37 Relativity
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RELATIVITY
LEARNING GOALS By studying this chapter, you w/ea: w /ea: •
•
The two postulates of Einstein's special eoy of relaivy, and wat motvates tese posates. Wy diffeen obseves can disagee abo wete two events are smltaneos How elativity pedis tat moving clocks n sow and expementa evdence ta conims ts
•
How e leng of an obje canges de o te obects moton How e velocy of an obe depends on te ame of efeence fom wic t is observed How te teoy o elaivity modi es te elatonsip between velocty and momenm How to solve pobems invovng wok and knetc enegy fo pat ces movng at elaivstc speeds. Some of te key concepts of insten's geneal eoy o elatvity
?At Brookhaven
National Laboratory in New Yor, atomic nuclei are acceerated to 99.995% of the uti mate speed limit of the universe-the speed of light. Is there aso an upper imit on the k gy of a partice?
hen the year year 1905 began, Albert Einsten Einsten was an unknown 25-year-od clerk n the Swss patent ofce. By the end of that amazing year he had pubished three papers of extraordnary importance One was an analyss of Brownan motion; a second (for which he was awarded the Nobe Prze) was on the photoeectrc eect In the thrd Ensten introduced his sp h v proposing drastc revsons n the Newtonian concepts of space and time The speca theory of reativty has made wderangng changes n our under standing of nature but Einsten based it on just two simpe postuates. One states that the aws of physcs are the same n a nertia frames of reference; the other
W
states that the speed of ight n vacuum s the same n al nerta frames These innocent-sounding propostons have farreachng impicatons. Here are three: ) Events that are smutaneous for one observer may not be smutaneous for another. 2) When two observers moving reative to each other measure a tme nterva or a ength, they may not get the same resuts (3) For the conservaton principes for momentum and energy to be vaid in al nertal systems Newton's second law and the equatons for momentum and knetc energy have to t o be revised Relatvity has mportant consequences n ll areas of physcs ncudng elec tromagnetsm atomc and nucear physics, and hghenergy physcs Athough many of the results derived n ths chapter may run counter to your intuton, the theory s in soid agreement wth experimenta observatons
37.1
Invariance of Physical Laws
Lets take a look at the two postuates that make up the specal theory of reativ ty Both postulates descrbe what s seen by an observer in an intil frame of nc, whch we ntroduced n Section 42 The theory s "specia n the sense that it appes to observers in such specia reference frames 1268
Invariace of Physcal Law
269
Einstein's First Postulate
Einstein's rst postulate, caled the pp tt, states: T p m t m f the laws difered that diference coud distinguish one inertial frame from the others or make one frame somehow more "coect than anoher. Here are two examples. Suppose you watch two chidren playing catch with a bal while the three of you are aboard a train moving with constant velocity. Your observations of the motion of the the bll, no matter how careuly done can't tell you how ast (or whether) the train is mov ing.Another This is because aws of motion the same in every inertial rame. exampeNewtons is the eectromotive orceare(em) induced in a coil o wire by a nearby moving permanent magnet. In the rame o reerence in which the coil is stationary (Fig. 37.1a) the moving magnet causes a change o magnetic ux through the coil and this induces an em. In a dierent frame o reerence in which the mgne is stationary (Fig. 37.1b) the motion of the coil through a mag netic eld induces the emf. According to the principle of relativity both o these rames of reference are equally valid. Hence the same emf must be induced in both situations shown in Fig. 371. As we saw in Chapter 29 this is indeed the case so Faradays aw is consistent with the principle of relativity Indeed ll o the laws of electromagnetism are the same in every inertial frame o reerence. Equaly signicant is the prediction o the speed o eectromagnetic radiation, derived rom Maxwell's equations (see Section 32.2). According to this analysis light and al other electromagnetic waves travel in vacuum with a constant speed now dened to equal exactlyis299,792458 m/s. useexact the approximate X 10 mis which with in one part within in(We 000often of the exact value.) As we wevalue wil c = 3.00 see the speed of ight in vacuum plays a central role in the theory of relativity Einsteins Second Postulate
During the 19th century most physicists beieved that light traveed through a hypothetica medium caled the ether just as sound waves travel through air. If so the speed o light measured by observers would depend on their motion rea tive to the ether and would thereore be dierent in dierent directions. he Micheson-Morley experiment described in Section 35.5 was an eort to detect motion of the earth relative to the ether. Einsteins conceptual leap was to recog nize that if Maxwells equations are valid in all ineria frames then the speed of ight in vacuum should also be the same in al frames and in all directions. In fact Michelson and Morley detected no ether motion across the earth, and the ether concept has been discarded Athough Einstein may not have known about this negative result it supported his bold hypothesis of the constancy o the speed of ight in vacuum. Es's sd psu ss: ss: d g uu d dd u
Lets think about what this means Suppose two observers measure the speed of light in vacuum One is at rest with respect to the ight source and the other is moving away from it Both are in inertial frames of reference According to the principle o relativity the two observes mus obtain the same result despite the fact that one is moving with respect to the other If this seems too easy consider the ollowing situation A spacecraft moving past the earth at 1000 m /s res a ssile straight ahead with a speed o 2000 m /s (reative to the you spacecraft) 37.2) What is problem the issiles speed reative the earth? Simple say; this(Fig is an elementary in reative velocityto(see Section 3.5). The coect answer according to Newtonian mechanics is 3000 m/s. But now suppose the spacecraft turns on a searchight pointing in the same direc tion in which the missile was red An observer on the spacecrat measures the
37.1 T w (a) v v v v .
CHPTER 37 Retivity
12 37.2 37.2
(a) New (a) Newto� to�iian mech echanic anicss ma makes kes cor corec ec p red p redcton ctonss about ab out elat elatvey vey sow sow-o -ovn vng g ob ob jects jects;; b b t makes akes c coec oectt pedc p edcttos os abo abot t te behavIOr of light.
(a) A
spaeship 5 movs with spd l m/s relati e an bseve n earh
A
miss M s fired wth speed I m/s llave h spa spaes eshi hip p.
b
A
gh beam s emited fm h spaceshp at spd .
Msse (
. m/s l /s + m/s �
NEWTONAN ECHANC HOD: HOD: Newia mehas es us rrey hat he mssie mves wih speed l m/s reative =
t th sver n earth eart h
5' �
igh bem L)
=
l
m/s
NEWTONAN ECHANC FA FA ewtnian mhanis tes us /ry tha the lght moves a sp sp d greae tha c elative o th sver n earth whh wd ntad Estei's sd pstae
speed of light emted by the searchligh and obains the value c. According to Einstein's second postlate, postlate, the moion of he light after has left l eft he sorce so rce cannot depend on the moton of the sorce. So he observer on earth who measres the speed of his same light mst also obtan he vale c, not c + ms This resl contradicts or elemenary notion of relative velocites and it may not appear o agree wth common sense. Bu "common sense is inuton based on everyday experience and this does no usaly inclde measremens of the speed of ight.
The Ultimate Speed Limit Einseins second postuate mmedaely mples the following reslt: p f v v
,
pd f v
We can prove hs by showng ha ravel a c mples a logical conradcion. Sppose hat the spacecraf S in Fig is moving a the speed of igh rela tive to an observer on he earth so that c. If he spacecraf rns on a head light the second postlate now asserts that the earth observer S measres the headlght beam o be also moving a c Thus this observer measres hat the head
uSls
lght beam and the spacecraft move together and are always at he same pont in space. Bt Einsteins second postulae also asserts ha he headligh beam moves at a speed c relative o the spacecraf so they cannot be at the same point in space This contradicory reslt can be avoded only f t s impossibe for an iner ial observer sch as a passenger on the spacecraf to move at c As we go through our discusson of relaivty you may nd yourself asking he quesion Ensten asked himself himself as a 16-year16-year-old old student "Wha "Wha woldI woldI see ifI were rav eling at the speed of lgh? Einsten realized only years later that his qesions basic aw was that he could not travel a c
The Galilean Coordinate Transformation Les restate ths argument symbolically usng wo inertia frames of reference labeled S for the observer on earh and S for the moving spacecrat as shown n Fig To keep things as simple as possible we have omtted the axes The -axes of the two frames lie along the same line b the orgin 0' of frame S moves relaive to the orign 0 of frame S wth constan velocty u along the com mon axs. We on earh se our clocks so that the two origins coincde a tme t so heir separation at a later ime t s ut
371 Invariance of hscal Laws
) 11 -----�p �p x
"y Oii ° d 0 de me,
0
The positon of particle P can be descrbed by the coordnates x and y in rame of eference S o by x' and in fame
x'
Frame 5' moves elave fm 5 with cn oy I ag e mm x x'x'-ai ai..
\"
0' -ut _ x
I
X'
aionss equaion se Many of the equ e wi wise rdia iae crd a frame frame c ierti ertia ur i se ur Chse sated in f ames ames as sate rence f ial refe reference inerial your iner dene yo re only only if you de chape are re in s ch derrived in de N CAUTION CAUTIO
on in in diecton the diect on mus ustt be the -diect cton stvee x -die the po postv instance nce,, the h. Fo insta aagrap raph. eding g paag the prec precedin ecion is is o the origin O. In Fg. 373 his di ecion he origin ves elav elavee to he mov hee oigin oigin 0' 0' mo whch h cion x-diecion hee posiiv posiivee x-die dene h must dene ef f reaiv reaivee to 0, you must movess to the e n sead 0 move igh; igh; if nsead to be o the eft.
Now think about how we descrbe the moton of a partcle . Ths mght be an exploratory vehcle aunched from the spacecraft o a puse of lght from a aser We can descrbe the position of ths partcle by usng the earth coordinates ( z ) n S or the spacecraft coordnates ( ' , ) n S. Fgure 373 shows that these are smply related by = x + ut
(Galean coordnate transformaton)
(371)
These equatons based on the famlar Newtonan notons of space and tme are called the Gallean coordinate transformaton If patcle moves n the -drecton ts nstantaneous veocty as meas dxldt, Its velocty ; as measured by ured by an observer statonary n S s an observer statonary n S s : = 'dt. We can derve a relatonshp between V and by takng the dervatve wth respect to t of the rst of Eqs (371) =
d d -= + u dt dt Now ldt s the velocity measured n S and dxdt is the velocty measured n S so we get the Gaiean veociy transformation for one-dmensonal moton v, = ;
+
u
(Galean velocty tansformat tansformaton) on)
2
.2
Although the notaton ders this result agrees with our dscusson of reative velocites n ection 35 Now heres the fundamental problem Appled to the speed of lght n vacuum c + u. Ensteins second postulate supported subse Eq. (372) says that c quenty by a wealth of expermenta evdence says that c = c. Ths s a genune nconsstency, not an iluson and t demands resoluton If we accept ths postu ate we are forced to conclde that Eqs (37) and (372) cannot be precisely correct despte our convncng dervaton. These equatons have to be moded to brng them nto harmony wth this princple The resoluton nvolves some very fundamental modcaons in our kne matc concepts The rst dea to be changed s the seemngly obvous assumpton that the the observers n frames S and S use the same time scae formally stated as t = t. Alas we are about to show that ths everyday assumpton cannot be cor
rectthe observers have derent scales. We must deneare thenot veloc ty n two frame S as ust = Idt not as tme ldt; the two quantites the same The diculty les n the concept of simutanei whch s our next topc A careful analyss of smutanety wll help us develop the appropriate modca tons of our notons about space and tme
122
CHPTER 37 Relativity s Your Undrsndng of Scon
As a highspeed spaceshp ies past you t res a srobe ight that sends out a pulse of ight in al directions An oserve aboard the spaceshp measures a spherca wave fron tha speads away from the space ship with the same speed c n al directions (a) What is the shape of the wave front hat you measure? i) spheical; (i) elpsoidal with the ongest axis of the elipsod aong the direction of the spaceships motion (iii) elipsodal wih the shotes axis of he elipsoid along he drecion of the spaceships moion iv) not enough infomation s given to decide ) Is he wave fron centeed on the spaceship
37.2 4 An event has a denite poston and tme-for insance, on he pavement direcly beow the center of he Eifel Tower at mdnght on New Year's ve.
Relativity of Simultaneity
Measuring times and time intervals invoves the concept of smlanet In a given frame of reference, an even is an occurrence that has a denite position and time (Fig 37.4). When you say that you awoke at seven 0' clock, you mean that two events (your awakening and your cock showing 7:00) occurred simultaneously The fundamenta probem in measuring time intervas is this In general, two events that are simutaneous in one frame of reference are not simultaneous in a second frame that is moving reative to the rst, even if both are inertia frames
A
Thught Experiment in Simultaneity
This may seem to be contrary to common sense. To ilustrate the point, here is a version of one of Einstein's thought exper iments-menta experiments that foow concepts to their ogica concusions. magine a train moving with a speed comparable to c, with uniform veocity (Fig. 37.5) Two ightning bolts strike a passenger car one near each end. Each bot eaves a mark on the car and one on the ground at the instant the bot hits. The points on the ground are labeed and in the gure and the corresponding points on the car are and . Stanley is stationary on the ground at 0 midway between and B. Mavis is moving with the t he train at 0' in the midde of the passenger car midway between and B . Both Stanley and Mavis see both ight ashes emitted from the points where the lightning strikes Suppose the two wave fronts from the ightning strikes reach Stanley at 0 simutaneousy. He knows that he is the same distance from and so Staney concudes that the two bolts struck B and simutaneousy. Mavis agrees that the two wave fronts reached Staney at the same time, but she disagrees that the ashes were emitted simutaneousy Staney and Mavis agree that the two wave fronts do not reach Mavis at the same time Mavis at 0' is moving to the right with the train so she runs into the wave front from before the wave front from catches up to her. However, because she is in the middle of the passenger car equidistant from and her observation is that both wave fronts took the same time to reach her because both moved the same distance at the same speed c. (Recal that the speed of each wave front with respect to either observer is c) Thus she concudes that the ightning bolt at B strck bo the one at ' . Stanley at 0 measures the two events to be simutaneous, but Mavs at 0 ' does not! Whether or not two events at d erent x-axis locations a simultaneous depends on the state of otion of the observe You may want to argue that in this example the ightning bots reay are simultaneous and that if Mavis at 0' could communicate with the distant points without the time deay caused by the nite speed of ight, she woud reaize this. But that would be erroneous; the nite speed of information transmission is not the rea issue. If 0' is midway between A and then in her frame of reference the time for a signal to trave from to 0' is the same as that from B to 0 ' . Two signas arrive simutaneously at 0 ony if they were emitted simutaneousy at and In this exampe they o not arrive simutaneousy at 0', and so Mavis must concude that the events at and B were not simutaneous.
