{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Final Exam Formula Sheet

Final Exam Formula Sheet - Principal efinitions nd...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Principal efinitions nd Equations f Ghapter D o 9 a vton's second (e.6e) ntions on the r dimensions. polar coordiinertial frame inertial frame r velocity Q, instant fo, the hef rame$ 'is inates(r', @) Let us now compare the equation of motion (9.69) for the inertial frane, with (9.7I ) for the co-rotating frame. The most important thing to recognizeis that, because g2:6, they are exactly the sameequationsfor r and /, although certain terms are disnibuted differently benveenthe two sides.In (9.69) for the nonrotating frarne, the only force terms on the left arc the real net force, with components4 and f;. On the right of (9.69), the accelerationcontains the centripetal acceleration-rl2 in its radial componentand the Coriolis acceleration2i6 inits @component.In (9.71) for the rotating frame, neither of theseadditional accelerationterms is prcsent (because we arrangedthat {' is zero), but insteadthey are reincarnatedon the force side of the equations(with opposite signs, of course)as the centrifugal force m{22r inthe radial equationand Coriolis force -2miQ in the d equation. Sincethe nno versionsof the equationsarethe same,it is clear that they areequally conect. In the inertial frame, the forces are simpler (no "fictitious" forces) but the accelerationsare more complicated; in the rotating foame,it is the other way round. Which frarne one choosesto use is dictated by convenience.In particular, when the observeris anchoredto a rotating frame (as we earthlings are), it is generally morc convenient to work in the rotating frame and to learn to live with the "fictitious" centrifugal and Coriolis forces. o 9 Principal efinitions nd E quatlons f C hapter D a F InertialF orcei n a n A ccelerating ut N onrotating rame b fo. Newton's ntrifugal and The motion of a body, as seenin a frame that has accelerationA relative to an inertial frame, can be found using Newton's secondlaw in the form mt :F * Fi,,,,ti.r,where F is the net force on the body (as measuredin any inertial frame) and F;"",t.r is an additional inertial force Fin"ni"t : (9.70) F6 is purely rodifference , since y' is 'component y the analog e tJ, so tbe the equatioo (9.7t) I sinethcy -tnA, tEq.(e.5)1 V V The A ngular elocity ector If a body is rotating about an axis specified by the unit vector u (direction given by the right-hand rule) at a rate crr(usually measuredin radiansper second),its angular velocity vector is defined as lEq.( e.21)l The " UsefulR elation" v The velocityof a point r fixed in a rigid bodythat is rotatingwith angular elocity aris Y :& rX f. Irq.(e.22)l 3 60 C hapter9 i F M echanicsn N oninertial rames i Time D erivativesn a R otatingF rame o If f rame 3 h as a ngularv elocity Q r elativet o f rame 3 ,,,t hen t he t ime d erivatives f a single v ector Q a s s een i n t he t wo f rames a re r elated b y (#),:(f),*exQ t [ E q . 9 . 3 0 r] Newton'sS econdL aw i n a R otatingF rame If f rame S h asa ngularv elocity Q r elativet o a n i nertialf rame E .,,t henN ewton'ss econd law i n t he r otatins f rame t akes t he f orm lt t : F* F. r . *F . r . . [Eq.