University of Malta G.F. Abela Junior College

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1 Univrsity of Malta G.F. Abla Junior Collg FIRST YEAR END-OF-YEAR TEST Subjct: Physics Dat: Friday 17 th Jun 2016 Lvl: Advancd Tim: 09:00-12:00 Dirctions to Candidats: Show ALL your working. Writ down units whr appropriat. Answr ALL qustions in Sction A Answr any FOUR qustions from Sction B You hav bn providd with two booklts. Us on booklt for Sction A and th othr booklt for Sction B Writ down your indx numbr on ach booklt. 1

2 List of Formula Th following quations and formula may b usful in answring som of th qustions in this tst Uniformly acclratd motion v = u + at s = ut at2 v 2 = u 2 + 2as s = ( u + v ) t 2 Mchanics Nwton s scond law: F = d(mv) dt Powr: P = F v Momntum: p = mv Circular motion and rotational dynamics Angular spd: ω = dθ dt = v r Angular acclration: α = dω dt = a r Cntriptal forc: F = mv2 r Torqu: τ = Iα Work don in rotation: τθ = ( ) 1 2 Iω2 Simpl harmonic motion Displacmnt: x = x sin(ωt + φ) Vlocity: v = ωx cos(ωt + φ) Acclration: a = ω 2 x Priod: T = 1 f = 2π ω m Mass on a light spring: T = 2π k 2 of 19

3 Ray Optics Rfractiv indx: η 1 sin θ 1 = η 2 sin θ 2 1η 2 = sin θ1 sin θ 2 = v1 v 2 1η 3 = 1η 2. 2η 3 Thin Lnss: 1 f 1 f = 1 u + 1 v = 1 u 1 v (ral-is-positiv) (Cartsian) Magnification: m = v u = hi h o m = v u = hi h o (ral-is-positiv) (Cartsian) Matrials Hook s law: F = k x Strss: σ = F A Strain: ε = l l Young s modulus: Y = σ ε Enrgy stord in a strtchd wir: E = 1 2 k( l)2 Stationary wavs Spd of wavs on strings: v = T µ Wav motion Two-slit intrfrnc: s = λd d Diffraction grating: d sin θ = nλ Singl slit diffraction: θ = λ a Diffraction by a circular aprtur: sin θ θ = 1.22 λ D Gravitational Filds Forc btwn point masss: F = GM1M2 r 2 Gravitational potntial: V G = GM r 3 of 19

4 Elctric Filds Elctric fild strngth: E = F +q = dv dr Uniform fild: E = F +q = V d Forc btwn point chargs: F = Q1Q2 4πε r 2 Elctric potntial: V = Q 4πε r Work: W = QV Currnt Elctricity Currnt: I = nav Mathmatical Formula Surfac ara of a sphr: S = 4πr 2 Volum of a sphr: V = 4 3 πr3 Surfac ara of a cylindr: S = 2πrh + 2πr 2 Volum of cylindr V = πr 2 h Logarithms: ln(x n ) = n ln x ln( kx ) = kx Rlationship btwn cosin and sin: sin(90 θ) = cos θ Small angls: sin θ tan θ θ (in radians) cos θ 1 4 of 19

5 List of Constant Th following physical constants may b usful in solving som problms found in this tst: Acclration of fr fall on and nar th Earth s surfac: g = 9.81 m s 2 Gravitational fild strngth on and nar th Earth s surfac: g = 9.81 N kg 1 Boltzmann constant k = J K 1 Molar gas constant R = 8.31 J K 1 mol 1 Avogadro s constant N A = mol 1 Coulomb s law constant k = 1 4πε = N m 2 C 2 Charg of an lctron = J Mass of an lctron m = kg Gravitational constant G = N m 2 kg 2 Prmittivity of fr spac ε = F m 1 Prmability of fr spac µ = 4π 10 7 H m 1 Planck constant h = J s Spd of light in vacuum c = m s 1 Elctronvolt: 1 V= J Unifid atomic mass unit: 1 u= kg 5 of 19

