LEBANESE UNIVERSITY FACULTY OF ENGINEERING
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1 Entranc Exa 3 PHYSICS Duraton: H 8 JULY Exrcs I: [ pts] Study of th oton of a partcl Consdr a hollow crcular sld (C of radus 5 c and locatd n a vrtcal plan. A O partcl (S, of ass g, can sld on th nnr surfac of (C. Intally, (S s at A, ts poston of stabl qulbru. W shft (S, n th postv drcton, by an angl o, thn w rlas t fro rst at th nstant t. At an nstant t, ts angular longaton s and ts angular vlocty s. Th horzontal plan through A s th rfrnc lvl for th gravtatonal potntal nrgy. Tak g /s ; sn (rad.. W nglct th forcs of frcton. a Dtrn, at th nstant t, th chancal nrgy of th syst ((S, Earth. 8 A (S b Drv th scond ordr dffrntal quaton n θ that dscrbs th oscllatons of (S. 4 c Dduc th valu of th propr (natural prod of ths oscllatons. t(s d Dtrn th t quaton of oton. -4. In fact, consdrng th sa ntal condtons, (S undrgos, at an nstant t, th forc of -8 frcton f -λv, whr λ s a postv constant. a Dtrn th xprsson of th powr of th frctonal forc at th nstant t. Dduc that th dffrntal quaton that govrns th oton of (S s wrttn as: + + g. b Th soluton of ths dffrntal quaton s of th for: (t A xp(-t/( cos (ωt-φ. Tak (t+t/(t, whr T s th psudo-prod. Dtrn th xprsson of and dduc th valu of. Exrcs II: [5 pts] Why s th sky blu? In 94, J.J Thoson proposd a odl for th hydrogn ato, n whch th lctron of ass, locatd at M, s lastcally lnkd to ts fxd nuclus locatd at O. Th ato s thus rducd to an lastc pndulu (, k, th lctron of ass undrgong th forc F -kom whr OMx and whr O s ts stabl qulbru poston. Th lctron ay thus ov along. Gvn: kg, k N/ and w nglct th wght of th lctron.. a Nglctng frctonal forcs, drv th dffrntal quaton of oton of th oscllator. b Dduc th xprsson of th propr angular frquncy and that of th propr prod T of th oscllator. c Calculat th valus of and T.. A lunous wav, ssud fro th Sun, s charactrzd by an lctrc fld E E cos(t+, whr blongs to th ntrval rd blu, ths two xtr radatons havng th followng wavlngths n vacuu: rd.8 and blu.4. W ntnd to study th acton of ths wav on th lctron of an ato of th atosphr, usng th All th Entranc Exa Sssons ar avalabl on
2 Thoson odl. Th lctron thus undrgos, at an nstant t, th lctrc forc F'-E - E cos(t+ addton, a frctonal forc of th for F - h v whr v dx.gvn:,6. -9 C; h - kg/s. All th Entranc Exa Sssons ar avalabl on and, n a Show that th dffrntal quaton n x s of th for: x + B x + x - D cos(t+. b Dtrn th xprssons of th postv constants B and D and calculat th valu of B. c Calculat th valu of rd and that of blu. 3. Th soluton of ths dffrntal quaton, n stady stat, s of th for x A cos (t. By gvng t two partcular valus, dtrn th xprsson of A n trs of..e 4. By gvng th consdrd ltng valus, show that th xprsson of A can b rducd to: A (ω -ω. 5. Knowng that th lctron ts, n all drctons, an lctroagntc radaton whos avrag powr s proportonal to th squar of th apltud of ts acclraton, a Gv th xprsson for th avrag powr P av n trs of,, E, and. b By coparng th two avrag powrs P rd and P blu, xplan why th sky