( 9.37)J where F i s t he n et f orce o n t he b ody ( as m easuredi n a ny i nertial f rame) a nd t he i nertial forces F .,,, a nd F .1 a re t he C oriolis a nd c entrifugal f orces, F . u , : 2 n ti x d L and Fci: n r(9 x r ) x g . [ E qs.( 9.35)& ( 9.36)] F Free- allA cceleration g r The o bservedf ree-fall a cceleration ( delined a s t he i nitial a cceleration, elative t o the e arth. f rom r est) i ncludes t he " true" g ravitational a ccelerationg n a nd t he e ffect o f the c entrifusal f orce B:8.+(S|xR)xS2. tEq.( e.44)l "Vertical" i s d efineda s t he d irectiono f g , a nd " horizontal" a s p erpendicularo g . t tiame).W hat valueo f a bou astronaut's f er SECTI O N. 2 9 9.1 * * ( a) C o ( dn- R ") : c, Frid - (2G,,1 numericalc or correspondin (magnitude n a 9.4 * * D o t he 9. 4. [ I n t his c i | - 3 el 2. l 9.5 * * R evieu Figure9 .-5 nd a !.f * ** l g1 / 1 up f rom t he l er of t he o ceani s and d escribe l t evaluate ,1.1 U7 t he l aw o f cosi keepc 1l t errns SECTI O N. 4 9 g.f * ( s) E xplr the s pecial as c 3,, a nd c ompar Problemsf or C hapter9 Sturs i ndiute t he u ppro-rinrutel evel o J d iJticultt',.fntme osiest( *) t o n tost d fficult ( ***). SEcTtoNs .r A ccelerationw ithout R otation 9.1 * B e s urey ou u nderstand hy a p endulumi n e quilibriumh angingi n a c ar t hat i s a ccelerating w forwardt ilts b ackward, nd t henc onsider he f bllowing:A h eliumb allooni s a nchored y a m assless a t b stringt o t he f loor o f a c ar t hat i s a ccelerating f brwardw ith a cceleration . E xplainc learlyw hy t he A balloont endst o t ilt . fttnrarcla nd l ind i ts a ngle o f t ilt i n e quilibrium.[ Hirrt: H elium b alloonsf loat g because f t he b uoyantA rchimedeant brce, w hich r esultsf rom a p ressure radienti n t he a ir. W hat i s o the r elationb etweent he d irectionso f t he g ravitationalf ield a nd t he b uoyantf orce?l 9.2* A donut-shapedpace tation( outerr adiusR ) a rrangesor a rtificialg ravityb y s pinningo n t he s s f axis o f t he d onut w ith a ngular v elocity r r.r. ketch t he f orces o n. a nd a ccelerations f, a n a stronaut S o ( a) a s s eent iom a n i nertialf rame o utsidet he s tationa nd ( b) a s s eeni n t he standing n t he s tation i p astronaut's ersonal estf iame ( which h as a c entripetal cceleration : ,rlR a s s eeni n t he i nertial r a A sEcTr oNs . 5 I 9.8 * W hat a re the N orth P ole 9. 9* A b ulletr colaritude . F i: e angular elocit v 0 : 4 0 d eg'l 9.10* * T he d e that t he a ngula sometimes allt c ! . ll * ** l p 1 li1 Lagrangiana pp (except hat i t c t a n oninertialrz f Principal efinitions nd Equations f Chapter10 D a o side the range between01and 02 to which the motion is confined, then @still cannot vanishand the motion is still as skerchedin Figure 10.12(a).On the other hand, if the angle 9o lies betweeng1and 02, then<| will changesign twice in each oscillation of d. In this case,the precessionmoves fint in one direction, then in the other, and the i overall motion is as sketchedn Figure l0.l2O). D Princlpal efinltions nd E quations f C hapter 0 a o 1 M CMa ndR elative otions L = L(motionof CM) * L (motion elativeo CM). r t o ( 10.104), (10.1 l s) Z3 (in magnihangesign, so 'sdecreasing). llatesbetrveen nller than 23, anglelies out- tEq.(10.9)l 7 : I(motion of CM) * T(motionrelativeo CM). tEq.(10.16)l t The M oment f I nertia ensor T o The angular momentumL and angular velocity ar of a rigid body are rclated by L:Ito IEq.