6 Sction A Answr ALL qustions in this sction. Each qustion in this sction carris 10 marks. You ar xpctd to writ down th qustion numbr in th margin of your answr booklt. Qustion 1 (a) Th flow of lctric currnt in a mtal wir is du to th movmnt of conduction lctrons. (i) What ar conduction lctrons (ii) Undr what circumstancs will th movmnt of th lctrons produc currnt flow? (b) Figur 1 shows a lngth L of conductor of cross-sctional ara A. Th conductor contains chargd particls which ar fr to mov from lft to right as shown. L cross-sctional ara A Figur 1 (i) Givn that thr ar n such particls pr unit volum and that th charg of ach particl is, driv an xprssion for th total charg of th particls in a lngth L of conductor of crosssctional ara A. (ii) If th particls ar ach moving with a drift vlocity v in th dirction shown, writ down an xprssion for th tim takn for all th particls to pass through th shadd ara. (iii) Us your answrs to (b)(i) and (b)(ii) to find an xprssion for th currnt I in trms of A, n, and v (c) A currnt of 0.5 A flows in a coppr wir of cross-sctional ara 0.95 mm 2. Givn that 1 m 3 of coppr wir contains conduction lctrons, calculat th man drift vlocity of th lctrons. 6 of 19

7 Qustion 2 (a) Exprss th joul (J) in trms of th SI bas units. (b) Th gravitational fild strngth clos to th Earth s surfac is 9.81 N kg 1. Exprss th unit of th gravitational fild strngth in trms of th SI bas units. (c) A track of lngth d has on nd on th floor and th othr nd at a hight h abov th ground. A trolly of mass m is placd at th uppr nd of th track, and runs frly down to th bottom. A studnt suggsts that th tim t takn by th trolly to rach th bottom of th track satisfis th quation 1 t 2 = ( ) 2g d 2 h. (i) Chck whthr th quation is homognous. (ii) Which on of th following statmnts follows dirctly from your answr to (c)(i): A: Th quation is homognous and dfinitly corrct. B: Th quation is not homognous and dfinitly incorrct. C: Th quation is homognous but it may still b incorrct. D: Th quation is not homognous but it may still b corrct. (4) (iii) Giv a rason for your answr to (c)(ii). Qustion 3 (a) A longitudinal mchanical wav W travls through a mdium. Th rsulting displacmnt y (in mtrs) of a particl P at a distanc x from th origin O at a givn tim t is givn by { } 28 y = 0.03 sin π(t 0.35x) 5 whr x is in mtr and t is in scond. (i) What is th amplitud of th wav? (ii) Calculat th frquncy of th wav. (iii) Find th wavlngth. (iv) Calculat th wavspd. (b) How would you adjust th quation givn in (a) to dscrib th displacmnt of particl P if th dirction of travl of th wav W is rvrsd? 7 of 19

8 Qustion 4 (a) Stat Nwton s scond law of motion. (b) A block of mass 4.8 kg is hld at rst at th top of an inclind plan making an angl of 17 to th horizontal. Th block is thn rlasd and slids down th inclind plan. Th lngth of th inclind plan is 2.6 m. Th avrag frictional forc opposing th motion of th block as it slids down th inclind plan is 6.3 N whras air rsistanc is ngligibl. (i) Calculat th initial gravitational potntial nrgy of th block rlativ to th bottom of th inclind plan. (ii) Calculat th total work don against friction as th block slids down th inclind plan. (iii) Hnc or othrwis find th spd of th block at th bottom of th inclind plan. Qustion 5 (a) Stat th conditions for static quilibrium. (b) Figur 2 shows a systm in static quilibrium. Th systm consists of a 20-kg brick, a 25-kg wight and inxtnsibl rops. Th rop QR is horizontal. P S T 1 T 3 35 o Q T 2 R 20 kg 25 kg brick Figur 2 (i) Find th tnsion T 1. (ii) Find th tnsion T 2. (iii) Find th tnsion T 3 and th angl θ which th rop RS maks with th vrtical. (4) 8 of 19