Relativty f mutanety
1273
5 A thought experimen in smutaneity Ligig i t d back a rai (poi A ' ad ) and i pi A d S.
A ly o
Td t t s s mv oward it c rom frot of t t d wy rm t i t - back t t
b
•
A
o
(c)
Bcs Ms g ! t o t ! t lgig t o i i !.
•
A
d
•
A
Saley ss t lti t t two poi m tim: t iv tt t t t lit g i t ot d the back o t t simu imull aeou aeou ll y. y . A POiI, lt rom li i i t o i s y t eaced aviis av
Furthermore, there is no basis for saying that Stanley is right and Mavs is wrong or vice versa. Accordng to the prncple of relatvity no inertal frame of reerence is more correct than any other in the formulaton of physical aws Each observer is COect in his or her own fme of reference In other words simltaneity s not an absolte concept Whether two events are smultaneous depends on the frame of reference As we mentioned at the beginning of this section simltaneity plays an essental role n measuring time ntervals It ollows that the time interval between o eents may be dent dent in derent derent frames of reference. So our next task is to learn how to compare time intervals in dierent frames of reerence. Ts Yr Udsdg c Staney, who works for the rail system shown in Fig. , has carefuy synchronized the cocks at a of the ra stations At the moment that Staney measures a of he cocks striking noon Mavs is on a high-speed passenger car raveing from Ogden vie toward North Haverbrook. According to Mavs when the Ogdenvie Ogdenvie cock strike noon what ime is it n North Haverbrook? i) noon; (i before noon (iii) afer noon
24
CHAPE 37
3703
We can derive a quantitatve relatonship between tme intervals in derent coordinate systems. To do this, let's consder another thought experiment. As before, a frame of reference moves along the common -axis wth constant speed reative to a frame As dscussed in Section 37.1, must be less than the speed of lght Mavs who is ridng along with frame measures the tme nterval between two events that occur at the sme pont n space Event 1 s
t
Phys CS
Relativity of Time Intervals
Rlivy fT
u
u
c.
when ash light been fromreected a light source Event d2 away, is when the ash returnsa to of havng from a leaves mirror a distance as shown n Fg 376a. We label the tme interva Lto using the subscrpt zero as a reminder that the apparatus is at rest, wth ero velocty in frame The ash of ght moves a tota distance 2d so the tme interva s
Lto
2d
373
-
c
The roundtrp time measured by Stanley in frame is a different interval Lt; n his frame of reference the two events occur at d erent points n space. During the time Lt the source moves relative to a dstance Lt (Fig. 37.6b) In the roundtrp distance is 2d perpendicular to the relatve veocty but the roundtrp distance in is the longer distance 2/ where
u
In writing this expresson, we have assumed that both observers measure the same dstance d. We wll justify ths assumpton n the next section. The speed of ight is the same for both observers so the round-trip time measured n is
4 We woud ike to have a relatonship between Lt and Lto that is independent of d To get ths, we sove Eq (373 for d and substtute the resut into Eq. (37.4), obtanng
-c � c (u) 2
t
=
Lto + 2
Lt 2
(37.5)
6 (a Mavis, in frame of reference S', observes a light pulse emitted from a souce at 0' and reected back along the same lne. b Ho� Stanley (in frame of reference S) and Mavis observe the same light pulse The positions of 0 at the times of departure and eturn of the pulse are shown
b
Mvs beve ligh pue eed fo sue 0 d eee k lg he e le e..
Mirr
Souce 0'
Mvs meue e interal O
? Snley see
Sey ee loge e el : Te gh pule a e pee n S, u ve gee dsne h i S S..
o he e Lg pue fllwg ign p
37 Relativty Tm Intervas Now we square this and solve for I t
25
It; the result is to to =
-
VI
U
:2
2
C
Snce the quantity - u2/ c is ess than t is greater than to: to: Thus Stanley measures a lnger round-trip time for the light puse than does Mavis.
Time Dilation We may generalize this important result. In a particuar frame of reference sup pose hat two events occur at the same point n space The time nterval between these events as measured by an observer a rest n this same fame (which we cal the rest fme of this observer) is to to Then an observer in a second frame moving with constant speed u relative to the rest frame will measure the time interva to be t, t, where (tme dilation)
We recal that no inerial observer can travel at u = c and we note that VI - / s imaginary for u > c Thus Eq gives sensible results only when u < c The denominator of Eq is always smaler than , so t is always arger than to to Thus we call this eect im ln Think of an old-fashioned penduum clock that has one second between ticks as measured by Mavis n the cock's rest frame; this is to to If the cock's rest frame is moving reatve to Stanley he measures a time between ticks t that is longer than one second In brief bservers measure any clck t run slw it mves relative t them (Fig Note that ths conclusion is a drect resut of the fact that the speed of light n vacuum is the same in both frames of reference The quantity V - / n Eq appears so often in relativity that t is given ts own symbol (the Greek letter gamma:
This image shows an explodg star cale a uv wth a sant galay. he brghtess of a typca superova decays a a certa rae. Bt speovae hat are movg away rom s a a sbstantia racion racion o o he speed of ligh decay more slowly, accordance wih Eq. (37.6) he ecayig sperova s a mong "clock that rs slow
In terms of this symbol we can express the time dlation formula Eq. as
t 1to
(time dilaton
(378)
As a further siplication u / c is sometimes given the symbol { (the Greek letter beta then l /� Figure shows a graph of as a function of the relative speed u of two frames of reference. When u s very small compared to c / is much smaler than and is very nearly equal to . In that imit Eqs and approach the Newtonian relatonship t to corresponding to the same time interval in all frames of reference If the relatve speed u is great enough that is appecaby greater than 1, the speed is said to be relativistic if the derence between and is negigibly small, the speed u is called nn nnlativistic. lativistic. Thus u 6.00 X 0 m/ s 0.200c (for which 02 is a relativistic speed but 6.00 X 04 m /s 0.000200c (for which 00000002 is a noeativistic speed
Prope Time There is only one rame of reference in which a clock is at est and thee are in nitely many n which it is moving Therefore the time interval measured between two events (such as two ticks of the cock that occur at he same point in a
8 he aty 1/ U2 C2 as a co co of the relatve spee u o two frames o reerece. As sped LI apprche the pd of g pprache y y
6 4
"=
� l l22 2
' .
. . . .
•
•
1 1 1 1 1 1 1 1 1 I
-= - - - __ __
o
0
0 Speed L
00
6
HR 37 Relativi
particular frame is a more fundamental quantty than the interval between events at dierent points We use the term propr m to descrbe the time nterval between two events that occur at the same point CAUTON
9 9 A ame of reerece picured as a coordae system wh a grd o sychro ized clocks.
I
-'88 88 8 8 8 I
-8 x I >F
·· .
The grid is three dmenoa; identcal pae f k e i rn d behid he page, cnected y grd lie peredur he age a ge..
Mea easu surin ring g tim time e int inter erva vals ls I is mp mpor ortan tan o o oee h ha a he he i ime me erv ervaal t i Eq E q. (3 (37 7.6) v vove ovess eve evens ns ha ha occu occurr at at d dfr fr ent spa space poi oints nts i i h hee fram framee of of refe refere rece ce S. Noe aso No aso ha a ay y di di erece erecess betw be tween een t an and d he pr proper oper me me to to are are nt caused used by d f fere rences n n he he mes mes re requr qured ed or or l lgh gh o o trav traveel rom hos rom hosee space space po pos o a a os oseerver a a rest rest n S. n S. We We ass assum umee ha ha our our obse o bserve rverr is abl ablee o cor correc rec for for d d erence erencess in g in gh h trans ranst t ime imes, s, jus jus as an asron asronomer who's who's ob observi servig g he su sun under undersands sands ha a a eve eve see owobserv o eah actuallly occ actua occred 500 500 s ago ago o he he su's su surface. rface. A Alea leaively we can use two observe ers, one s saonary aonary a a the locaio locaio of of he he rs rs even an and he oher oher a he he secod secod,, each wi with hs or or her ow ow clock. W e can synchro synchroze hese hese wo cloc clocks who whou dculy dculy as log log as hey hey are a res in res in the the same same frame of referec referece. For exa example, we we could se send a lgh lgh puse sm smaneously ou sly to the two clocks clocks from a a po poit it mdway mdway betwee betwee them. W hen he n the pulse pulsess arrve, arrve, the the observ ob servers ers se their their cocks cocks o o a prearra prearraged ged ime. (Bu (Bu note note hat hat clocks clocks ha ha are synch are synchroiz roized ed one rame o reerence are nt general synchroned any oher rame)
In thought experments, it's ofen helpfu to imagine many observers with synchronized clocks at rest at various points in a particular frame of reference We can picture a frame of reference as a coordinate grid with ots of synchronzed clocks distributed around it, as suggested by Fg379 Only when a cock is moving relative to a given frame of reference do we have to watch for ambiguities of synchronization or simultanety. Throughout this chapter we will frequently use phrases like "Stanley observes that Mavis passes the point 5.00 m y 0, 0 at time 200 s hs means that Stanley s using a grd of clocks in his frame of reference, like the grid shown in Fig 379 to record the time of an event We could restate the phrase as "When Mavis passes the point at x = 5.00 m = 0 Z = 0 the clock at that ocation in Stanleys frame of reference reads 2.00 s We wi avoid using phrases ike "Stanley sees that Mavis is a certain point at a certain time, because there is a time deay for light to travel to Stanleys eye from the position of an event
Time Dilation DEFY the relevant ncepts:
The concep o me dlao s used whenever we compare he me nervals beween evens as measred by observers n deren neral rames o reerece
SE P the prblem usng the ollowg seps: 1 o descrbe a me nerval, you mus rs decde wha wo evens dene he begnnng and he end o he nerva You mus also deniy he wo rames o reerence n whch he me ierva s measured. Determe wha he arge varable s.
EEE the lutin as ollows ] I may proems volvng me dlao he me nerval bewee eves as measured n one rame o reerence s he
prper ime to to The proper me s he me erval ewee wo eves n a rame o reerence n whch he wo eves occur a he same pon n space The dlaed tme t s he longer me erval bewee he same wo eves as measured a ame o reerence ha has a speed u relave o he rs rame. The wo evens occur a dere pos as measured n he second rame You wll need o decde whch rame he me nerval s to ad n whch rame s t t and t, t, and he solve or 2. Use Eq (376) or (378) o relae t he arge varable
EVLE yur anwer:
to, Noe ha M s never smaller ha to, ad u s never greaer ha your resus sugges oherwse you need o rehnk yor calculao -
Time dition at O99c Hgh-eergy suaomc partcles comig rom space eract wth atom the earh's upper amosphere prodcng usabe part ce led n A muon decays wh a mea feime o 0 0- as measred n a frame o reerece which i s a rest f a muon s moving a 0.990c (abou 97 X 108 m / s) relave o he earth wha will you (a observer o earh) measue s mea lietme to be?
j
DEFY: hs prolem conces the muons eme whch is he me erval beween wo evens he producion o the muo ad is subseqen decay hs leme s measured by wo der e oservers: one who observes he mo at res and anoher (yo) who observes i movg at 0990c
.
33 Rlavy of m ras ras
177
ET UP: Let be you fame o refeence on eath, and le be
EVALUATE Or resu predicts tha the mean etme of he muon
he muon's frame o reerence. The arget variable s he nerva beween hese events as measured n
in the earh ame s about seven imes onge than n the muons rame to. Ths prediction has been vered experimen ay; ndeed i was the rst experimenta conmation o the ime dilaion fomua, Eq 36
EEUTE: The tme interva between the wo events as measued n 220 X 10-6 s, is a proer tme since the two evens occur at he
same position relative to the muon Hence Ho = 220 X 106 s. The muon moves eatve to the earth between he two events, so the wo events occu at different postions as measued in and he time nterva in that rame is t (the targe varabe) From Eq 376
_?· = 2 X I-6 S 0.990) \/
Examl
=
56 6X X 10 s -
Time dilation at jetliner speeds
An aane ies rom San Fancisco to New York (about 4800 k, o 480 X 10 m) a a steady speed o 300 m/ s (about 60 mi/h) How much ime does the rip take as measued by an obseve on the ground? By an observer in the pane?
,
DENTFY Here we are nterested n what our two observers DENTFY
measure o the time interval between the apane depating om San rancisco and andng in New York.
ET UP The arget variables ae the ime ntervas beween hese evens as measued in the rame o reerence o he ground and n he ame of reference of the aiane S'
om Eq. 376,
=
100 00X 0 160 X 104s)\ 1
The radica cant be evauaed with adequate pecision wih an ordinay caculato. But we can approxmae it using he bnomal theorem (see Appendx B):
(J - 00 X )
=
I
(100 X 0) + ...
