3 beep 4 beep 5 beep 6 beep 7 beep 8 beep 9 beep 1 beep 11 beep 1 beep 13 EINSTEIN: SPECIAL RELATIVITY SPACE AND TIME SPEED O LIGHT IS CONSTANT INDEPENDENT O THE MOTION O SOURCE OR OBSERVER c = 3.x1 8.s -1 TIME DILATION Tie Dilaion 1 - / c ie ineal easued by obseing oing clock ie ineal easued by obseing saionay clock pope ie he ie ineal beween wo eens occuing a he sae poin in space w... a clock a es w... ha poin. dilaed ie ineals hey ae he ie ineals on oing clocks w... a saionay obsee. All ie ineals easued on oing clocks ae longe copaed wih he saionay clock oing clocks un slowe saionay clock Tie is a elaie quaniy: diffeen obsees can easue diffeen ie ineals beween he occuence of wo eens. This aises because he speed of ligh is a consan and independen of he oion of he souce of ligh o he oion of an obsee. An obsee waching a oing clock sees he passage of ie on he oing clock o be slowe han he passage of ie on hei own clock. oing clock saionay clock: 8 beeps hae occued oing clock: 6 beeps hae occued Tie ineals ae no absolue. This is a iolaion of a fundaenal conceps in Newonian physics whee ie is an absolue quaniy. oing clocks un slow 34

35 EINSTEIN: SPECIAL RELATIVITY SPACE AND TIME SPEED O LIGHT IS CONSTANT INDEPENDENT O THE MOTION O SOURCE OR OBSERVER c = 3.x1 8.s -1 TIME DILATION 1 9 / c =.994 Tie Dilaion 1 1 c / c Newonian physics ok / c Conside a ain wih elociy =.9c w... a saionay fae of efeence. In he saionay fae of efeence, he duaion of an een was 1. s. Wha would be he duaion of he een as easued by an obsee waching he oing clock? = 1. s =? s =.9c s 1 (.9 c ) 1 1 c c oing fae of efeence saionay fae of efeence.9 s To an obsee on Eah, he ie aken fo he een is.9 s. The obsee in he ain, easues a ie ineal of only 1. s. The Eah obsee sees ha he ain clock has slowed down. I is essenial ha you undesand ha his is no an illusion. I akes no sense o ask which of hese ies is he eal ie. Since no pefeed efeence fae exiss all ies ae as eal as each ohe. They ae he eal ies seen fo he een by he especie obsees. Tie dilaion ells us ha a oing clock uns slowe han a clock a es by a faco of 1/{1 (/c)}. This esul, can be genealised beyond clocks o include all physical, biological and cheical pocesses. The Theoy of Relaiiy pedics ha all such pocesses occuing in a oing fae will slow down elaie o a saionay clock. 35

36 EINSTEIN: SPECIAL RELATIVITY SPACE AND TIME SPEED O LIGHT IS CONSTANT INDEPENDENT O THE MOTION O SOURCE OR OBSERVER c = 3.x1 8.s -1 LENGTH CONTRACTION ain a es w... obsee ain in oion w... obsee Lengh conacion L L L 1 - / c L L L easued by obsee in saionay fae of efeence by obseing oing objec. L is he pope lengh as easued by an obsee who is a es o he objec. How long is a ain? I depends on he elaie oion of he obsee and he ain. ain is shoe in diecion in oion bu jus as high and wide as i was a es This is a eal diffeence in lengh of he objec when i is oion elaie o an obsee. o a peson in he ain, hee is no conacion in lengh. L =? oing fae of efeence saionay fae of efeence L =? 36

37 EINSTEIN: SPECIAL RELATIVITY SPACE AND TIME LENGTH CONTRACTION / TIME DILATION / MUON DECAY Muons ae unsable paicles wih a es ass of 7 ies ha of an elecon and a chage of ±1.6x1-19 C. Muons decay exponenially ino elecons o posions wih a half-life of 1/ = 1.56x1-6 s as easued in hei fae of efeence. When high enegy paicles such as poons called cosic ays ene he aosphee fo oue space, hey ineac wih ai olecules in he uppe aosphee a a heigh of abou 1 k, ceaing a cosic ay showe of paicles including uons ha each he Eah s suface. The uons ceaed in hese cosic ay showes ael a =.98c w.. o he Eah. Exponenial decay: A = N paicles Afe ie, N paicles eaining decay consan = log e () / 1/ N N e Newonian (classical) poin of iew Half-life 1/ = 1.56x1-6 s Decay consan = log e () / 1/ = 4.44x1 5 s -1 Speed of uons =.98c = (.98)(3.x1 8 ).s -1 =.94x1 8.s -1 Disance aelled by uons o each Eah s suface = 1x1 3 Tie o each Eah s suface = 1x1 3 /.94x1 8 s = 3.4x1-5 s Pecenage of uons eaching Eah s suface = (1)(N/N ) = 1 e - = (1){exp[(-(4.44x1 5 )(3.4x1-5 )]} =. % Hence, fo a Newonian poin of iew, os uons would no be able o each he Eah s suface fo he uppe aosphee whee hey ae poduced. Howee, expeiens show ha a lage nube of uons do each he Eah s suface in cosic ay showes. Special elaiiy lengh conacion Muon s fae of efeence, he disance fo uppe aosphee o Eah s suface conaced L = 1x1 3 =.98c L =? L L c x x / Tie fo uons o each Eah s suface = (1.99x1 3 /.94x1 8 ) s = 6.77x1-6 s Pecenage of uons eaching Eah s suface = (1)(N/N ) = 1 e - = (1){exp[(-(4.44x1 5 )(6.77x1-6 )]} = 5 % Many any oe uons can each he Eah s suface hen pediced by Newonian physics hae o ejec Newonian physics and accep Einsein s posulaes: space and ie ae no absolue quaniies. 37