s blu. Exrcs III: [5 pts] ubdu-strontu datng So grantc rocks, durng thr crystallzaton, hav dtand an aount of rubdu, a radoactv sotop of rubdu, of radoactv constant,4 - yar -, and anothr aount of strontu ford of stabl sotops ( and (. A nuclus dcays nto a nuclus.. Gv, wth justfcaton, th typ of th dcay of a nuclus.. Calculat th radoactv half-lf t T of th rubdu sapl. 3. and N ( ar rspctvly th nubr of rubdu atos prsnt at th currnt nstant t and that of th atos that wr prsnt at th nstant t, nstant of rock foraton. Show that th nubr N*( of strontu atos ford fro th nstant t untl th nstant t has th xprsson: N*( ( λt N ( s th ntal nubr of strontu- nucl prsnt n th sapl. Gv th xprsson of th total nubr of ths nucl prsnt n th sapl at th currnt nstant t n trs of, N (, and t. 5. By asurng xprntally th ratos u and v n th v, nrals of two dffrnt grantc rocks (grant A, grant B, w obtan th adjacnt grant A two graphs.,9 a Why was th sotop usd as a rfrnc? b Show that w can wrt: v au + b, whr: a ( λt,8 -. grant B c Dtrn th valu of a for ach of th two grantc rocks. u Dduc th approxat ag of ach of th two rocks.,7 3 d Why ddn't w us th carbon-4 of half-lf of 573 yars for datng ths rock?
3 Exrcs IV: [8 pts] Chargng a capactor and oton of a rod A q M K Th crcut of th adjacnt fgur conssts of two horzontal and paralll Laplac s rals connctd to an dal gnrator of f E 6 V, a capactor (C of capactanc C. F and a rsstor of rsstanc 5. Th rals, D - + bng horzontal and sparatd by a dstanc c, ar placd n an N x upward vrtcal agntc fld and of agntud B. T. x A tallc rod MN, of ass. kg, can ov wthout frcton on th rals whl ranng prpndcular to ths rals. Th two rals and th rod ar of nglgbl rsstanc. At th nstant t, (C bng dschargd, w clos K. At an nstant t, th crcut carrs a currnt, (C s chargd by q and has, across ts trnals, th voltag u MA u C. MN, locatd by ts x-coordnat and undrgong th acton of th Laplac s forc, has a vlocty V of algbrac valu V Th crcut s thus orntd n th drcton of. dx. a Gv th drcton of F and ts agntud F as a functon of th currnt. b Show that th xprsson of th voltag across th trnals M and N of th rod s thn wrttn as u NM + BV.. a Applyng Nwton's scond law, show that V k u C, and dtrn th postv constant k. b Applyng th law of addton of voltags, drv th dffrntal quaton: E C du C + B C u C. 3-a Th soluton of ths quaton s of th for u C a - b -t/. Dtrn th valus of th constants a, b and. b Dduc th xprssons, as a functon of t t, of V and. c Dtrn x as a functon of t t knowng that, at th nstant t, x. d Dtrn th nstant t at whch th stady stat s practcally rachd. Dtrn th charg Q of (C, th abscssa x of MN and th natur of oton of th rod startng fro t. B B All th Entranc Exa Sssons ar avalabl on
4 Entranc Exa 3 Soluton of PHYSICS Duraton: H 8 JULY Exrcs I : a M.E(t I' + gh ' + g( cos dm. E b Frcton forcs ar nglgbl M.E(t constant + g( sn ; + gsn For sall, sn (n rad + g c Th dffrntal quaton s of th for: + T g. 5.4 s g d Th t quaton of oton cos(t + and sn(t + For t : cos( ; and sn( or (rad For ; and for s rjctd cos(t All th Entranc Exa Sssons ar avalabl on