(�)) (�
The emaining erms are o he orde o 0 or smaer and can be dscaded The approximae resut or o s
to
=
050 X 0) 60 X 0s ) (I (I
EEUTE The wo evens occu at dieren postons San ran cisco and ew York) as measured n so the ime interval meas ured by ground observes coresponds o in Eq 376 To nd
The proper ime measured n the airplane s very sighy ess (by ess than one pat in 10 than he ime measured on the ground.
we simpy divde the distance by he speed:
EVAUATE We dont noice such eects in eveyday ie But
/
=
480 X 10 m 300 m/ s
=
60 X 10 S
(about 4 hos)
In the arplanes rame San Fancsco and New York passing under he pane occur a the same poin (he position of the plane). The tme inerva in the airplane is a proper time, coresponding o n Eq 36 We have s2 (300 m/ s2 (3.00 X 0 m/s) m/s)
pesen-day atomic cocks (see Section 3 can attain a precision o about one part in 10 A cesium cock aveing a ong dsance n an ariner has been used to measure hs efec and hereby ve y Eq 36 even at speeds much ess han c.
00 X 0 -
Exam
Just when is it proper?
Mavis boards a spaceshp and then zips past Saney on eah at a eatve speed of 000c At the insant she passes, both start tmers. (a) At he instant when Staney measures that Mavis has aveled 00 X 107 m past him what does Maviss timer read? (b) At the insan when Mavis eads 0.400 s on her ime, what does Staney ead on hs?
·t
are when Mavis passes Stanley and when Saney measues Mavis as havng aveed a distance o 900 X 107 m the arge variabes ae the tme inervals between these two evens as measued in and in The wo events in part (b) are when Mavis passes Staney and when Mavs measures an eapsed ime o 0400 s the taget variable is the time inerval between these wo events as measued n S As we wil see understanding his exampe hnges on undesanding the diference between these two pars o evens
DENTY This pobem nvoves ime daton or wo di dirnt rnt
EEUTE a) The wo events, Mavis passing the earth and Mavis
sets of events the sarting and stopping o Mavss tmer and the starting and stopping of Staneys timer. ET UP: Let be Staneys rame o eerence, and et be Mavss ame o eerence. The two evens o nteest in pat (a)
eaching x = 900 X 107 m as measured by Staney, occur at dier en postons n Saneys rame but at he same poston in Mavs's rame ence the time interva in Saney ame S is !t while the time interva in Mavis's ame S is the prope time As measued Cninued
128
CHPR 37 Reliviy
by Stanley, Mavis moves at 0.600 0.600(300 08 ms) .0 0 m/s and traves the 900 07 m n a ime 6r (900 X 07 m/( 0 X 0 m/s 0500 s From Eq 36 Mavis's timer reads an eapsed time of
1 tYI - u2 /C2 0.00 sY - (0 (06 600 00 0400 s b I is emptingbut wrongto answer that Staneys tmer reads 000 s. We are now consdering a dier pair of evens the starting and the reading of Staneys timer hat boh occ at the same point in Staney' earh frame S. These two evens occ a dierent positions n Mavis's frame S', so the ime interva of 000 s that she measures between these evens s eua to 6t.
In her frame Staney passes her a time zero and is a distance behind her of ( I 0 0 m/s(000 s) 20 X 07 m a ime 0.400 s The ime on Saneys timer is now the t he proper time: =
Mo
tYI -
U /2
0 400 sY sY11 - (0 (06 600 00
0.30 0.3 0 s
diference between 0.00 s and 0320 s sti tro EVALUATE: I the diference bes you consder the foowing: Staney taking proper account of the transit tme of a signa from x = 900 07 m says that Mavis passed that point and his timer read 0500 s at the same insant Bu Mavs says ha those wo events occurred at dierent positions and were nt simtaneosshe passed the point at the insant his tmer read 030 s. his exampe shows the reaivty of simutaneity
The Twin Paadox Equatons (376) and (37.8 fo tme dlaton suggest an apparent paradox called the w paradx Conside dentcal twn astonauts named Eatha and Astd Eartha emans on eath whle he twn Astd takes o on a hgh-speed tp though the galaxy Because of tme dlaton, Eatha observes Astrid's heatbeat and al othe e pocesses poceedng moe slowly than he own Ths to Eatha, Astid ages moe slowly; when Astd etus to eath she s yonge has aged less than Eatha Now here s the paadox: Al netal ames ae equvalent Can't Astid make exacty the same aguments to conclude that Eartha s n fact the younge? Then each twn measues the other to be younge when theye back togethe, and that's a paadox To esolve the paadox, we recognze that the twns ae identcal n all respects Whle Eatha emans in an appoxmatey netal ae at al tmes Astd must cc wth espect to that inetial fame dring pats o he tp n oder to leave tun aound and etun to eath. Eatha's refeence fae s always appoximately netal Astd's s oten far rom netal Ths thee s a eal phys cal deence between the ccumstances o the two twns Cae analyss shows that Eatha s coect when Astd etus she s younge than Eatha
standi ng on the Ts You Us of Sco Samir who is standing ground starts his sopwach at he nstant tha Maria es past him in her space ship at a speed of 0600 same instant Maria starts her a As at the measured n Samirs frameAtofhe reference what is the readng onstopwatch Marias sopwatch instant that Samirs sopwach reads I 00 s? 0.0 s; ii ess than 0.0 s; ii more than 0.0 s b As meased in Maras frame of reference what is the reading on Samirs Sa mirs sopwatch at the instan tha Marias stopwatch reads 00 s? i 00 s ii ess than 00 s iii ii i more than than 00 s
374
v
t
Physc
Rlatvy f Lngh
Relativit of Lengt
Not only does the tme nterval between two events depend on the observes fame of reeence but the sc between two ponts may also depend on the obsever's rame of refeence. The concept of smutanety s involved Suppose you want to measue the length of a movng ca One way s to have two asss tants make maks on the paveent at the postons of the font and ea bumpes Then you measue the dstance between the maks But you assstants have to make the maks h sm m I one maks the postion of the font bumpe at one tme and the othe maks the poston of the ea bumpe hal a second late you wont get the ca's tue ength Snce weve leaned that sultanety sn't an absolute concept we have to poceed wth cauton
34 Relivy of Legh
279
Lengths Parallel to the Relative Moton To develop a relationship beween lengths tha are measured parallel to he direction of motion n various coordnate systems, we consider another thought exper iment. We attach a ight source to one end of a ruler and a mirror o the other end. The ruler is at rest in reference frame ' and its length in this frame is o (Fg 37 lOa). Then the time o required for a light pulse to make he round trip from source to mirror and back s
2o o
=
c
(37. (3 7.9)
This is a proper time interval because departure and retu occur at the same point in In reference frame the ruler is moving to the rght wih speed durng his travel of the ight pulse (Fig 37Ob) The length of the ruler n is I, and the time of travel from source to mirror as measured in is [ Durng this interval the ruer with sorce and mirror attached moves a distance [ The toal length of path d from source o mirror is not but rather
30 The light pulse travels wth speed , so t is also true that
371 Combining Eqs. (37.1) and (37 l) to elimnate d we nd
c
-
31
Dividng the dstance I by c does n mean that lght travels with speed c but raher tha the disance the pulse travels in is greater than t. n the same way we can show that the time for the return trip from mirror to source is -
-?
(a)
=
c
+
3m
aVi
So urce
io
� l
The ler is sttnay n Mvs Mv s's fme eeence .. The lght puse tav tav ls sne 0 m he lgh surce th mir
r
/
The rle ms t peed u Ses ae eeee The gh puse raes a stce 1 (e ength e e measured S) pu a addnl dsane I �([ m e ght source te mrr.
A ruler is at es in Mavis's fame 5 A light plse is emjtted from a source at one end of the ruler, reected by a mrror at the othe end and retuned o the sorce position. () Motion of the lght pulse as measured n Stanleys rame 5
28
C HAPTE R 37 Relativy The
total It I, +It +It2 It c-u+c+u c(lu/c2) It It It time
for the round trip, as measured n S s
=
l_
l_
37.14
We also know that and are reated by Eq. (376) because is a proper time in S Thus EQ (379) for the round-trip time in the rest frame S of the ruler becomes
Finally combning Eqs (3714) and (375) to eliminate obtain
l lR 2 l
1 -? c y
l
It
(5)
and simpifyng we
(ength contraction)
-u2/c2
6
shorter
l
[We have used the quantity y lvl dened in Eq (377)] Thus the length measure measured d in S in which the ruer is movng is than the ength measured in its rest frame S UION
L s his is not an optca usion! The rer reay s shorer in reference rame than t is in ' A ength measured in the frame n whch the body is at rest (the rest frame of the body) s called a prpr gh; thus L is a proper ength n S and the length measured in any other frame moving reatve to S is This eect is caled gh r When is very smal in comparison to y approaches Thus in the imt of sma speeds we approach the Newtonan reatonship L = This and the corre sponding result for time diation show that Eqs (37) the Gailean coordnate transformation are usualy sufciently accurate for relative speeds much smaler than f is a reasonable fraction of however the quantity V can be appreciaby less than . Then can be substantially smaler than and the eects of ength contraction can be substantia (Fig 37.1).
u
The speed at whic electrons traverse he k beam ine of the Stanfod Linea Acceerator Center s sower han c by ess than 1 cm/s. As measured in the reference frame of such an eectron, the beam ine (whch exends from the top o he bottom of his photograph) is ony about 5 cm ong'
Beam lie
c
c u
less than l. l
c l
-u2/C2
Lengths Perpendicular to the Relative Motion We have derived Eq. (3716) for engths measured n the drection to the relative motion of the two frames of reference engths that are measued r to the drection of moton are contracted To prove ths consider two dentical meter sticks One stick is at rest in frame S and ies along the pos tive -axis with one end at 0, the origin of S The other is at rest in frame S and lies aong the positive -ais with one end at 0 the origin of S Frame S moves in the positive direction relative to frame S Observers taney and Mavs at rest in Sand S respectively station themselves at the 50-cm mark of ther stcks At the instant the two orins coce te two sticks e aong he same line At ths instant Mavs makes a mark on taney' taney ' stick at the point that t hat coincides with her own 50-cm mark and tanley does the same to Maviss stick uppose for the sake of argument that taney observes Maviss stick as longer than his own Then the mark tanley makes on her stick s its center In that case Mavis will think tanley stick has become shorter since half of ts length coincides with than haf her sticks length length o Mavis observes movng stcks getting shorter and tanley observes them getting longer. But this mplies an an asymmetry between the two frames that contradicts the basc postulate of relativit relativityy that tells us al ineial frames are equivalent We conclude that consstency with the postuates of relativty relativty requires that both observers measure the rulers as having the ength even though to each observer one of them is stationary and the other is
endicular
not
less
same
parallel pe
below
8
C H A PT E R 37 Ravy
at rest We wnt to nd he length I mesured by observers on erh. From E 3716 f
= 10)
-
:=
(0.9 .990 902 400m)V - (0
rest o mease he length I, wo observers wh synchron synchronzed cocs coesure the postons of the two ends of the space p ltneosly n the earth's reerence frame, as sown n Fg. 373. ese wo measements will ppear smtaneous to n observer n e spaceshp)
56.4 m
VL Tis answer mkes sense: he spaceship s shorter n
frme n wch t s in moon thn n a frme n whch s a
-
Hw fr r r h bsrvrs? Te two observers menoned n Exmple 37 re 56.4 m aprt on he eJh How p does the t he spcesp spcesp crew mese them o be? . DNFY he two sets of observers are the same s n xm ple 37.4, bt now he dsnce beng mesred is te separaton beween the wo erh observers. S Te dstnce between the erth observers as mesued on rrr engh, snce he wo observers are at rest n the eart s r
ert rame (Thnk of lengt of ppe 564 m ong ha exends from 0 to O2 n Fg 37.13. s pipe s a res in he erth frme, o leng is a oer lengt he earh is movng relave to te spacesp a 0990c, so the spaceshp crew will mease a dsnce sorer thn 56.4 m beween he wo erth observers Te vale t tey mesre s or target varbe. U: W
V Ths nswer does sy that the crew measres ther
spcesp o be boh 400 m long and 796 m long he observers on earh measre he spceshp to hve contraced ength of 56.4 m becse tey are 56 m pa wen tey measre the pos pos ons of the ends a wh tey measre to be the sme nstant. (Vewed from te spaceshp frame, te observers do not measre hose postons smltaneously. Te crew hen measres he 56.4 m proper lengh to be contraced o 796 m Te key pon s that the measremens made n xmple 374 (n whch he erth observers mese te dsance between te ends of te spceshp) re dffer dffer ent from those made nbetween ths example (n observers) whch the spceshp crew meases the disnce the erth
= 5 m and = 0990c te length I hat the
crew bers mese s 1
=
lo� - < =
56.m)V (0 (09 990 90) )
7.96 m
-
Conceptual Exampe ..6 Mvig wh As ws stated n ample 37, mon hs, o averge, a proper fetme o 220 0 s and a dlted lfetme of 56 X 0 s n a frame n wch ts speed s 0990. Mlpyng consn speed by tme to nd dstnce gves 0.990(300 108 108m/s) m/s) 1-6 s = 653 m and 0.990(300 08 m/s) X (5.6 X 10 s = 630 m Interpre tese tw dstances 1 ') U wil ey wil they ers,, th ervers obsserv t ob .99 90c pt vess 0.9 on move age muon verage If an ver y cay n to deca hen nd he onee po ted d at on ete asur uree t to be cre meas pe, ampe, y. For exam wy m w 63 nt 463 point her poi noher t no laerr t - s lae 10.6 X 10 5.6 5 n nta and nd a mounta p of a to top th t t evel el wth d ev ed creee d be cr cou ud is is mon co
heen mov h ovee st str rg g t down o o dec ecaay t se se 4630 30 m belo low w.