38 EINSTEIN: SPECIAL RELATIVITY SPACE AND TIME MUON DECAY Muons ae unsable paicles wih a es ass of 7 ies ha of an elecon and a chage of ±1.6x1-19 C. Muons decay exponenially ino elecons o posions wih a half-life of 1/ = 1.56x1-6 s as easued in hei fae of efeence. When high enegy paicles such as poons called cosic ays ene he aosphee fo oue space, hey ineac wih ai olecules in he uppe aosphee a a heigh of abou 1 k, ceaing a cosic ay showe of paicles including uons ha each he Eah s suface. The uons ceaed in hese cosic ay showes ael a =.98c w.. o he Eah. oun clock ie ineal as easued by Eah obsee = 6.77x1-6 s =.98c Exponenial decay: A = N paicles Afe ie, N paicles eaining decay consan = log e () / 1/ N N e Eah based clock ie ineal as easued by Eah obsee = 3.4x1-5 s Newonian (classical) poin of iew Half-life 1/ = 1.56x1-6 s Decay consan = log e () / 1/ = 4.44x1 5 s -1 Speed of uons =.98c = (.98)(3.x1 8 ).s -1 =.94x1 8.s -1 Disance aelled by uons o each Eah s suface = 1x1 3 Tie o each Eah s suface = 1x1 3 /.94x1 8 s = 3.4x1-5 s Pecenage of uons eaching Eah s suface = (1)(N/N ) = 1 e - = (1){exp[(-(4.44x1 5 )(3.4x1-5 )]} =. % Hence, fo a Newonian poin of iew, os uons would no be able o each he Eah s suface fo he uppe aosphee whee hey ae poduced. Howee, expeiens show ha a lage nube of uons do each he Eah s suface in cosic ay showes. Special elaiiy ie dilaion Eah obsee: uons aels a disance 1 k a a speed.98c. Tie o each Eah s suface = 1x1 3 /.94x1 8 s = 3.4x1-5 s Moing clock s un slow obseed ie ineal on uon s clock is c x x / s s Pecenage of uons eaching Eah s suface = (1)(N/N ) = 1 e - = (1){exp[(-(4.44x1 5 )(6.77x1-6 )]} = 5 % Sae answe as using lengh conac Noe: had o find and no. 38

39 MASS ENERGY E = c RELATIVISTIC MASS Newonian echanics, if a foce is applied, an objec will acceleae and is elociy will incease indefiniely. Special Relaiiy, as he elociy appoaches he speed of ligh, he sae foce poduces less and less acceleaion, gadually educing o zeo. This is because he ass of he objec is inceasing and acceleaion and ass ae inesely popoional when he foce is consan ( = a). If his did no happen he elociy would becoe infinie. / 1- c Whee does his exa ass coe fo? The applied foce is sill doing wok (W = s) so he objec is gaining kineic enegy (since is inceasing). This addiional enegy is coneed ino ass accoding o he equaion E = c, ahe han coninually inceasing he elociy of he objec. Newonian physics ok / c 1- c = easued ass of he oing objec o = easued ass of he objec a es (es ass) = elaie elociy beween he obseed objec and he obsee c = speed of ligh The ass of an objec is a elaie quaniy, i depends on he elaie elociy of he objec w... an obsee. 39

40 MASS ENERGY E = c Einsein s faous equaion E = c equialence of ass and enegy Enegy can be coneed ino ass and ice esa: * A paicle and is anipaicle collide, all he ass is coneed ino enegy. * Mass is coneed ino enegy in a nuclea fission eacion. * When a body gies off enegy E in he fo of adiaion, is ass deceases by an aoun equal o E/c. In Special Relaiiy, he Law of Conseaion of Enegy and he Law of Conseaion of Mass hae been eplaced by he Law of Conseaion of Mass-Enegy. When ass inceases as a body gains elociy effeciely liis all an-ade objecs o ael a speeds appoaching he speed of ligh. The close a body ges o he speed of ligh, he oe assie i becoes. The oe assie i becoes, he oe enegy ha has o be used o gie i he sae acceleaion. To acceleae he body up o he speed of ligh would equie an infinie aoun of enegy. Clealy, his places a lii on boh he speed ha can be aained by a spacecaf and heefoe he ie i akes o ael fo one poin in space o anohe. 4