9 Qustion 6 (a) Stat th principl of suprposition as applid to wavs. (b) In Young s doubl-slit xprimnt, a pair of narrow paralll slits S 1, S 2 act as cohrnt sourcs. A typical xprimntal arrangmnt is shown in Figur 3. Th colour filtr bing usd allows only grn light to pass through and blocks th othr componnts of whit light. shild singl slit travlling microscop S 1 S 2 whit light sourc Young's doubl slit colour filtr prspx rul Figur 3 (i) What do you undrstand by th trm cohrnt sourcs? (ii) Writ down a brif dscription of th intrfrnc pattrn obtaind. narrow bam of monochromatic light (633 nm) S 1 S 2 H-N lasr Young's doubl slit Figur 4 (c) In an altrnativ stup Young s slits ar illuminatd dirctly using an H-N lasr which mits monochromatic light of wavlngth 633 nm. Th slit sparation is 0.5 mm. Th narrow lasr bam is incidnt normally on both slits as shown in Figur 4. Th fring pattrn is obsrvd on a scrn at a distanc of 3 m from th doubl-slit. Calculat th fring sparation. (d) A studnt claims that carrying out Young s doubl slit xprimnt is not nough to prov that light is a transvrs wav. Do you agr? Giv a rason for your answr. 9 of 19

10 Qustion 7 A pndulum bob of mass 200 g is attachd to on nd of a string of lngth 0.95 m. Th bob movs in a horizontal circl with constant spd in such a way that th string is inclind at an angl of 25 to th vrtical as shown in Figur 5(a). Th point C rprsnts th cntr of th circular path through which th pndulum bob is moving. O 25 o string circular path (horizontal) m = 200 g C pndulum bob (a) (b) Figur 5 (a) By using Figur 5(b) as rfrnc and ignoring air-rsistanc, draw a fr-body forc diagram for th pndulum bob at th instant shown. Clarly idntify ach of th two forcs acting on th pndulum bob. (b) Find th valu of ach forc acting on th pndulum bob. (c) Calculat th radius of th circular path. (d) Calculat th spd of th bob. () Find th angular vlocity of th bob in rad s 1. (f) Find th priod of th motion. 10 of 19

11 Qustion 8 (a) Explain th diffrnc btwn transvrs wavs and longitudinal wavs. (b) Giv an xampl of a transvrs wav. (c) Sound gnratd by a vibrating tuning fork can b dscribd as bing a longitudinal harmonic wav. A particular vibrating tuning fork gnrats a sound wav of frquncy 512 Hz which travls in air with a spd of 330 m s 1. Calculat th wavlngth of th sound wav producd by th vibrating tuning fork. signal gnrator loudspakr L microphon M dual-trac oscilloscop optical bnch Figur 6 (d) Rfr to Figur 6. A studnt carris out an xprimnt to find th spd of sound in air. Sh first conncts a signal gnrator to a dual-trac oscilloscop and to a loudspakr L. Thn sh mounts a microphon M on an optical bnch and conncts it to th sam dual-trac oscilloscop. Th signal gnrator is thn turnd on and th frquncy st to 725 Hz. Two tracs appar on th scrn. Sh notics that th top trac (which rprsnts th signal bing fd into th loudspakr) is fixd whras th bottom trac appars to shift as th microphon is movd towards or away from th loudspakr. Sh movs th microphon to a point P clos to th loudspakr whr th signal rcivd by th microphon M is in phas with th signal fd into th loudspakr L. Thn sh movs th microphon M furthr away from th loudspakr L to th nxt position Q whr th two signals appar to b onc again in phas. Th studnt rpats th xprimnt thr tims and finds that th avrag distanc btwn P and Q is 45.4 cm. Us this data to calculat th spd of sound in air. () In a thundrstorm, a studnt timd th dlay btwn a flash of lightning and th thundrclap that followd it. If th dlay was 3.6 s and th spd of sound in air is 330 m s 1, calculat how far sh was from th lightning striks. 11 of 19