owever n observer movn t an average mon wl say hat t trveled ny 653 I becse exsted or ony 0 0 s To show ha ha answ s copeey consstent
consder he montan. Te 630m dsance s s heght, a proper length n the drecton o moon. Relve to te observer travelng wth ths mon he mountain moves p 0990 wth the 4-m engh contrcted to
=
=
1
-
= C
630m)Vl - (0990)
653 m
hus we see tat lengt contracon s conssten wth tme da ton e same s true for an eecn movng t speed u n lnear cceerator (see Fg. 371 Compared o he vales measred by physicist alongsde acceerator an observer rdn along wthstandng te elecon wold he measre the cceeror's engt and the me to travel tat length to boh be shorter by a actor of V I - U2/C2
-
37 The L
How an Object M
v g
18
Near c Wuld Appea Appear
Let's hjnk a abou e viual appeaance of a movng ee-dimenonal ody. f we coud see e poon of all poin of e ody mulaneouly, i WGd appea o nk ony n he decion of moion. Bu e don't ee all e pon mulaneouly; lig fom poin fae fom u ake longe o reac u an doe lgh fom poin nea o u, o e ee e fae pon a e poion e ad eale me. Suppoe e ave a ecangula od fae paallel o e oodnae plane Wen e look endon a e cene of e cloe face of uc a od a e, e ee only a face. (ee e cene od n compue-geneaed Fg. 3714a) Bu en a od i moving pa u oad e ig a an appeciable facon of e peed of lig, e may ao ee i le ide becaue of e eale-ime eec u decied. Ta i e can ee ome poin a e couldn ee en e od a a e ecaue e od move ou of e ay of e ig ay fom oe poin o u. Conveely, ome ig a can ge o u en e od i a e i locked by e moving od. Becae of all al l e od in Fg. 37.4 and 37.l4c appea roaed and dioed.
4 Computer simuato of the appea ance of an aray of 2 rods wth sqae cross sectio The centeigores od is viewed end-on he smuaion coo chages in the aray caused by the Dopper eect see Section .6.
a Ara a s
•
Test Your Understnding of Section 4
A mniature spaceship is yng past you, movng horizotally a a substatia facton of the speed of lgh A a ceran insant, you obseve that the ose and ai of he spaceshjp agn exacly with the two ends of a meter stick hat you hold in your hands. Ran the folowig distaces n ode fo ongest to shotes: (i) he pope legth of the meter sic; i he proper length of the spaceshp ii the legth of he spaceshp measued in your frame of efeece (v) the ength of the mete stic meased n he spaceship's fame of efeece
•
•
•
b) Aa movn o a D
37.5 Th Lornz Transformaions In ecion 37. e dicued e Galilean coodnae anfomaion equaion, Eq. 37.). Tey elae e coodinae x z of a poin in fame of refeence o e coodinae x, Zl ) o e pon in a econd fame 5 Te econd ame move conan peed u elaive o 5 in e poive diecion along e common x-x-ai. Ti anfomaion alo aume a e ime cale i e ame in e o fame of efeence a epeed by e addiional elaionip t = . Ti Galilean anfomaon a e ave een, i valid only n e lm en u appoace zeo We ae no eady o deive moe geneal anfoma ion a ae conen e pnciple of elaviy Te moe geneal ela ionip ae called e
The rent Crdinate ransfrmatin Ou queion i i: Wen an even occu a pon x y z ) a ime , a oeved in a fame of efeence a ae e e coodinae x l ) and ime t of e even a oeved n a econd fame moving elave o 5 i con an peed u n e + xdecion? To deive e coodinae anfomaion e efe o Fig. 3715 (ne page) ic i e ame a Fig. 37.3 A efoe, e aume a e ogin concde a e niial me 0 . Ten in S e diance fom 0 o 0' a ime t i ill ut. e coordin te x i a pper length in S o in S i i conaced y e faco ( 3716). 6). u e diance diance x fom 0 o P, a een in /y \1 - U2 U2/C /C2 2 a n Eq. (371 S i no imply x
u + x a n e Galilean coodnae anfomaion u {2 377 X u + x I ?
M -
c-
I I
Arra mov o a Dg
(
_ -
j\
\ i \ 1 ' /N
CHPTER 37 Relativy
1284
5 A d , ' is h, so / d (x UI) =
-
y Fram m ovs relative t frae S w city u lng he cmmn x-. x-. ''-a as
0'
Org 0 a ond a me I
= =
s
x ' � I ------- . _-lP
T
y
Tle 1.e ent nt a/o];! a/o ];!(( j" �l �l is te ptm ;];] an t s easued • i the w (x z. n fr Sa San n
e
a
y. ) r
1 x 0 t /1 1-
-0+
_
Song hs quon for x' w obn
X
x - u
3718
Equon s p of h Lonz coodn nsformon; noh p s h quon gng t' n ms of x nd To obn hs w no ht h pncp of qus th form of h nsfomon fom S o S' b dnc o h fom o S Th on dnc s chng n h sgn of h oc componn u Thus fom Eq mus b u h
x'
R
-ut' + x
2
I -c
39
W now qu Eqs nd o mn hs gs us n quon fo ' n ms of x nd t W h gbc ds fo ou o wok ou h su s
'
- ux/ ux/c2 c2
30
V - U2 / C
As w dscssd prous nghs ppndcu o h dcon of moon no ffcd b h moon so ' y nd z' = . Cocng hs nsfomon quons w h
x' y' z'
y z
(Lonz coodnt nsfomon)
372
'
Ths quons h Lorenz coordinate ransformaion, h sc gnrzon of h Gn coodn nsfomon, Eqs nd t ' For us of u h ppoch zo h dcs n h dnomnos nd pproch , nd h ux/c2 rm ppochs zo. In hs m Eqs bcom dnc o Eqs ong wh t t' n gn hough boh h coodns nd m of n n n on fm dpnd on s coodns nd m n noh fm
Space and ime have become inerined; we can no longer say hat length and time hae absolute absol ute meanings ndependen of he .am .amee of reference Fo hs r son w f o m nd th hr dmnsons of spc coc s foudmnson n cd p nd w c (x y Z ) ogh h p of n n
37 he Lorentz asfomations
The Lorentz Velocity Transformati Transformation We can use Egs. to derive the relativistic eneaization of the Gaiean veocty transformation, Eg We consider ony one-dimensiona moton aon the axis and use the term "veocity as bein short for the -component of the veocity. Suppose that in a time dt a partice moves a distance dx, as meas ured in frame . We obtain the correspondin dstance dx' and time dt' in by takin dieentas of Egs. : =
dx' y(x - U t) dt' = =y(t y(t -u x/c2 We divide the rst eguation by the second and then divide the numerator and denomnator denomna tor of the resut by b y t to obtain
x' dt'
dx u t u dx c2 t
Now dx/dt is the veocity in and dx' / dt' is the veocty ; in so we nay obtain the reatvistic eneraizaton
Lorentz veocity transformaton
(3 22)
When u and are much smaer than c the denominator in Eg. approaches , and we approach the noneativistic esut � = -u. The oppo ste extreme is the case c; then we nd =
c -u uc/c22 uc/c
=-
x
]
_
c(-u/c) =c 1 -u/c
Ths says that anythin movin with veocity =c measured in S aso has veocty � c measured in despte the reative moton of the two frames So Eg. is consistent with Enstein's postuate that the the speed of iht in vac uum s the same in a inertia ames of reference =
The principe of reativity tes us there is no fundamental disinction between he two frames frames and . Thus the expression for for n terms of : must have the same form as Eg with V chaned to � and vice versa and the sin of u reversed. Cayn out these operations with Eg. we nd
=
+
ujc2
Lorentz veocity transformation
72)
This can aso be obtaned aebraicaUy by sovin Eg. for Both Egs. and are Lorentz veloci tnsfomation for one-dimensiona motion CAUTION cc c c cd cd Keep in mnd tha he Lorenz ransfoatio equations given by Eqs. (3721), (3.22 an (373 assme ta frame ' is movig n the positive x-diecio wt velociy u elative to fame . Yo shol always se p yor coordinate system to olow tis coveton
When u is ess than c the orentz v eoci y trnsformatons show us that a body movin with a speed ess than c in one frame of reference aways has a
85
* 3 7 6 Th Dr f f gn W
Exampe 88
1287
Rtiv vocitis
(a) A spaceship movng away from the earth wh speed 0.900e fires a robot space probe n he same recton as ts moion, with speed 0700e elave to the spaceshp Wha s he probe's veocity reatve o he earth? b) A scoutshp tries to catch p wh he spaceshp by raveng a 0950e reatve o the earth Wha s the veocty of the scoushp reatve o the spaceship?
the veocty of he probe reatve to S; the arget varabe n part b) s the veocty of the scoushp reatve o S.
EXECU (a) We are given he veocity of the pobe eatve to the spaceshp, v; = 0700e We se Eq veocy v, reatve to the earth:
v: u 0700e 0900e v = 1 uv:/e 1 + 0900e0700ee
i
IDENIFY:: Ths exampe uses he Lorenz velocity transformation IDENIFY E UP et the earth's rame of reerence be S and e he space shps frame of reference be S' (Fg 3716) The reative veocty of the two frames s
u
=
y
u, 090c
u
000c
u; 000 �
Scoutship
Robo spce --Spaceship 0 --------- - --x prbe
=
0982e
(b) We are given the veocy of the scoutshp reatve o he eah v 0950e We use Eq 3722) o determine ts veocy v elave to the spaceshp
0700e. The target varabe in par (a) is
The spaceshp, robot space probe, and scoutshp
(3723) to etemne ts
v' =
v - u
l
-
uvJe2 uvJe2
0950e - 090 0900e 0e _ 0345e 0.900e 0.900e 0950ec?
VLU Its nsructve o compare our results to what we wou have obtane had we se he Galean veocy transfor maon forma Eq 37.2) n part (a) we wou have found he pobe's veocty reatve o the eath o be v = v u 0.700e + 0900e = 1600e. Ths value is geater han the speed of ght an so must be incorect n part (b) we woud have fond the scosps veocy relative to the spaceship to be v = the greater reatvscay v -u v =0.95 0.950e - s 0900e 0900 e = seven 0.050etimes vaue 0.30e amost han thecorec nco 45e rec Galean vaue
Y Udd S 5
(a n frame S events P and P2 occur at the same , - an -coordnates but even P occurs befoe event P. n frame S whch event occurs rst? b) n fame S events P3 an P4 occu at the same tme l and he same - and -coornaes but event P3 occurs at a less posve -coornate than even P• n rame S' whch event occurs rst?
I
*376 The Doppler Effect
for Electromagnetic Waves An additional mportant consequence of relatvisc kinematics s the Dopper efect for eectromagnetc waves. In our previous discusson of the Doppler eect (see Secton 168) we quoted wthout proof the formua, Eq. 630) for the freqency sht that resuts from moton of a source of eectromagnetc waves rela tive to an observe We can now derive that resut Here's a statement of the problem A source of ght is movng wth constant speed toward Stanley who s statonay n an nertal frame (Fg 7l7) As measured n its rest frame the source emts light waves wth frequency fa and perod T Il. What s the frequency f of these waves as receved by Stanley? Mving sorce emi waves of reqcy fo fo Frs wve t mtd ere.
Soce el scond sco nd wave e hr
:= I _� - �
Potio o rst wv e h i h h cn rt emit emit Soay eer eet e eecyI > l b. taley
A
The Dopper eect for lght A gh source movng a speed . reatve to San ey emts a wave crest then traves a dis ance uT toward an observer and emts the next crest n Staneys reference rame S, the secon crest s a dstance behnd the rst cres
Rlivic Monu
Example 37.9
A jet
1289
from a black hole
A number of galaxies have spemassive black hoes at heir cen tes (see Secton 12.8) As maea swis aound such a back hole, it is heated becomes ionized and generates strong mag netc eds. The esing magneic foces see some of he mateia into hgh-speed jets tha bast ot of the gaaxy and into integaactic space (Figre 3719) The be ight we obseve from he e in Fg. 3.19 has a eqency of 666 X 04 Hz, bt in the fame of efeence of the e materia he igh has a fe qency of 5.55 X 10 H (in the infaed egion o he eeco 37 This image shows a fastmoving et 5000 ight-yeas in engh emanating from he center of he gaaxy M87. The igh rom the et is emted by fast-moving eectons spiraing aond magnetc eld ines (see Fg. 2.16)
magnetic spectum) At wha speed is the et mateia movng owad s?
i
DENTFY: Ths pobem nvoves he Doppe efec for eecro magnetc waves ET UP: The fequency we obseve is f 6 6 66 X 104 Hz and the feqency in the fame of the souce is fa= 555 X 10 Hz Snce f > fa the souce is appoaching us and therefore we se Eq (3.25) to nd he taget variabe u.