(10.42)l whereL andormustbe seen s3 x 1columns ndI is the3 x 3 momentof inerda a a d a e a a lensor,whose iagonal ndoff-diagonal lements redefined s Irr:D^n1] + " !), e tc. a nd I ry: - l r noxnyo, tc. e " * r. ( 10.37) ( 10.38)I & Principal xes A A principal axis of a body (about a point O) is any axis through O with the property that if ar points along the axis, then L is parallel to or; that is, L: tso n a ) 'to lEq.( 10.65)l for somereal number1..For any body and any polnt O, there are three perpendicular principal axesthrough O. lSection 10.4and Appendixl Evaluated with respect to its principal axes, the inertia tensor has the diagonal brm es, and and02. vhile d ':[ir l ] ( tEq. 10.7e)I 4 08 o Chapter1 0 R otationalMotionf R igidB odies E Euler's quations If L d enotest he r ate o f c hangeo f a b ody's a ngular m omentum a s s eeni n a f rame f ixed in t he b ody ( body f rame), t hen i t s atisfiesE uler's e quations i rr . ) ' . L * t oxL:f. h , .. Itr.-.' A Euler's ngles b o The o rientation f a r igid b ody c an b e s pecified y the t hreeE uler a nglese . Q , l l 1& i n F igure1 0.10. defined [Sections0.9 l 0.l0l p ivot i s f The L agrangianbr a r igid b ody s pinninga bouta fixed t, : \ xr1$2s in2g+ e tl + i xt<rl,* 6 coso)z - M gRcos0. ' .. J ' - - -: l 0 .t " l l l .9 ' Stars intlicate t he appro.rimate let'el o f d fficultv, f rom e usiest ( *) t o n utst d ifficult ( ***). lf'.i .tlill ' -h . ' - t K t 10.2* T o i llustrate he r esult( 10.1 8),t hat t he t otal K E o f a b ody i s j ust t he r otational E r elative o p oint t hat i s i nstantaneouslyt r est,d o t he f ollowing: W rite d own t he K E o f a u niform w heel( mass a any o M r adius R ) r olling w ith s peedu a long a f lat r oad, a s t he s um o f t he e nergies f t he C M m otion a nd p the r otationa bout t he C M. N ow w rite i t a s t he e nergyo f t he r otationa bout t he i nstantaneous oint o f contactw ith t he r oad a nd s how t hat y ou g et t he s amea nswer.( The e nergyo f r otation i s { /ro:. T he momento f i nertia o f a u niform w heel a bout i ts c enteri s I : ! .M R 2.T hat a bout a p oint o n t he r im i s I,:: M R2. ) a 10.3* F ive e qual p oint m asses re p laceda t t he f ive c ornerso f a s quarep yramid w hoses quareb ase is c entered n t he o rigin i n t he.r.l'p lane,w ith s ide t , a nd w hosea pexi s o n t he z a xis a t a h eight H o s abovet he o rigin. F ind t he C M o f t he f ive-mass ystem. 10.4* * T he c alculationo f c enterso f m asso r m omentso f i nertia u sually i nvolvesd oing a n i ntegral, most o fien a v olume i ntegral, a nd s uch i ntegralsa re o fien b est d one i n s phericalp olar c oordinates (defined ack i n F igure4 .16).P rovet hat b , r n ,,1 - i /a f( ). l 0 . l* T her es ult ( 1 0 .7 ),th a tfm o r' u :0 ,c a n b e p araphrasedtosaythattheposi ti onvectoroftheC M to relativet o t he C M i s z ero,a nd, i n t his f orm, i s n earlyo bvious.N evertheless, b e s urey ou u nderstand i p r esult, rovei t b y s olving( 1 0.4)f or r j a nd s ubstitutingnto t he s um c oncerned. the f ' l ':r .r l tn Jcr , ,1 r r sEcroN 1 0.1 P ropertieso f t he C enter o f M ass t \ [ Eq.