12 Sction B Answr ANY FOUR qustions from this sction. Each qustion in this sction carris 25 marks. You ar xpctd to writ down th qustion numbr in th margin of your answr booklt. Qustion 9 (a) A car of mass 750 kg on a straight road starts from rst and acclrats at 3.5 m s 2 for 15 s. It thn travls for 2 minuts at constant vlocity, and finally dclrats uniformly, coming to rst aftr a furthr 25 s. (i) Sktch a vlocity-tim graph for th whol 160 s. Labl th vlocity and tim using th appropriat numrical valus. (ii) Find th total distanc travlld in th 160 s priod. (iii) Calculat th avrag spd for th whol journy. (iv) Calculat th rsultant forc acting on th car during th first 15 s of th motion. (v) Calculat th rsultant forc acting on th car during th last 25 s of th motion. (4) (b) A tnnis ball of mass 58.5 g moving at 24 m s 1 hits a smooth vrtical wall at right angls and bouncs off along th sam lin at 18 m s 1. Th tnnis ball is stimatd to b in contact with th wall for s. (i) Dfin impuls. (ii) What is th magnitud of th impuls of th wall on th ball? (iii) What is th avrag forc xrtd by th wall on th ball? (iv) What is th avrag forc xrtd by th ball on th wall? Explain. (c) Figur 7(a) shows a loadd suprmarkt trolly of mass M about to link with stationary stack of two mpty trollys. Th mass of an mpty trolly is 7.5 kg. All trollys ar idntical whn mpty. Th spd of th loadd trolly bfor th trollys link togthr is 2.35 m s 1. Th spd of th linkd trollys aftr th collision is 1.25 m s 1. (i) Calculat th nt mass of th objcts insid th loadd trolly. (4) 12 of 19

13 Figur 7 (ii) Hnc or othrwis, find th mass M of th loadd trolly. Qustion 10 (a) A mass m is attachd to th lowr nd of a light hlical spring suspndd from a fixd point. Th mass is gntly pulld downwards 35 mm from its quilibrium position and thn rlasd from rst. Th mass-spring systm starts to oscillat in simpl harmonic motion. Th tim takn for 20 oscillations is 12.5 s. Assum that th oscillations ar undampd. (i) Calculat th priodic tim of th oscillations. (ii) Calculat th angular frquncy ω of th oscillations. (iii) Calculat th stiffnss constant of th spring if th mass m is 200 g. (iv) Estimat th spd of th mass m as it passs through its quilibrium position (v) Calculat th acclration of th mass m at th instant it is rlasd from rst 13 of 19

14 (b) A glidr P of mass m lying on a horizontal air-track is attachd to two hooks by mans of two idntical springs of stiffnss constant k as shown in Figur 8. Th two springs ar labld R and S. Th glidr is first gntly displacd from its quilibrium position and thn rlasd so that it starts to oscillat along th horizontal. Throughout th motion of th glidr, both springs rmain xtndd. At on instant, th glidr is at a distanc x from its rst position O as shown in Figur 9. spring R glidr spring S Figur 8 air-track spring R O x glidr spring S Figur 9 air-track (i) Whn th glidr is in its quilibrium position as shown in Figur 8, th xtnsion in ach spring is X. Writ down an xprssion for th magnitud of th forc which ach of th springs xrts on th glidr whn it is in its rst position. (ii) By arbitrarily choosing a sign convntion to indicat vctor dirction, find xprssions for th forcs xrtd on th glidr by th spring R and th spring S rspctivly at th instant shown in Figur 9 in trms of X, x and th stiffnss constant k. (iii) Hnc, find an xprssion for th rsultant forc acting on th glidr at th instant shown in Figur 9. (iv) Driv an xprssion for th acclration a of th glidr at th instant shown in Figur 9 in trms of th mass m of th glidr, th stiffnss constant k and th displacmnt x of th cntr of th glidr from its quilibrium position. (v) By using your answr to (iv), xplain why th oscillations of th glidr ar simpl harmonic. (Giv two rasons) (vi) Hnc, driv an xprssion for th priodic tim T of th oscillation of th glidr in trms of its mass m and th stiffnss constant k. (2,2) 14 of 19