EXECUTE: We need to sove q. 37.25) fo u. That takes a ite ageba; we' eave t as an execise fo yo o show ha the est is
f fo l l - e If + 66 666 X LO Hz /555 X 10Hz Hz u=
We have fifo we nd
=
u=
20
so
12.0 - I = 0.986e : ) -e 12. 12 .0 +
EVALUATE Becase the feqency shif s qie sbstantia woud have been eroneos o use the appoximate expresson f fl l ule Had yo ied o do so yo wod have fond c = c6 c6.. 66 X O H 5.55 0Hz/5.55 X 0Hz 0Hz= 110c This rest cannot be coec because the et matea canno rave faster than ight
37.7 Rivisi Mnu Newton's laws of motio have the same form al ertal fames of eference Whe we use transformatos to change from one nertal frame to aother, the laws should be invariant (uchagng) But we have just learned that the prcple of elatvty forces us to replace the Gallean transformatons wth the moe genera Lorentz trasformatos As we wl see ths requres correspondg genealzatons the laws of moton and the dentos of mometum and eergy. The prncple of conservaton of mometum states that whn two bodis intract, th total momntu is constant povded that the et external force actg o the bodes an erta refeece frame s zero for exampe f they form an so lated system teacting only wth each other). If conservaton of momentum s a vald physcal aw t must be vald in ll n erti al frames of referece Now hees the problem: Suppose we look at a collson n one nertal coordnate system S and d that momentum s coserved hen we use the Lorentz trasformaton to obtan the veloctes n a secod iertal system S'. We d that f we use the
Newtonan deito of momentum e momentum s not conserved n m) the second system! If we are convced that the prncple of relatvty and the Lorentz transformaton ae corect the only way to save momentum coservato s to generalze the dfntion of momentum. We wont deve the correct elatvstc geeralzato of momentum but here s the result. Suppose we measue the mass of a partcle to be m whe t s
at rest reatve to s We often call
m
the rest mass We wll use the term
1290
H A PTE R 37 elativy
. Graph of the magnitude of he momentum of a partice of est mass as a unction o speed . Also shown is the Newtonan predction, whch ives corect resuts ony at speeds much ess than c
material patce for a partice tha has a nonzero rest mass. When such a pa cle has a velocity V, ts rsc momnum p s (reatvstic momentum)
Relativic momnu boms nfnt u poh
When the paticle's speed v is much less than c this s approximately equa to the Newtonan expression p = mv but n general the momentum is greater in mag ntude than m Fgure In fact, as v approaches c the momentum approaches innity
p 5/1c
I
3nlc
HAPPENS!� II I I I
:
DOESN''T DOESN
2
I
HAPPEN
4m
Relativity Newton's Second Law, and Relativistic Mass
mc I
' - -L
o
. .4c .6 .8
3727)
u
What about the relativistic generalization of Newtons second aw? n Newtonian mechancs the most general form of the second aw s
d F=-
Nwonn m inorly ds h momnum boms nnt oly i bome ii.
378)
dt
That s the net force F on a patce eqas the tme rate of change of ts momen tum Experiments show that ths rest s sti vad n reatvstic echancs, pro vided that we use the reatvstc momentum gven by Eq . That s, the relativisticay correct generaizaton of Newtons second aw s
3729) Becase momentum s no longer drecty proportiona to veocty, the rate of change of momentum s no longer drecty proportona to the acceeraton As a resut, constant foe does not cause constant acceeon. or exampe, when the net force and the velocty are both aong the -axs, Eq . gves
-
(F and along the same line)
3730)
where a is the acceeraton, also along the axis Solvng Eq . for the accel eration a gives
a=F 1_U 1_ U 32 17 C
We see that as a particle's speed ncreases the acceeraton caused by a given force contnuousy decreases As the speed approaches c the acceleraton approaches zero, no matter how great a force is appled Thus it s impossble to acceerate a patcle wth nonzero rest ass to a speed equal to or greater than c We again see that the speed of ight n vacm represents an utmate speed imit Equation . for relatvstc momentum is sometmes nterpreted to mean that a rapdy movng partce undergoes an ncrease n mass If te mass a zero veocity the rest mass) is denoted by 17, then the "relatvstc mass m s gven by
ndeed we moecues consder the of a contane), system of particles as rapdy movng when dealgas n moton a stationary the tota such rest mass of the system s the sum of the relatvistc masses of the partcles, not the sm of thei rest masses However, f bndy apped the concept of relativstc mass has its ptfals s Eq . shows, the relativistc generalizaton of Newtons second aw is not =
F
m, and we w show n Section . that the reatvstic knetic energy of
77 Ratiis Momnu
9
K = rel
a partcle s not � m 2. The use of relatvistc mass has ts supporters and detractors, some qute strong n the opnons. We will mostly deal wth ndvd ual patcles so we wll sdesep he controversy and use Eq as the gen eralzed denton of momentum wth m as a constant for each paicle ndependen of ts state of moon We wl use he abbrevaton
1
y V -V = 2 C C2 2 We used hs abbrevaton in Secton with u replaced by u, the relative speed of wo coordnate systems. Here u s the speed of a parcle n a partcua coord nate system-that s the speed of the partcle's rest ame wth respect o that system In terms of y Eqs and 0 become
p = ymv
3731 31 )
(elatvstic momentum)
v
(F and along he same lne)
3732)
In near accelerators (used n medcne as wel as nuclear and elementary partcle physics; see Fg the net foce F and the velocty v of he acceler ated patcle are along the same stragh lne But for much o the path n mos circular acceleatos the partce moves n unform ccuar motion a constan speed Then the ne force and velocty ae perpendcular, so he force can do no work on the parce and the kinetc energy and speed reman consant Thus the denomnator n Eq 9 is consant and we obtan
.
v
m a yma (F and perpendcular) 3733) 33) 1 (I _ V 2/C 2/C ) 2 Recal from Section 4 hat f he particle moves n a ccle the net foce and acceleraton are drected nward along the radus r and a /r /r.. What about the geneal case in whch F and v ae neither along the same line no pependicular? Then we can resolve the net orce F at any nstant nto com ponents paalel to and perpendcuar to V. The esultng acceeraton wll have cOesponding components obtaned fom Eqs and Because of the deent y3 and y factors the acceleaton components wll not be popor tonal to the net foce components That is unless the net foe on a relativistic particle is either alog the same lie as the partce's eoci or perpendicular perp endicular to it the t he etforce and acceletio vectors a not parael. parael.
=
F
Relatiistic dynamics of an electron
An electro res mass 911 10-3 kg charge 160 10- C) is moving opposie o an eecric eld of magiude 5.00 10 N/C Al other forces are egligible i compariso to he elecric eld force a) Fnd he magiudes of momentum ad of acceeraio a he isas whe v 00 090 ad 099 b) ind the coresponding acceeraios if a ne oce of he same magide is perpedicuar o he veocy.
E
I Oc,
. In addiion o he expessios from this sectio fo rel ativistic momeum and acceeraio, we need he relaionship beween elecric force ad elecric ed from Chaper 21 par a) we use 7.1 o determie he magnde of momentum and . o deermine he magnde of accel
DNTY: ST U
eatio due o a force along the same le as he velociy. pa b he orce is perpendcuar o the velocy so we se 7 o deermie he magiude of acceeration
g
EXECUTE: EXECUT E:
V VV//
(a) T o d bo boh h he m mag agiude iude of mome momem
magide of acceeraion we eed he values of'
nd h h
o each o e ach of h hee three sp speeds eeds W e nd ' ' = 1.00 1.002. 2.29, 29, ad ad 709. he he
vaues o he momentum magnude are
P (1.00(9 'W X 0-3 kg0010(00 08m/s = 7 X 1O- 1O-4kgm 4kgm/s /s a OOO .9911 911 10 10-- kg(090 kg(090(00 (00 X 10 m/s P2 .9 56 0 kg ° m/s at = 090
u
CHAPTER 37 Relatity
1292
P3 = (7.09)(9l1 0- kg)(099)(00 8 m/s) 19 X 102 g· m/s at U = 099c
he aeleraions at he wo higher speeds are smaller by a a tor o y
= \qE= (60 O-C)(500 x 0N/C) = 800 0-4 N From Eq 7 a = ym When u = O OO O O and y = 00,
hese aeeraions are arger than the orresponding ones in par a by ators o y2
From Chaper , he magniude of he ore on he eleron s
(I
=
? 800 0-4 88 0 /s (.00)3(9.1 X 0-3 kg) =
The aeleraions a the wo higher speeds are smaler by ators o y: hese las wo aeleraons are only8.% only8.% and 08%, respeivey o the vaues predied by nonrelavisi mehanis b rom E 7. a = jym if F and v are perpendiuar. When u = 00 I Oc and y = 00
=
800 0 N X 0 m/s2 = 88 (.00)(9. 0-3 kg)
EVALUATE: u ress n par a show ha a higher speeds, he relaivsi vaues o momenum dier more and more rom he nonrelaivis values ompued sing p = mu Noe hat he momenm a 099c s more han hree times as grea as a 090 beause of he inrease in he aor y ur resuls aso show that he aeleration drops o very uikly as u appoahes c A the Sanord Linear Aeleraor Cen ter an essentially onstant eleri ore s used o aelerate ele trons o a speed ony slighly less han c I he aeleraon were onsan as predied by Newtonian mehanis, hs speed would e aaned aer the elerons had traveled a mere 15 m In a eause of the derease o aeleraon wih speed, a pah lengh o km s needed
Aording o relativsti mehanis when you doube he speed o a parile, the magniude o its momenum inreases by i a ator o ; i a faor greaer han ; iii a ator between and that depends on he mass o the parle s Your Undrsndng of Scon
37.8 Relativistic Work and Energy When we developed the relationship between work and kinetic energy in Chapter 6, we used Newton's aws of motion When we generalze these laws according to the principe of reativity we need a corresponding generaization of the eqation for kinetc energy.
Relativistic Kinetic nerg We se the workenergy theorem beginning with the denition of work. When the net force and dispacement are n the same direction, the work done by that force is W f x. We sbstiute the epression for F from Eq. 0 the appicable relativistic version of Newton's second law. In moving a partce of rest mass m from pont X to point X2
(37.34) To derive the generazed epression or kinetic energy K as a uncton of speed we wold ike to convert ths to an integral on u. To do ths, rst remember that the kinetic energy of a particle eqals the net work done on i in moving it from rest to the speed u: K = W. Thus we let the speeds be zero at point X and at pont X' SO as not to confuse the variable of integraton with the na speed we change to Ux in Eq. That is, s the varying x-component of the velocity of the par ice as the net force accelerates it from rest to a speed We aso reaize that x and are the innitesima changes n X and respectivey n the time interval . Becase x/ and a we can rewrte a x in Eq. as ad x
-d x
,
=
d-
x
= A
v
v
7. Riv k d
93
Making these substitutions gives us
fv
K= W = 0 (1
mVrdv r _
� /e2)3 2
(37.35) 35)
We can evaluate this integal by a simple change of variabe; the nal result is (relativistic kinetic energy)
336) 36)
As v appoaches e, the kinetic energy approaches innty. Iq. is J coect, it must also approach the Newtonian expresson K = !mv2 when u • is much smaller than e (Fig. 21 To verify ths we expand the radical using the binomial theorem in the orm
Graph of the knec energy of a parce of es mass m as a function of parce speed u Aso shown s the Newtonan pedcto, whch gves correc reslts ony at spees ch ess tan c
(1+x)"= 1+nx+ +nx+n(nn(n- 1)x2/ 1)x2/+ + n ou case n
=
K
-� and x = -v2/e2 and we get
Relativsc knet knet energy beome fn s pphes
t
... . . �
HAPPN5!� Combning this with K =
(I
-
1)me2, we nd
u -1 m v2 -+ -+ K = +-+ e2 8 e4
(I
1 ? v4 u+ + + = u+ 2 8 e2
)?
(3737 37
When u is much smaller than e al the erms in the series inq inq except the rst are negigiby small and we obtain the Newtonian expression mv2
Rest Energy and E = mc Equation for the kinetic energy
o a moving partice incudes a term me2 VI - u2 u2/e /e22 that depends on the motion and a second energy term me2 that is independent o the motion t seems that the kinetic energy o a paricle is the dierence between some total energy E and an energy me2 that it has even when it is at rest. Thus we can ewriteq ewriteq
as (total energy o a partice)
(338
For a partice at rest K = 0), we see that E = me2 The energy me2 associated with rest mass m rather than motion is caled the res energ o the particle. here is in act direct experimental evidence hat rest energy eally does exst. he simplest exampe is the decay o a neutra pion. his s an unstable subatomc partcle o rest mass "; when decays t dsappears and electromagnetic adiation appears I a neutal pion has no kinetic energy beore its decay the tota enegy o the radiaton ater its decay is ound to equa exactly mTe2 In many other undamenta paticle transormations the sum o the rest masses o he patices changes. n every case there is a corresponding energy change consistent with the assumption o a rest energy me2 associated with a rest mass Histoically the principles o consevaton o mass and o energy devel oped quite independenty. The theory o relativty shows that they are actually two special cases o a singe boader conservaton principle the principl f
HAPPN
I 'mc-
?
�=
7
-
Newnn e ny red ne eergy eme eme nnte ny v eme n nn nee.