( 10.106)] 1 f Problemsor C hapter 0 I a vl rt: JJJJ J, ( & [Eqs. 10.87) ( 10.88)] f d f I r :dr / s ino o I d O \ r .e.et. ra s [Think about t he m a l lv o l u me d V e n c l o s e d b e tween ndr - f d r,0 a nd0 * d 0,andQandQ + d Q.] If t he v olume i ntegralo n t he l eft r uns o ver a ll s pace,w hat a re t he l imits o f t he t hree i ntegralso n t he right? o 10.5* * A u niform s olidh emisphere f r adiusR h asi ts f lat b asei n t he " rl p lane,w ith i ts c entera t t he oriein. U se t he r esulto f P roblem 1 0.4t o f ind t he c entero f m ass.[ Comment:T his a nd t he n ext t wo lll. l0 ' r ar . Ft nJ i t r n r , t r . r ni nt cgr al r r acnt cr . J t t h i l( l. ll " r ar t n tas s . 1/. r nd innera ndo u : pher eo t 'r ad * 10. 12 * . \ r trian,elesara p Find i ts m on explain t s t r r i * 10. 13 * A t l the h orizonta ,r. its CM is down t he e qu fiom t he d ow is t raditional pendulum." w the " equivale 10.14* * A s t (6000k g) a nd 5 86 T Chaoter1 4 C ollision heorv t I I I I Calculatin do/dll do/dA Rltl RzA If y ou c an f ind t versa),t hen d .n t 0 n/2 € 0 rrl2 (b) (a) o Figure1 4.17 ( a) T he d ifferentialc rosss ectionf or s cattering ff a h ard i sphere s i sotropici n t he C M f iame. ( b) I n t he l ab f rame,i t i s p eaked n i the f orwardd irection. The R uthe The d ifferentia by t he R utherf and t he l ab c ross s ectionc an b e w ritten d own i mmediatelyf rom ( 14.56) ( with 1 : 0 .5). T he t wo c ross s ectionsa re p lotted a s f unctions o f t heir r espective anglesi n F igure l 4.n.t: A s w e a lreadyk new.t he C M c rosss ectioni s i sotropic. The l ab c ross s ection i s m arkedly s kewed i n t he f orward d irection. T he C M a n The l ab f rame the C M f rame two f ramess at 1 o D a Principal efinitions nd E quations f C hapter 4 The S cattering ngle a nd l mpact P arameter A i i The s catteringa nglei s t he a ngleI b y w hich a p rojectile s d eflectedn i ts e ncounter w b with a t arget. he i mpact p arameteri s t he d istance b y w hich t he p rojectile ould T 1l have m issed t he c enter o f t he t arget i f i t h ad b een u ndeflected. lSection 4. l TheC ollision ross ection C S The c ross s ection o n. f or a p articular o utcome " oc" ( elastic s cattering,a bsorption, reaction,f ission)i s d efinedb y Nn. : N in.trturQr. 1 [Sections 4.2& 1 4.31 N whereN *- i s t he n umbero f o utcomes f t he t ypec onsidered, ;n.i s t he n umbero f o ( p of incident rojectiles,ndn ,",i s t hed ensity number/area) t argets. a The D ifferential rossS ection C The d ifferentialc rosss ection9 rr,Q) d9 finedb y Problems - ( f or s cattering a d irection 0,d) i s d ein rr E quation 14.56)g ivest he l ab c rosss ection s a f unctiono fthe C M a ngle0 .,n. o e xpresst ( i T a T o as a n e xplicit f unction o f 6 116,ne w ould h avet o s olve ( 14.53)f or 0 .,,,i n t ermso f 0 1u6.o m ake t he i a plot o f F igure l ,l. l 7(b). a m uch s impler p rocedure s t o t reatb oth d 1,6 nd ( doldQ)66 a s f unctions p e of t he p arameter .n,a nd m ake a p arametric lot w ith d .n,r unning f rom 0 t o z . 1 sEcTloN 4.2 * 14.1 A b lue cm. F ind t he c r a pancake, s s e will h it a b erry 14. 2* ( a) Ac barn= l 0-28 r 14.3* A b ean and t he l iquid :eenb y t he i nr 14.4* * T he c Il' l Oeo f t hes Problems C hapter 4 f or 1 N".(into f)): Nin"ztu, d i l d 9. f t<t, llard t E q . (la . l7) l Calculating D ifferential rossS ection t he C If you can find the scattering angle 0 as a function of the impact parameter b (or vice versa).t hen do b ldbl d a : t l" ol d e l IEq.