15 Qustion 11 (a) Nwton s univrsal law of gravitation stats that any two bodis in th Univrs ar attractd to ach othr by a forc which is dirctly proportional to th product of thir masss and invrsly proportional to th squar of thir sparation. (i) Clarly distinguish btwn th univrsal gravitational constant G and th acclration du to gravity g. (ii) Exprss th units of G in trms of th bas SI units. (b) Dfin th gravitational potntial V of a point in th fild. (c) Th mass M E of plant Earth is 81 tims th mass M M of th moon. Th distanc btwn th cntr of th Earth and that of th moon is m. Earth P Moon x E Figur 10 (i) Sktch a lablld graph to show how th gravitational fild strngth g du to th Earth varis with hight h abov th Earth s surfac. (ii) Rfr to Figur 10. At a point P which is at a distanc x E from th cntr of th Earth, th combind gravitational fild strngth du to th Earth and th Moon is zro. Calculat th valu of x E. (d) A satllit S of mass 750 kg is in a circular orbit at a hight of 2600 km abov th Earth s surfac. Th radius R E of th Earth is 6400 km and th mass M E of th Earth is kg. (i) Show that th orbital spd v of a satllit at a hight h abov th Earth s surfac is givn by: whr G is th univrsal gravitational constant. GM E v = (R E + h) (ii) Hnc or othrwis calculat th orbital spd of th satllit S. (4) 15 of 19

16 (iii) Calculat th kintic nrgy of th satllit. (iv) Calculat th gravitational potntial nrgy of th satllit. () Kplr s third law of plantary motion is simplifid by taking th orbits to b circls round th Sun. It stats that T 2 is dirctly proportional to R 3 whr R dnots th radius of th orbit and T dnots th priod in which a plant dscribs its orbit. According to svral obsrvations and masurmnts, th orbits of th Earth and of Jupitr ar vry narly circular with radii km and km rspctivly. Jupitr s priod round th sun is yars. Show that ths figurs ar consistnt with Kplr s third law. Qustion 12 (a) Snll s law dscribs th rlationship btwn angls of incidnc and rfraction. (i) Stat Snll s law. (ii) Dfin th rfractiv indx for glass in trms of th spd of light. (iii) Explain what is mant by th critical angl for two mdia on ithr sid of a common intrfac. (iv) Stat th two conditions undr which a light will undrgo total intrnal rflction whn striking a boundary btwn two givn mdia of diffrnt optical dnsity. (v) A ray of light travlling through a transparnt mdium X is incidnt on a plan intrfac and gts rfractd into air. Th spd of light in mdium X is 66% of th spd of light in air. What is th angl of rfraction in air if th angl of incidnc is 25? (vi) Calculat th valu of th critical angl if light is to strik th intrfac btwn mdium X and glass of rfractiv indx (b) Th shap of a lns dtrmins how rays which ar paralll and clos to th principal axis ar dviatd. Lnss may b ithr convrging or divrging. Draw a lablld ray diagram to illustrat th maning of th principal focus (focal point) of a divrging lns. Clarly labl th principal focus (focal point) using th lttr F. (4) (4) 16 of 19