94
CHAPTR 37 tivit v f d y In some physical phenomena, nether the sum
Although the contro room of a nucear power pant is very compex, the physica principe whereby such a pant operates is a smpe one: Part of the rest energy of atomic nucei is converted to therma ther ma energy, energy, which in turn is used to produce steam to drive eectric generators.
of the rest asses of the partcles nor the total energy othe than est energy is separately conserved but there s a ore general conservaton principle: In an solated system when the sum of the rest masses changes thee s always a change n 1/ ties the total energy othe than the rest energy. This change s equa in magntude but opposite n sign to the change n the sum of the rest asses. This ore general mass-energy consevaton law s the fundamental pnciple nvolvedundergoes n the generaton ofapower through reactons. When a uraniu nucleus sson n nuclear reactonuclear the sum of the rest asses of the resuting fragents is h the rest ass of the parent nucleus. An aount of energy s eleased that equas the mass decrease Ultplied by Most of this energy can be used to produce steam to opeate turbnes for electric powe gener ators (Fg 22 We can also relate the total enegy E of a partcle (knetc enegy plus rest energy) directly to its oentu by cobining q. 2 for relatvistic oentu and Eq. 8 for total energy to elmnate the partcle's velocity. The simplest procedure s to rewrite these equatons n the followng fors: and Subtractng the second of these fro the st and reaangng we nd (total energy rest energy and moentum)
Agan we see that for a particle at rest = 0, E = quation 9 also suggests that a partcle may have energy and moen tum even when t has no rest mass In such a case = 0 and
E
(zero rest mass)
(37.40)
In fact zero rest mass partces do exst. Such patcles always tavel at the speed of ght in vacuu One exaple s the h the quantum of eectromagnetic radiation (to be discussed in Chapter 8 Photons ae etted and absorbed du ng changes of state of an atoic or nuclea system when the energy and momen tu of the syste change.
Example
nrgtic ctrons
(a) Find the rest energy of an eectron m = 0 0-31 kg q = - = 602 09 C in joues and n eectron volts (b) ind the speed of an eectron that has been acceerated by an electrc ed from rest, throgh a potentia increase of 200 V typca of TV picture tubes or of 500 MV a votae ray machne)
1
DNIFY: his proem uses the ideas of rest enery reatvstic
From the denition of the eectron vot in Section , 1 eV = 602 0-19 . Using tis we nd
m? =
1 5 X
eV eV 602 X 0-19 0 eV 05 eV
X
0-14
(b) In calclations sch as ths it is often convenient to wor with the uantity dened from the modied :
inetic enery, and (from Chapter eectrc potenta enery =
We usetothe to stated nd thetota restenergy energy and ndrelationship te speed that S Eq U ives me22the me
U a) Te rest enery s m2 = 0 01 kg 2 X = 1 X 01 J
Sovin this for u, we et
u=e 08 m/s
- ( l fy ) 2
The tota energy of the acceerated eectron is the sum of its rest enery m and the inetc enery V"a that it gains from te
37.
work done on i by the electic ed in moving rom poin a o point b: E=ym= ym= nC2 + eV'
or
V,)( y=I+c An eecron acceleraed hrough a potenia ncease of Vw=0.0 = 0.0 kV gans an amount of energy 00 keY, so fo this eecton we have 00 103 eV or 0511 06 eV = 109
y= + and u
c\ (/092 (/092= = 0. = 85 07 m/s
Newtonian Mechancs and Reltivity
1295
Repeaing the calcuaton for Vba =500 = 500 V we nd VbalnC 98 y= 08 and u 0.99. =
VALUATE These esuts make sense: With V V= = 00 kV he
added kinetc enegy of 00 keV keV is less than 4% of the res energy o 05 e eV and the nal speed is abou onefourth of the speed of light. With Vba = 500 V he added kinetc kinet c enegy of 500 eV is much greate than the rest energy and the speed is close o The eecron ha accel eaed fom rest through a poential increase of 500 V had a kneic energy of 500 e By convention we cal such an eec ton a "500eV electron A 5.00eV eecron has a rest energy of 0511 eV as do al electrons) a ineic energy of 500 eV and a toa energy of 5.5 e Be careful no to confuse hese d eent enegies. CUN
hee elec eees
A r Two protons each with M= M= 0- kg) are intiay mov ng wh equa speeds in opposite directions They coninue o exist
In hs colision the kinetic energy o wo proons is tans fomed into he rest energy of a new partice a pion
afe a headon collison tha). also produces a neua of mass 02 n = .40 =.40 g Fg f he proons andpion the pion ae at rest ater he colsion nd he nital speed of the potons Enegy s conserved in the colison
BFOR Proton
1
Proton
IDENTFY: This poblem uses he idea of relaivisic otal energy
whch is conserved in the collision ST UP We equate the unknown) toa enegy of he two poons before the collision to the combined rest energes of the two po tons and the pion ate the collsion. We then use E. 8) o solve for he speed of each proon EXECUT The oal energy of each proton before he colsion is yM By conservation o enegy (yM (yM
(M2 + 40 X 02 kg -- 0 y 1 + - + M ( 10 10 27 g
AFTR +
Pio 2.40 X
EVALUATE The iniial kineic energy o each proton s
so u
c\ ( fy2
0.0
- kneic I M2 energy ythe 00M2 The rest energy o= a proton is 98These eV so is ( (98 eV eV= 5 eV are " 5eV proons) ou can verify that the rest energy of the pion is wice this o 5 e All the kineic enegy lost in ths completey neastic colision is transformed into the rest energy of he pon
A proton s acceeraed fom res by a constan orce that always points in the drection of the paice's moton Compared to the amount of knetic energy tha he proon gains during he st meer o its travel how much kinetic energy does the proon gan durng one meer o tavel whle t s moving at 99% of he speed o ight? i ght? ) the same amoun; ii) a greater amount ii) a smaler amoun Test Your ndersanding of ection 8
I
37.9
108 kg)
Newtonian Mechanics and Relativity
The sweeping changes equied by the pinciple of elativity go to the vey oots o Newtonian mechaics, incdng the concepts o length and ime he euations o motion and the consevation principles. Ths it may appea that we
1296
C H A PTE R 37 Relativy
h dsrod h foundons on whch Nwonn mchncs s bu. In on sns hs s ru, h wonn formuon s s ccur whnr spds r sm n comprson wh h spd o gh n cuum. n such css m don, ngh conrcon, nd h modcons of h ws of moon r so sm h h h r unobs unobsrb rb. . In fc, r on of h prncps prncps of Nwonn mchncs sus s spc cs of h mor gnr rsc formuon. Th ws of Nwonn mchncs r no wng; h r incomplete. Th .4 Withou nformaton ro osde the spaceship, e asb'onat cannot dstingish situaton () fom sitaton (). (a) An astonau spcip
s
abo o drop he wa in a
apprximatey r mng of rsc mchncs. Th corrc whn spdscs r sm n compson o c, nd hr bcom xc corrc n h m whn spds pproch zro. Thus r dos no comp dsro h ws o wonn mchncs bu geneizes hm. won's ws rs on r sod bs of xprmn dnc, nd woud b r srng o dnc nw hor h s nconssn wh hs dnc. Ths s comon p n h dopmn of phsc hor. Whnr nw hor s n pr conc wh n odr, sbshd hor h nw mus d h sm prdcons s h od n rs n whch h od hor s suppord b xpr mn dnc. Er nw phsc hor mus pss hs s cd h d l
The General Theory of Reativity b In gavy avy-- fr ee pace. the no aeleae its s e wa wah h wa a a = g a it
() O e ear ' uae, e wah aeeae owwar a = a i he
oor
A hs pon w m sk whhr h spc hor of r gs h n word on mchncs or whhr further gnrzons r possb or ncssr. For xmp nr ms h occupd prgd poson n ou dscus son. Cn h prncp of r b xndd o nonnr rms s w? Hrs n xmp h usrs som mpcons of hs quson A su dn dcds o go or Ngr s wh ncosd n rg woodn box. Dur ng hr r f sh cn o ough h r nsd h box Sh dosn' f o h oor bcus boh sh nd h box r n fr f wh downwrd ccron of 9.8 m /s2. Bu n rn nrpron, from hr pon of w, s h sh dosn f o h oor bcus hr gron nrcon wh h rh hs suddn bn ud o. As ong s sh rmns n h box nd rmns n fr f sh cnno whhr sh s ndd n fr f or whhr h gron nrcon hs nshd. A smr probm occurs n spc son n orb round h rh. Objcs n h spc son seem o b wghss bu whou ookng ousd h s on hr s no w o drmn whhr gr hs bn ud o o whhr h son nd s conns r ccrng owrd h cnr of h rh. gur 3724 mks smr pon or spcshp h s no n r bu m b ccrng r o n nr rm o b rs on h rh's surc Ths consdrons form h bss o Ensns l y f lvy f w cnno dsngush xprmn bwn unform gron d prcur ocon nd unform ccrd rfrnc frm, hn hr cnno b n r dsncon bwn h wo. Pursung hs concp, w m r o rp rsn an gron d n rms of spc chrcrscs of h coordn ssm. Ths urns ou o rqur n mor swpng rsons of our spc-m concps hn dd h spc hor of r. n h gnr hor of r h gomrc proprs of spc r cd b h prsnc o mr (g. 3725 Th gnr hor of r hs pssd sr xprmn ss ncud ng hr proposd b Ensn. On s hs o do wh undrsndng h roon of h xs of h pn Mrcur's pc orb cd h precession of he periheion Th prhon s h pon o coss pproch o h sun.) A scond s concs h pprn bndng of gh rs from dsn srs whn h pss nr h sun Th hrd s s h gravitaona red sh h ncrs n w-
3. Newtonian echaics ad Relaviy
1297
5 A wo-dimensional representaio of cved space. We magne he space (a pane) as being isoe as shown by a masse objec (he sun) Ligh fom a isan sa so e oows he disoe sface on is way o he eah The dashe lne shows he ec ion fom which he gh appears o be comng The eec s gealy exaggeaed; o he sun, he maxmm deviaion is ony 000048°
length of ight proceeding outward from a massive source. Some detals of the general theory are more dfcult to test, but this theory has played a central role in investigations of the formation and evouton of stars black holes and studies of the evolution of the universe. The genera theory of reatvity may seem to be an exotic bit of knowedge with ]jttle practcal application In fact this theory plays an essential role in the global positoning system (GPS) which makes it possibe to determne your position on the earth's sace s ace to within a few meers sing a handheld recever (Fig. 3726 he heart of the GPS system is a collection of more than wo dozen satelites in very precise orbits. Each satelte emts carefully timed radio signas and a GPS receiver simultaneously detects the signals from several satel]jes. he recever then calculates the time delay between when each signal was emtted and when it was received and uses ths information o calculae the receivers position. o ensure the proper ting of the signals, its necessary to include cor rectionsto due the specia theoryasofwel reaivity the saelites arethe moving reatve thetoreceiver on earth as the (because general theory (because satel ltes are higher in the earth's gravitatonal ed than the receiver he corrections due to relativity are smal-less than one part in l0 9 -but are crucial to the superb precision of the GPS system
uses ado 6 Aobiing GPS eceie fom he GS saees osgnas dee mine s posiion. To accoun o he eecs of eaiviy he eceive mus be une o a sighy highe fequency (10 MHz han he fequency emed by he saees 09999995 M).
CHAPTER
SUMRY
Invariance of physical aws. simutaneity: simutaneity: All of the fundamental laws of physics have the same form in all inerial frames of reference. The speed of light in vacuum is the same in al inertial rames and is independent of the motion of the source Simultaneity is not an absolute concept; evens ha are simultaneos in one frame are not necessaiy simltaneous in a second frame moving relaive o he rst Time diation diation If wo events occur at the same space point n a particular frame of reference, the time interval between the events as measred in that frame is called a proper ime interval If this frame moves with consan velocity u relative to a second frame, the time interva between the events as observed in the second frame is onger than M his eect s called tme dilatin (Examples
)
=
1
S ight() � = " � /S'' - uL l s /S us"!s !s 100 mls =
c
�
V - =2: 2:/
37137.3)
Length contraction If two points are at rest in a partic lar frame of reference, the dstance L between the points as measured in that frame is caled a proper length f this frame moves with constant velocity u relative o a second frame and the distances are meas
� .
L beween red to the motion, the distance the poinsparallel as measured in the second frame is shorter than his effect is called length contraction See Exampes 374376) The Lorentz tansformations The orentz coordinae transformations relate the coordinates and time of an event in an inertial frame to to the coordinates and time of the same event as observed in a second nertal frame ' moving at velocity u relative to the rst For one dimensional moton a particles velocities VI in and in are related by the orentz velocity transformation See Examples and
377
x- ut = "(x ut) VI U2 U2/C /C22 y' = Y z' = Z t x/c c ux/ / d t " ( - ux VI - u u/ /
y
x' =
?
37.8.
F
r--
.
_
_
/If
p
=+ \
x O
' + x
he Doppe effect fo eectromagnetic waves he Doppler effect is he freqency shft in light from a souce due to the relative motion of source and observer For a surce moving toward the observer with speed u, Eq gives the received frequencyf in terms of he emie frequenyf a. See Example
3725)
f
uf u
Saionay b b d ligh q
u
Moving source emt ligh f fqy
> o·
1 "0 "0
379.
Reativistic momentum and enegy: enegy: For a particle of rest mass m moving with velocity the reativistic momenum s given by Eq or and the relaivisic kinetic energy K is given by Eq The
p
V, (3727) (373 3736
£ is The total energym the sum the inetic energy and the res energy totaof energy can also be expressed n erms of the magnitde of momentum p and rest mass m. See Exampes 3703712
=
Inv
VI / /
-
mv
K
Relasc kec 1 nergy , :
[
/ ESN
/ HAPPEN
Nwan ki nergy
�o- --
298
Answers t Tes Your Undersanding Qesins
199
N mchc h pcl h f lvy The special theory of anics ics echan nian mech of Newto New tonian ciples ples of pinci s. All the the pin chanic anics. ian me mech tonian n of New N ewton lizatio ation genera neraliz ty s a ge reativity reativi are present as limiting cases when all the speeds are small compared to e. Futher generalizaion to nclude nonnetial frames of reference and heir relationship to gravtatonal elds leads to the gen era theory of relativity
Key Terms
special theory of relativity, 6 principle of reativity 6 Gallean coordinate transformation simutaneity event, time dilaton poper tme, 6
twin paradox proper length 0 length contraction 0 Lorentz transformaions, spacetime spacetime coordnates rest mass,
Answe to Chapte Opening Question
? .
o. Whie the speed of light c s the utimate "speed limit for any partce, there is no upper limit on a particle's kinetic energy (see Fig 3721). As the speed approaches e a small increase in speed coresponds to a large increase in kinetic energy
Answes to est ou Understanding Questions 37 :
You too wil measre a sphercal wave front that epands at the same speed e n all diectons. This is a consequence of Einsteins second postulate he wave front that you measure is no centered on the current position of the spaceship; rather it is centered on the point P where the spaceship was located at the instant that it emitted the ight pulse For eam ple suppose he spaceshp is moving at speed e/ 2. When yor watch shows that a time I has elapsed since the pse of ligh was emtted your measrements will show that the wave font is a sphere of radis et cenered on P and hat the spaceship s a dis ance e/ 2 from P.