5 a P f v v v ' Th dffrntal quaton dscrbng th oton of (S s gvn by: dm.e P + g( sn ' + ' + g( sn. + + g(' + ' + g b Th coffcnt (t T (t A (tt A t cos[ (t T ] cos[ t ] (tt t T ; constant t n( T T.3 kg/s All th Entranc Exa Sssons ar avalabl on
6 Exrcs II: a No frcton, consrvaton of chancal nrgy: v + kx constant Drv both sds wth rspct to t: x'x'' + kx'x ; x'. x'' + k x, s th dffrntal quaton b Th gnral for of th dffrntal quaton s: x'' + wth th propr (natural angular frquncy: k x, and th propr (natural prod T : T. c Th valu of :.48 6 rad/s 3 9. and T s. a Accordng to Nwton's scond law: F d P d v x'' x'' x'' + h h x' x'+ x'' + B x' + k x - kx E + F ' hx' cos(t+ ; x D cos(t+ wth : + kx k E cos(t+ b B and D. B s -. 8 c 3 c rd rad/s 8 3 and blu rad/s h E k. All th Entranc Exa Sssons ar avalabl on
7 3 x' Asn(t and x'' A cos(t. By rplacng ach varabl by ts xprsson n th dffrntal quaton, w obtan: A cos(t B Asn(t + A cos(t B Asn(t + For t A + For t Acos(t E cos(t+ Acos(t D cos(t+ A D cos( BA D cos ( D cos ( + D sn ( D A [( A B D ( + D sn( + B ] 4 For th two xtr radatons <, as wll B << ( D E A ; Thus A. ( ( 5 a Th squar of apltud of th acclraton s: (Aacc [ A] E (. E Thus th avrag powr Pav ct ( blue b Thus: Pblu ct ( blu rde Prd ct ( rd P ( blu blu rd. 7 th sky s blu P ( rd rd blu All th Entranc Exa Sssons ar avalabl on
8 Exrcs III: T n + a z p W know that a ; z 4.88 yars a z p -λt Nubr of dsntgratd nubr of ford, th sson s -. λt N*( λt N*( ( λt 4 N*( + ( λt + All th Entranc Exa Sssons ar avalabl on
9 5 a Snc th sotop ovr t. b t ( s stabl and ts nubr dos not vary Thus v au + b a ( λt and b c For th grant A : aa ( (7 For th grant B: ab ( (9 v grant A.8.7 grant B u 3 For A : ta ( ta ta.794 ta yars For B : tb ( tb tb.465 t B yars d Th carbon 4 (as othr sotops s usd to dat sapls whos ags do not xcd T. Thus th carbon 4 dats at axu 57 yars. All th Entranc Exa Sssons ar avalabl on
10 Exrcs IV: a Th forc F s horzontal, drctd to th rght and of agntud F B. b Th agntc flux through th crcut s: B S BnS Bx ; th nducd f : d B dx BV Th voltag across M and N of th rod s thn wrttn: unm BV. snc gos out fro th pont M ; so th postv pol of quvalnt gnrator s connctd to M. dp dv a By applyng Nwton's scond law: F g N ; Aftr projcton along th drcton of oton, w fnd: F B C du C dv dv ; BC du C V BC uc + ct BC At t, V and uc ct, thus: V b By applyng th law of addton of voltags, w obtan: uc. und unm + uma + uad E + BV + uc; du E C C B C + u C + uc duc B C+ EC ( u c All th Entranc Exa Sssons ar avalabl on
11 3 a At t, uc a b and uc a a a t t ; du C a B C B C E C + a a E.6 a 5.94 V B C (... C and.495 s B C So: uc 5.94[ ] BC b V V.594[ C du C.t uc V. 5.94[.t ] (n /s t.t ]..t (n A dx.594 c V.594[ ] x.594 t +. At t ; ct.t ct.97 x.594 t +.97[ ]..t t.t t + ct d Th stady stat s rachd for : t s Th charg Q of th capactor: Q CuC C Th abscssa x of th rod: x [ -5 ] Startng fro th nstant t th oton s unfor, snc V bcos constant. All th Entranc Exa Sssons ar avalabl on
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