(r4.23)l The R utherford ormula F l in The d ifferential c ross s ection f or s catteringa c harge q o ff a f ixed c harge Q i s g iven by the Rutherford formula i6) ( with :spective sotropic. do ( k qQ \' de: \[email protected]/D ) ( IEq. lasDl The C M a nd L abC rossS ections The l ab f rame i s g enerally nderstoodo b e t he f ramei n w hich t he t argeti s a t r est; u t the C M f rame i s t hati n w hich t heC M i s a t r est.T he d ifferential rosss ectionsn t he c i two f ramess atisfy :(#)"^|trffi| (#),^, ( tEq. la.as)l ls e ncounter :ctile w ould 1 ection 4.ll Problemsf or C hapter1 4 ( Starsindicatethe approximateevel of dfficulty, from easiest *) to mostdfficult (***). l absorption, SEcTloN1 4.2 T he C ollision C ross S ection 4.2 & 1 4.31 : n umbero f 0, d ) i s d e- it . T o e xpress t T o m ake he ,. a n6 sf unctions 14.1* A b lueberryp ancakeh as d iameter1 5 c m a nd c ontains6 l arge b lueberries, acho f d iameterI e (number/area)of berries in the cm. Find the cross section o of a blueberry and the "target" density n,u. pancake,as seenfrom above.What is the probability that a skewer,jabbed at random into the pancake, will h it a b erry ( in t erms o f o a nd r e,-, a nd t hen n umerically)? 14.2* ( a) A c ertainn ucleush as r adius5 f m. ( l f m - l 0-ls m .) F ind i ts c rosss ectiono i n b ams. ( l barn= 1 g-zs D o t he s amef or a n a tomo f r adius0 . I n m. ( 1 n m = l 0-e m .) - 2.;(b) 14.3* A b eamo f p articlesi s d irectedt hrougha t ank o f l iquid h ydrogen.I f t he t ank's l engthi s 5 0 c m and t he l iquid d ensity i s 0 .07 g ram/cm3.w hat i s t he t argetd ensity ( number/area) f h ydrogena toms o seenb y t he i ncidentp articles? 14.4** T he c ross s ectionf or s catteringa c ertain n uclearp article b y a c opper n ucleusi s 2 .0 b arns. If l Oe o f t hesep articlesa re f ired t hrough a c opper f oil o f t hicknessl 0 p m, h ow m any p articlesa re R 1 C hapter 5 S pecial elativity i a nd,s ince .u : 1 , t hec urrent,hisi s e xactly hes ame st heB f ieldi n ( 15.147). ) t t a f ' T he r emarkableeatureo f t his d erivationo f B i s t hat i t m ade n o r eference ' t o A mpere'sl aw. G auss'sl aw i n t he f rame S ', c ombinedw ith t he L orentz o ; t ransformation f t he f ields, h as g iven u s t he r esultt hat w e n ormally s eea s sraw' . 1Tp"i: *:.1"_":,t:,:::l:'"f " With t his s triking e xample o f t he b ehavior o f t he e lectromagneticf ield u nder t he Lorentz t ransformation, I m ust e nd o ur b rief f oray i nto r elativistic e lectrodynamics. You c an e xplore a f ew m ore a spectsi n t he p roblems a t t he e nd o f t his c hapter, a nd after t hat y ou c ould r ead t he e xcellent b ooks o f G riffiths a nd J ackson.3r The inverseLore a variables ndc ha The V elocitY The v elocities o f are r elated b Y t hr , 't url-u 1 o Principal efinitions nd E quations f C hapter 5 D a Four-Vecto TimeD ilation If w e r ewrite t hr four-vectorsx = If t wo e vents,a s o bserved i n f rame S o, o ccur a t t he s ame p lace a nd a re s eparatedb y a t ime A ro, t hen t he t ime b etween t hem a s m easuredi n a ny o ther f rame E i s A.t-yA .to lEq.