2 : n Mavss frame of reference the two events
(the gdenville clock striking noon and the North Haverbook clock striking noon) are not sinultaneous. Figure 375 shows tha the event toward the front of the ral car occurs rst. Since the rail car is moving toward North averbrook that clock struck noon before the one on gdenvile. gdenvile. So according to Mavis it is after noon in North Haverbrook 37 33 : he statement hat moving clocks rn slow refes to any cock that is moving relatve o an observer Maria and her stopwatch are moving relative to Sami, so Samir measures Maias stopwatch to be running slow and o have ticked of fewer seconds than hs own sopwatch Samr and hs topwach ae moving relative to Maria so she likewise meases Samis stopwatch to be runnng slow. Each observer's measuement is cor rect for his or her own frame of refeence BOlh observers conclude that a moving stopwatch runs sow This is consistent with the prin cipe of relativity (see Section 371), which states that the laws of physics are the same in all inertial fames of reference 374 : You measure the rest length of the stationary meer stick and the contraced length of he oving spaceshp to both be meter The rest length of the space ship s greate than the contracted ength that you mease and so
a =g
relativistic momentum, 0 total energy rest energy, cOIespondene principle 6 general theory of relativty, 6
must be greater than 1 mete. A miniature observer on board the spaceship would measure a contracted length for the meter stick of less than 1 meter. Note that in your rame of reference the nose and tail of the spaceship can simultaneously align with the two ends of he meter stick, since in yor frame of reference they have the same lengh of I meter. n he spaceships frame these two alignments cannot happen simutaneously because he mete stick s shorte han the spaceship Section 372 tells us that this shoudn't be a sur prse; two evens hat ae simultaneos to one obsever may not be simltaneous to a second observer moving relative to the rst one. 3 : > b 4 a) he last of Eqs. (3721) tels s the times of the two events in : I I ' = YI - ux Je and 1 2' = Y ( - UX/ e ) . n frame the two events occur at the same -coordinate so XI X and event P occurs before event P2 so ' 1, Hence you can see that < 1 and event PI happens before P2 in frame too hs says that if event P happens before P n a fame of reference where the two events occur at the same position hen P happens before P2 n any oher fame moving el ative to (b) n fame 5 the two events occr at deren coordinates such tha 3 X4 and events P3 and P4 occur at he ux))/e ) same time so 14 ence you can see that f) f)'' Y - ux s greater than ' Y 1 - uX/e ) so even P happens befoe P in frame This says that even though the two events are simuta neous in frame , they need not be simultaneous in a frame moving relative to . 37 7 : Equation (3727) tels us that the magnitude of momentum of a partice with mass /1 and speed is p V1 - /e. f inceases by a factor of 2 the nmera to m increases by a factor of 2 and the denominator V - /C decreases Hence p ncreases by a factor greate than 2. (Note that in ode o doble the speed, the initial speed must be less than e/2 hat's because he speed of light s the ltimate speed limi.) 378 : As the proton moves a disance the constant force of magnitude F does work W Fs and increases the kineic energy by an amount !K Fs This is true no matter what the speed of the proton before moving this distance hus the con stant force increases the proton's kinetic energy by the same amount during the rst meter of travel as dring any sbsequent meer of travel (ts tre that as the proton approaches the ultmate speed imt of e the increase in the protons peed is less and less with each subseqent meer of travel That's not what the quesion is asking however)
CHPTER Rltivy
PROBLEMS
o d wk mnphom mnphom
Discussion Questions Q37.1. You are standing on a tran platform watchng a hgh-spee train pass by. A igh insie one o the rain cars s tue on and then a ite later it is rne o a) Who can measre the proper time nterval or the raton o he lght: yo or a passenger on the train? ) Who can measre the proper engh o the train car yo or a passenger on he train? c) Who can measure he proper lengh o a sign atache o a post on the rain platorm, yo or a passen ger on the rain? In each case explain yor answer Q37.2 f simltaneity is no an absolute concept does ha mean tha we mst discard the concep o causality? event is to cause even ms occr rst s i possible that n some rames appears o be he case o B an in others, appears to be he cause of ? Explain Q373. A rocket is moving o the rgh at � the spee o light rea ive o he earh A ligh blb in he cener o a room inse he rocket sudenly s on Call the light hiting he ron en o the room event A an the ight hitting the ack o he room event (Fig 377) Which event occurs rst, or or are they simta neous as vewe y a) an asronat riding in he rocke and b) a person a rest on the earth?
Fgr Qeston Q373
how do massess paricles sch as photons and neurnos acqure his speed? Can't hey sart rom rest an accelerate? Explan Q37.11. he spee o light reaive o stil waer is 225 0 m/s the waer is moving past us he speed o light we measue depends on the speed o he waer Do hese acts violate Enseins secon posuae? Expain Q3712. hen a monochromatc lgh sorce moves owar an oserver ts wavelength appears to e shorter than the vae meas ured when he sorce s a res Does hs conract the hypohess that the spee o light is he same or al observers? Expan Q33 n prncple does a ho gas have more mass than the same gas when i is cold? Explain n pracice wod hs be a measa le eec? Explain Q314. Why do yo think he deveopmen o Newonan mechan ics preceded the more rened relativistc mechancs by so many years?
xercises ection Relativity of imutaneity 371 Suppose he two ghtning bots shown n g 375a are simulaneos to an observer on the train Show hat hey are not simulaneous to an observer on the grond hich ightnng srke does the gron observer measre o come rs?
ection Reatvity of Time Intervas 37.2. he posiive mon (f+, an nstable paricle lives on aver
speed o light were J 0 m /s instead o 3.00 08 m/s?
age 2.20 0-6 s measre in is own rame o reerence) eore decayng a) such a partcle s moving wth respec to the labo ratory with a speed of 0.900, what average ietme is measured n the laboraory? b) What average dstance measre n he abo raory oes the paice move beore ecayng? 373 How ast ms a rocket trave relaive to the earh so that me in the rocket "slows down to half is rae as measure by earth-based oservers? Do presen-day jet planes approach sch speeds?
Q37S. The average ife span in the Unied Sates is abou 70 years
374 A spaceship ies past Mars with a spee o 095c reave o
Q374 Wha o you hink would be erent in everyay lie i he
Does this mean that t is mpossible or an average person to travel a isance greater than 70 ighyears away rom the earh? A lightyear s the disance lght ravels n a year) Expan Q376. Yo are holding an eiptcal servng plater. How woud you nee to travel or the serving platter to appear rond to anoher observer? Q37.7 wo events occur at he same space pont n a paricuar inerial rame o reerence and are simltaneos n that frame s i possble ha hey may no be smulaneos n a eren nera rame? Explain Q37.8. A hgh-spee train passes a train platorm Larry s a pas senger on he train Aam s sanding on the train platorm an Davd s rng a bicyce towar he plaorm n he same directon as the train is travelng Compare the ength o a rain car as meas ure by Larry Aam and David Q39 he heory o relativiy sets an upper limit on the speed ha a paricle can have Are there aso lmits on the energy and momentum o a paricle? Expain Q37.0 A studen asserts that a material particle ms aways have a speed sower han ha o light an a massess particle ms aways move a exacly the speed of ight s she correc? so
he srace o the panet When he spaceship is rectly overhea a signal light on the Maran srace blnks on and then o An observer on Mars measres that the signal ligh was on or 75 0 fS a) Does the observer on Mars or he pilo on he spaceshp meas ure the proper tme? b) What is the raion o the ligh plse measre by the plo o he spaceship? 37.5 he negaive pion ( 7- ) is an nsae parice with an aver age letime of 260 0- s measred in the rest frame o the pon a) I he pon s mae o rave a very hh speed reave o a laboratory its average lieime is measure in he aoraory to be 4.20 X 0 s Caclate he speed o he pion expressed as a fraction o c. b) Wha distance, measre in the laoraory does the pon rave drng ts average ifeime? 376 As yo pilot yor space tlity vehcle at a constant speed oward he moon, a race pio ies past you n her spaceracer at a constant spee o 000c relatve o you. A the nstant the space racer passes you boh o yo sar imers a zero a) A he nsan when yo measre that the spaceracer has traveled 1 0 0 m pas you wha does he race plo read on her timer? b) When he race pilot reas he vae caclated in part a) on her mer wha oes she measre to be yor disance rom her? c) A he instant
Exercises when the race pilot reads the value calculated n part (a) on her timer, what do you read on yours? 377 A spacecraft ies away from the eath with a speed of 4.S0 X 106 m/s relative to the earth and then returns at the same speed. The spacecraft carries an atomc cock that has been care fully synchronized with an identical clock that remains at rest on eah The spacecraft returns to its startng point 35 days (l year) later as measured by the clock that remaned on earth. What s the dierence n the elapsed times on the two clocks measured in hours? Which clock the one in the spacecraft or the one on earth shows the shorest elapsed time? 378 An alien spacecraft is ying overhead at a great distance as you stand in your backyard You see its searchight bnk on for 0190 s The rst ofcer on the spacecraft measures that the searchlght is on for 12.0 ms (a) Which of these two measured times is the proper time? (b) What is the speed of the spacecraft relative to the earh expressed as a fraction of the speed of light ?
runway as measured by a pilot of a spacecraft ying past at a speed of 4.00 X 10 m / s relative to the earth? (b) An observer on earth measures the time interval from when the spacecraft is directly over one end of the runway until t is directy over the other end What result does she get? (c) he piot of the spacecraft measures the time it takes hm to travel from one end of the nway to the other end What vaue does he get?
Seco The orez rasfomao (37.21) Solve Eqs to obtan and in terms of and t 34show that the resulting transformationt has the sameform as and the original one except for a change of sgn for 3715 An observer in frame S' is movng to the right + direction) at speed 0.00e away from a stationary observer in frame S. The observer n S measures the speed ' of a paricle movng to the right away from her. What speed v does the observer n 0400c; (b) ' 0900e S measure for the partice if (a) ' 0990e? (c) ' 36 Space piot Mavis zips past Stanley at a constant speed rea tive to him of O.SOOe. Mavis and Staney start timers at zero when the front of Mavis's shp s directly above Stanley When Mavis reads 500 s on her timer, she turns on a brght light under the front of her spaceship (a) Use the orentz coordnate transformation derived in Exercise 37.14 and Example 377 to calcuate and as measured by Stanley for the event of turnng on the light (b) se
Seco Relavy of Legh 379 A spacecraft of the rade Federation ies past the planet Cor uscant at a speed of 0.00e. A scientist on Coruscant measures the length of the moving spacecraf to be 740 m he spacecraft ater lands on Coruscant, and the same scentist measures the ength of the now stationary spacecraft What value does she get? 3710 A meter stick moves past you at great speed Its motion rela tve to you is paallel to its long axis If you measure the ength of 0304S m ) -for the movng meter stck to be .00 f ( ft example by comparing comparing it to t o a foot ruer that is at rest relatve to you-at what speed is the meter stick moving reative to you? 37 Why A A W Bombadd by Muon? Muons are unsta ble subatomic partcles that decay to electrons with a mean ife time of 2.2 J hey are produced when cosmic rays bombard the upper atmosphere about 10 k above the earth's surface, and they travel very cose to the speed of lght The probem we want to address s why we see any of them at the earths suface (a) What s the greatest distance a mon could travel durng ts 2.2Js lifetme? (b) Accordng to you answer n pa (a) t would seem that muons could never make it to the ground But the 2.2-Js lifetme is measured n the frame of the muon and muons are mov ng very fast At a speed of 0.99ge. what s the mean ifetime of a
muon by an observer rest this on the eah? Howwhy far would as themeasured muon trave in this tme?atDoes result explain we nd muons n cosmic rays? (c) From the pont of view of the muon, it still Lives for ony 22 , so how does t make it to the ground? What s the thckness of the 0 km of atmosphere through which the muon must trave, as measured by the muon? t is now clear how the muon is able to reach the ground? 37 An unstabe pacle s created in the upper atmosphere from a cosmic ray and travels straight down toward the surface of the earh with a speed of 0.99540c relatve to the earth. A scientist at rest on the earths suface measures that the partcle is created at an altitude of 45.0 km (a) As measured by the scientist, how much tme does it take the partice to travel the 45.0 k to the surface of the earth? (b) Use the lengthcontraction formula to caculate the distance from where the partice is created to the surface of the earth as measured n the particle's frame (c) n the pacles frame how much time does it take the patice to travel from where it is created to the suface of the earth? Calculate this time both by the time dilation fomula and from the dstance calcuated n part (b) o the two results agree? 373 As measured by an observer on the earth, a spacecraft run way on earth has a length of 300 m (a) What is the ength of the
the time diation formula, Eq (37 to calculate the time inerval between the two events (the front of the spaceship passing over head and turning on the light) as measured by Stanley. Compare to the vaue of t you calcuated in par (a) (c) Multipy the time inter val by Mavis's speed both as measured by Stanley to calcuate the dstance she has traveled as measured by hm when the light turns on Compare to the value of you calculated in part (a) 3717 A pursuit spacecraf from the planet Tatoone is attempting to catch up with a rade Federation cruiser As measured by an observer on Tatoone, the cruiser is traveing away from the planet with a speed of 000e. he pursuit shp s traveng at a speed of O.SOOe relative to atooine in the same direction as the cruiser. (a) For the pursuit ship to catch the cruiser, shoud the speed of the cruiser relative to the pursuit shp be positve or negative? (b) What is the speed of the cruiser relative to the pursuit ship? givesvelocity the transformaton for only the 378 Equaton 3723 component of an objects Suppose the obect consid ered in the derivation also moved n the /direction Find an expression for uy in terms of the components of , , and e, which represents the transformation for the component of the velocty (Hint: Apply the orentz transformatons and relationships ike Ux dxd, ; cL'dt', and so on, to the components) 3719 Two paices are created n a highenergy accelerator and move o in opposite directions The speed of one particle, as measured n the aboratory, is 050e, and the speed of each pat ce reative to the other is 0950c What s the speed of the second particle, as measured n the aboatory? 370 Two particles in a highenergy accelerator experiment are approachng each other headon each with a speed of 0.9520e as measured in the laboratory What is the magnitude of the veocity of one partcle reative to the other? 37 Two partcles in a hghenergy accelerator experiment approach each other headon with a relative speed of 0.S90e Both patcles travel at the same speed as measured in the laboatory. What is the spd of each partcle as measured n th laboratory? 37 An enemy spaceship is movng toward your starghter with a speed, as measured in your frame, of 0400e The enemy ship =
102
CHAPTR 37 Rv
res a missile toward you at a speed of 0.700 eative to the enemy shp (Fig. 3728) (a) What is the speed of the missie ea tive to you? Express your answer in terms of the speed of Light. (b) If you measure that the enemy ship s 8.00 106 an away rom you when the missie is red, how much time, measured in your frame, w it take the missie to reach you? iure Exercise 3722
Em
�
Stfht
An imperia spaceship, moving at hig speed reative to the panet Arrakis, res a rocket toward the panet with a speed of 0.920 eatve to the spaceship. An observer on Arakis measures ha te rocket is approaching wit a speed of 0.30. What is the speed of the spaceship reative to Aakis? s the spaceshp moving owad or away from Arrakis?