17 (c) An illuminatd objct of hight 4 cm is placd at a distanc of 36.0 cm from a divrging lns having a focal lngth of 12 cm. (i) Dtrmin th distanc btwn th illuminatd objct and th imag. (ii) Find th imag hight. Qustion 13 (a) A ball is thrown vrtically upwards with a spd of 16 m s 1. Ignor air-rsistanc. (i) How long dos it tak for th ball to rach maximum hight? (ii) Calculat th maximum hight rachd by th ball abov th point of projction. (b) A ball moving at 2.2 m s 1 rolls off a tabl as shown in Figur 11. Th tabl is 1.25 m high Assum that air-rsistanc is ngligibl. Figur 11 (i) Find th tim it taks for th ball to rach th floor. (ii) Find th horizontal distanc d btwn th bas of th tabl and th point at which th ball hits th floor. (iii) Find th magnitud of th vrtical componnt of th vlocity of th ball just bfor it hits th floor. (iv) Calculat th spd of th ball just bfor it hits th floor. (v) Find th angl which th vlocity of th ball maks with th vrtical just bfor it hits th floor. 17 of 19

18 (c) Wayn kicks th ball towards goal in an attmpt to lob th kpr, but th ball just clars th crossbar as shown in Figur 12. Whn th ball is passing right abov th crossbar, it is at th highst point of its path. Th ground is lvl and th highr dg of th crossbar is 2.54 m abov th ground. Th ball lavs Wayn s boot with a vlocity of 9.0 m s 1 at an angl x to th horizontal. Air-rsistanc is ngligibl. Figur 12 (i) Find th angl x. (4) (ii) Calculat th spd of th ball as it passs ovr th crossbar. (d) A lad sphr is projctd with a vlocity of 22 m s 1 at an angl of 40 to th horizontal as shown in Figur 13. Th lad sphr rachs maximum hight and thn rturns to th original lvl. Figur 13 (i) How long dos it tak for th sphr to attain its original lvl? (ii) How far from th point of projction dos th lad sphr attain its original lvl? 18 of 19

19 Qustion 14 (a) Dfin tnsil strss. (b) Coppr is a ductil matrial. (i)what do you undrstand by th trm ductil? (ii) Sktch a strss-strain graph to illustrat th bhaviour of coppr until it fracturs. (iii) On your graph mark th rgion whr th bhaviour is lastic. (c) A coppr wir of cross-sctional ara 2.75 mm 2 and lngth 1.95 m hangs vrtically from a fixd support. Whn a 1.75-kg mass is attachd to its lowr nd th wir is xtndd by 0.10 mm. (i) Calculat th tnsil strss. (ii) Calculat th strain. (iii) Calculat th Young modulus of coppr. (d) A stl wir XY, lngth m and cross-sction ara 0.60 mm 2, is joind to an coppr wir YZ, lngth m and cross-sction ara 0.90 mm 2. Th total lngth incrass by 1.5 mm whn a tnsil forc F is applid along th compound wir XYZ. If th Young modulus of stl is Pa and that of coppr is Pa, find: (i) th xtnsions of XY and YZ ; (ii) th common tnsil forc F applid; (iii) th lastic potntial nrgy stord in th compound wir. () A rubbr spcimn is rpatdly strtchd and rlasd. Its tmpratur riss. (i) Why dos th tmpratur of th rubbr spcimn ris and what is this ffct calld? (ii) Draw a strss-strain graph to dscrib th bhaviour of th rubbr spcimn during a singl loading-unloading cycl. Assum that th lastic limit is not xcdd. (4) (1,1) c UoM Junior Collg Physics Dpartmnt of 19

Chapter 6: Polarization and Crystal Optics

Chapter 6: Polarization and Crystal Optics Chaptr 6: Polarization and Crystal Optics * P6-1. Cascadd Wav Rtardrs. Show that two cascadd quartr-wav rtardrs with paralll fast axs ar quivalnt to a half-wav rtardr. What is th rsult if th fast axs ar