( ls.l l )l agree to arTange become " rotatic is a ny s et o ffou transform t his v o t wherey : l /Jl - P 2, P : V l c,and V i s t he s peed f 3 r elativeo S u. Length ontraction C The I nvari If, a s o bservedi n f rame 3 o, a b ody i s a t r est a nd h as l ength / o, t hen i ts l ength m easured in a f rame S t raveling w ith v elocity V i n t he d irection o f t he l ength i s I : l oly. The s calarP ro ( t E q . 1 5 . 1 5 )l p Lengths erpendicular V a reu nchanged. to andi s i nvarian itselfi s o ftenI T The L orentz ransformation The L ight( The c oordinateso f a ny o ne e vent a s m easuredi n t wo f rames ( in s tandardc onfiguration) a re r elated b y t he L orentz t ransformation: The l ight c on i equivalentlY x':y(x-Vt) y':j7t : 7 ( t E q . 1 5 . 2 0 )l t' :y (t-V x/c2). 3rChapter 1 2 o f D avid J . G riffiths, I ntroduction t o E lectrodvnarrics,( third e dition, P rentice e Hall, 1 999) i s a t a pproximatelyt he l evel o f t his b ook b ut n aturally e mphasizes lectrodynamics ( much m ore h eavily.J . D . J ackson's lassicalE lectrodynamicsthird e dition, J ohn W iley, 1 998)i s C a g raduatet ext, w hich y ou c ould t acklea fter r eadingC riffiths' b ook. The R elati Light f rom a ungle0(0=r in rneasured t P rincipal efinitions nd E quations f C hapter1 5 D a o t ( 1 5. 1 47). , reference re L orentz The i nverseL orentz t ransformation i s o btained b y e xchanging p rimed a nd u nprimed variablesa nd c hanging t he s ign o f V . rlly s ee a s The V elocity-Additionormula F eld u nder t he trodynamics. chapter, a nd ( Thev elocities f a s ingleo bjecta sm easured t wo f rames in s tandard onfiguration) o in c arer elated y t he v elocity-addition ormula b f , "{ tl-- u ,-V l - u*Vfc2' UT' y(l - u ,V l c2) , uz a nd y(l - u ,V/ cz1' lEqs.( | s .26)& ( t s .27)l Four-Vectors separatedy b 3is lEq.( 15.1l)l If w e re w ri te th e c o o rd i nates(x,y,z)as ( xt,xz,-rj )andi ntroduce x q:ct, t henthe four-vectorsx : ( xr, x 2, x 3,x a) l abel p oints i n a f our-dimensional s pace-time. I f w e agreet o a rranget he c omponentso f r i n a 4 x I c olumn, t hen L orentz t ransformations become " rotations" o f t he f orm x ' : I \x.where A i s a 4 x 4 m atrix. A f our-vector is a ny s et o f f our n umbers,Q : ( Qy Q z,e t,4a) ( one s et f or e achi nertial f rame) w hich transform t his w av. q' : l tq' 1 ISection 5.8] The I nvariant calar roduct P S ;thm easured The s calar p roduct o f t wo f our-vectorsx a nd - y i s d efined a s .r . _ v: - {1,v1 x ZlZ t r :y3 - r a)+ f Eq.( 15.15)l ( tEq. 1s.50)l p andi s i nvariant ndera ll L orentzt ransformations.he s calar roduct f a v ectorw ith u T o itself i s o ftenw ritten? sx - x : x 2. The L ight one C I c onfigura- lq . ( 1 s. 2 0 )l ion, P rentice rtrodynamics iley, 1 998)i s The l ight c one o f a p oint Q i n s pace-time c onsists o f a ll l ight r ays t hrough Q ; equivalently,t c ontainsa ll p oints P w ith ( x p - x g )2 : 0 . i [ Section 1 5.l 0] The R elativistic oppler ffect D E Light f rom a s ource ravelingw ith v elocityV r elativet o f rame S i s o bserved t a n t a angle0 ( 0 = a ngleb etween a nd t he r ay o f l ight). I f t he f requency f t he l ight, a s V o measured t he s ource's estf ramei s r r.ro f requency bservedn S i s in r t he o i (/) : (Do y(l - B cos9) ( tEq. ls.6a)l 665 666 1 R Chapter 5 Special elativity M Mass,F our-Ve...
View Full Document

{[ snackBarMessage ]}