*Secton 6 The Doppler Eect fo Electomagnetc Waves *4 Rewrte Eq. 3.25 to nd the reative veocity it between
the source observer in termsofof igt the ratio of the eectromagnetic observed frequency andand theansource frequency What reative veocity wi produce (a) a 5.0% decrease in frequency and (b) an ncease by a factor of 5 of te obseved ight? *5 Tel It to the Judge (a) How fast must you be approac ng a red trafc ight = 75 nm for it to appea ap pea yeow = 575 nm? Express your answer in terms of te speed of ight (b) If you used this as a reason not to get a ticket for running a red ght, how much of a ne woud yo get for speeding? Assume that the ne is $100 fo each komeer per hou that you speed exceeds the posed imit of 90 /h * Sow that when the source of eectromagnetic waves moves away from us at 0.00, the frequency we measure is haf he vaue measured n the rest frame of the source Section Relatvstic Momentum (a) A partice wit mass m moves aong a straight ine under the acton of a force F directed aong the same ine. Evauate the deivative in Eq. (3.29) to show that the acceeration a dv/dt of the parice is given by a / (I v2/ ) 3 (b) Evauate the deivative in Eq (37.29) to nd the expression fo the magni tude of the acceeration n terms of F, m, and / if the force is per pendicuar to the veocity. When Should you Use Reativiy? As you have seen, reativistic cacuatons usuay invove te quantty y. Wen y is apprecaby greater than 1 we must use eativistic formuas instead of Newtonian ones. Fo what speed v (in terms of ) is the vaue of y (a) 10% greater than I; (b) 10% greater than I (c) 00% greater than ? 9 (a) At what speed s the momentum of a partice twice as great as the resut obtained from the nonreativistic expression ? Express your answer in terms of the speed of ight. (b) A force is appied to a partice aong its diection of motion. At wat speed is the magnitude of force required to produce a given acceeration twice as great as the foce required to produce the same acceera tion when he partice is at rest? Express you answer n tems of e speed of ght. -
Relativistic Baseball. Cacuae he magntude of the
force required to give a 0145-kg basea an acceeration a = 1.00 m/s in the directon of the baseba's initia veocity when ths veocity has a magnitude of (a) 100 m/s; (b) 0.900 (c) 0.990 (d) Repeat parts (a), (b), and (c) if the force and acce eration are perpendicua to the veocity Secton 8 Reatvstc Wok and Enegy What is the speed of a partce whose kinetc energy is equa to (a) its est energy and (b) ve times its rest energy? Annihilation In potonaniproton annihation a proton and an aniproton (a negatvey charged proton) colide and disap pear, producing eectromagnetic radiation. If each partice has a mass of 1.7 10 kg and they are at rest just before the annhi ation nd the tota energy of the radiation Give you answers in oues and in eectron vots. A proton (rest mass O kg has tota energy that is .00 times ts rest energy. What are (a) the kinetc energy of te proton (b) the magnitude of the momentum of the poton (c) the speed of the proton? 4 (a) How much work must be done on a partice with mass to accelerae it (a) fom rest to a speed of 0090 and (b) fom a speed of 0900 to a speed of 0990? (Express the answers in terms of .) (c) How do your answers in parts (a) and (b)
compare? 5 (a) By wha percentage does your rest mass increase when you cimb 30 m to the top of a tenstory buiding? Are you aware of this ncease? Expain (b) By how many grams does te mass of a 2.0g sping with force constant 200 N/cm change wen you compress it by .0 cm? Does the mass incease o decrease? Woud you notice the change in mass if you wee hoding the spring? Expain. A 0.0kg person is standing a res on eve ground. How fast woud she have to rn to (a) doube he ota energy and (b) increase her tota energy by a factor of O? An Antimatter Reacto When a partice meets its antipatice, they annihiate each other and their mass is converted to ight energy Te Unied States uses approximatey 1.0 X 10 ] of energy pe year. (a) If a ths energy came from a fuurstic anti matte reactor, how much mass of matter and antimatte fue woud be consumed yeary? (b) this fue had the density of iron 7.8 g/cm and were stacked in bricks to form a cubica pie, how high woud it be? (Before you get your hopes up antimatter eactos ae a log way in the futuref they ever wi be feasibe) A J ("psi) patce has mass 5.52 0 kg. Compe the rest enegy of the patice in MeY 9 A patice has rest mass 4 0 kg and momentum 210 10 10 18 kg m/s (a) What is te tota energy (kinetic pus est enegy) of te partice? (b) Wha s the kinetic energy of the partice? (c) What s te ratio of he knetc energy to e ret energy of te partice? 4 Starting from Eq (3.39), sow that in te cassica imit pc 1C) the energy approaces the cassica kinetic energy !mv pus te rest mass enegy . 4 Compue the kinetic enegy of a proton (mass 17 10 kg using both the nonreativisic and eativistic expres sions, and compute the ratio of the two resuts (reativistic dvded by nonreativisic) for speeds of (a) 8.00 10 m/s and (b) 2.85 0 m/s. 4 What is the kinetic energy of a proton moving at (a) 0100 (b) 0.500 (c) 0.900? How muc work must be done o (d) increase
a nd e) inease the pro the proton's speed fom 000c to 0.500c and ons speed from 0500c to 0900? f How do the last two esults ompare to results obtaned in the nonelativsti imit? 4 a) Though what potentia dfeene does doe s an eletron eletron have to be aelerated, starting from rest to ahieve a speed s peed of 0980c? b) What is the kineti energy of the eletron at this speed? Express your answe in joules and in eletron vots 44 44 Creating a Particle. wo protons eah with rest mass M = 67 0- kg are initialy moving with equa speeds in opposite heanpotons ontinue exist after a oll patile son that dietions aso produes see to Chapter 44). he est mass of the is m = 9.5 X 0 kg a) the two protons and the To are al at rest after the olision, nd the nitia speed of the protons, expressed as a fraton of the speed of ight b) What is he kneti energy of eah proton? Expess Ex pess you answer in Me. ) What is the rest enegy of the , expessed in MeV? d) Disuss the reationship re ationship between the answers to parts b) and ). 45 Find the speed of a patile whose relatvsti relatvsti kinei energy is 50% geate than the Newtonian vaue va ue for the same speed 46 Energy of Fusion. In a hypothetal nuear fuson ea to, two deuterium nuei ombine or "fuse to form one heium nueus The mass of a deuterum nuleus expressed n ato mass unts u), s 20136 u; the mass of a helium h elium nuleus s 4005 u u = .6605402 0 kg. kg . a) How muh energy energy s reeased when 0 kg of deuterim undergoes fusion? b) he annual onsumption of eletia enegy in the United States is of the order of 0 0 J. How muh deuterum must eat to pro due ths muh energy? 4 The sun produes energy by nuea fuson reations n whih matte s onverted into energy By measurng the amount of enegy we eeive fom the sun we kow that it is podung enegy at a rate of 38 0 W. a) a ) How many kiogams of mat ter does the sun ose eah seond? Approximatey how many tons of matter is this? b) At this rate, how ong would it take the sun to use up al its mass? 48 A O.OOg O.OO g spek of dust dus t is aeerated from est to a speed of 0900 by a onstant 00 0 N fore. a) If the noneativis ti fom of Newtons seond law 2F = a is used how fa does the objet tave to reah its nal speed? b) Using the oret rela tvsti teatmen t of Setion 37.8distane how farisdoes the objet reah itsteatment na speed? ) Whih geater? Why? tavel to
Probles 4 4 Ater being produed in a olision between elementary pa pa tiles, a positive pion (7+ must trave down a 20kmong tube to eah an expeimental aea. A 7 patle has an aveage ife time measued in its rest frame) of 260 0- s the 7+ we ae onsidering has this fetime a) How fast must the + travel f it is not to deay before it eahes the end of the tube? Sine u wll be very ose to c, write u = (1 - D)c and give you answe in terms of rather than u) b) The 7 + has a rest enegy of 39.6 MeV. What is the total energy of the 7+ at the speed alu lated in part a)? 50 A ube ube of metal wth sides of length a sits at rest in a rame S wth one edge paael to the axis heefore in S the ube has volume 3 Fame S' moves along the axis with a speed u. As measued by an obsever in frame S what is the voume of the metal ube? 51 The staships of the Solar Federation are marked with the symbol of the fedeation, a irle, while starships of the
303
enebian Empire ae maked wth the empies symbol an eipse whose major axis is 40 times longer than its mino axs ( 140 in Fg 3.29) How fast eatve to an observe does an empre ship have to travel for its markng to be onfused wth the marking of a federation shp? =
Figure 3729 Problem 375
datn
Emp
5 A spae probe is sent to the viinity of the star Capea,
whih is 422 ightyears from the earth A lightyear is the dis tane light travels in a year) he pobe travels with a speed o 099O An astronaut reruit on board is 9 years old when the probe leaves the eath. What is her bioogial age when the probe reahes Capela? 5 A partile partile is said to be xtLy Lativitic when its kineti energy is muh greater than its rest energ energy. y. a) What is the speed of a partile expressed as a fraton of c) suh that the tota enegy ten tmes the rest energy? b) hat s the perentage diferene between the eft and right sides of Eq 3.39) if c s neg eted for a partile partile with the speed alulated in part a)? 54 Everyday Time Dilation. wo atomi oks are are fuy synhronized ne remains in New York, and the other s oaded on an airiner that tavels at an average speed of 250 m/ and then etus to New York When the pane p ane returns the eapsed time on the ok that stayed behind is 400 h By how muh wi the readings of the two oks dier and whih lok wil show the shorer elapsed time? Hint: Sine u , you an simplify V - U2/ C2 by a binomia b inomia expansion.) 55 he Large Hadron Collider (LHC) Physiists and engineers fom aound the world have ome together to build the argest aeerator in the world, the Large Hadron Coider HC) at the CERN aboratoy in Geneva, Switzerland S witzerland The mahine wi aeleate protons to kneti energes of 7 eV n an undergound ng 27 on kmthe n HC, rumferene Fo the latest newsspeed and moe willnfor mation visit www.ern.h) a) What pro tons reah in the LHC? Sine u is very ose to , write u = - c and give you answer in terms of . b) Find the relativisti mass, , of the aeleated protons in tems of ther rest mass 56 A nuea bomb ontainng 800 kg of plutonium explodes The sum of the rest masses of the produts of the exposon is ess than the ogina ogina rest mass by one part in 10 a) ow muh energy s eeased in the exploson? b) f the exploson takes plae n 400 s, what is the average power developed by the bomb? (c) What mass of wae oud the eleased enegy ift o o a heiht heiht o 1.00 k? 5 erenkov Radiation. The Russian physist P A erenkov disoveed that a haged partle traveing in a soid with a speed exeedng the speed of light in that materal adiates eletomag neti adation his is anaogous to the soni boom produed by an airaft moving faster than the speed of sound in a see Setion 69. erenkov shared the 958 Nobel Pize fo this dis overy) What is the minmum kneti energy in eletron volts) that an eletron must have whie traveling nside a slab of rown gass ( n 52 n order to reate this eenkov radiation?
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