final exam - solutions

final exam - solutions - ms 123 $6.9. waxlvzw “71qu ac......

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Unformatted text preview: ms 123 $6.9. waxlvzw “71qu ac... git.“ L1 '10“le (23 pts) Problem 1: Multiple choice conceptual questions. Choose the best answer and fill in the appropriate bubble on your bubble { sheet. You may also want to circle the letter of your top choice on this paper. 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. Which of the following is the best resolution of the barn paradox, as discussed in class and analyzed for homework? (Lee. is the one” running with the ladderj Cathy is the one at rest relative to the barn.) CE; Cathy sees theladdner fit entirely within the barn, but Lee does not. b. Lee" sees. the ladder fit entirely within the barn, but Cathy does not. c. Each of them sees the ladder fit entirely within the barn. LQ8)C’ 9‘( L254 i ‘C Caug‘erir‘fD In théelativ‘ity context, which of the following would be an eXample of an “event”? . A light beam hits a sensor. _ . V .1 h Aw. b. ’ Alrightbeam traVels'through space. swim? ale LWQWJ an; “L (k, L 0. Bill travels on a very fast train. t,qu .9» +2.... a“ d. Ted observes Bill traveling on a fast train. i Frodo isveating breakfast ona train which moves at 2>I<l_1087.m/s. Sam is sitting at a picnic table near the train tracks, also eating breakfast. Which pair of statements is correct (note, “slow motion” refers to the eating motions only, not to the overall train speed of 2x108 m/s, which is obviously anything but slow): a. To Frodo, it looks like Sam is eating in fast motion..To Sam, it looks like Frodo is eating in slow motion. b. To Frodo, it looks like Sam is eating in fast'motion. To Sam, it looks like Frodo is eating in fast motion. 3Q To Frodo, it looks like Sam is eating in slow motion. To Sam, it looks like Frodo is eating in slow motion. d. To Frodo, it looks like Sam is eating in slow motion. To Sam, it looks like Frodo is eating in fast motion. '1th W Db‘kl‘fk '37mm—ciiim’i 1M 5m aJkgrg What is the maximum momentum that a particle with mass m and velocity v can have? a. mv ‘ . _ 6. 27710 b. 1720 ,f X m v, f. 2771012 c. mcv \0 \ There is no maximum d. 27711) Kb Q“ h" arb'irw'l7 A reference frame in which objects which do not experience forces do not accelerate is called a(n) reference frame. ' a. accelerating e. Lorentz b. depressed f. null c. Einsteinian g. proper relativisitic inertial h. Suppose Dr. Colton slams a book down on a desk, twice. The two book slams are (in Dr. Colton’s frame) separated by 10 seconds. To an observer in a rocket moving at 0.70 relative to Dr. Colton, the two slams will be: I a. separated by less than 10 seconds _ W”, I OJ « l ” [A 51g“, M pm (133 separated by more than 10 seconds Q {NJ UM Sui (‘9 4M ‘ a c. separated by exactly 10 seconds at Emily is moving at 0.90 relative to Joshua. David is moving at 0.8c relative to Joshua. Emily’s speed relative to David will be: a. less than 0.10 Moi 9,13? VJ. sq “m L (Q more than 0.lc &Vmu : < ) c. exactly O.lc [ x e a so O“wa :5 A! A ballet dancer (mass m) stands on her toes during a performance with area A in contact with the floor. What is the pressure exerted by the floor over the area of contact if the dancer is jumping upwards with an acceleration of a? a. mg 9/ b. m(g+a) E F: 9 mg I c. m.(g—a) « m3 1vvxq L 2 d. mg/A Q 772(g+a)/A N L .m (j #0») f. m(g—a)/A a W N 2 Milli: ) 9mg 8 Wk » We ’ A A Phys 123 Final Exam — pg 2 1.9. boat is on a lake. If an anvil (that sinks) is pushed from the boat into the water, will the overall water level of the lake rise, fall or stay the same? (compared to when the anvil was in the boat) (lg/gs; i W : an.me WW 4» was,“ I a \\ c. staythe same li’\ wile! ’~ AWQWH VolvVWL 1.10. Water (no viscosity, incompressible) flows from a little pipe into a big pipe with no height change. The flow speed (m/s) in th little pipe will be in the big pipe. \ . greater than A! v 1. [An Vb #3 § Wm. H A ; to .6} “lamb b. the same as ‘ c. less than 1.11. Water (no viscosity, incompressible) flows from a little pipe into a big pipe while also increasing in height. (That is, the water is flowing uphill.) The volume flow rate (mg/s) in the little pipe will be in the big pipe. . greater than . the same as \hCVL : Cmsifie’d’ L: ikv) 0. less than Q’i CVU‘S‘J'L émflgfii i‘ Limdfl) 1.12. A gas in contact with a thermal reservoir undergoes an isothermal expansion. The gas and the thermal reservoir are isolated from the rest of the universe. Which of the followmg is true. «wt; \Nkm’ (will; 30-8 a. The entropy of the gas will increase. The entropy of the reservoir Will increase. ‘ . . - - \7 Q >‘/{ 'i‘ P \ v\L) The entropy of the gas Will increase. The entropy of the reservon Will decrease. Q01 N“ n CW? v c. The entropy of the gas Will increase. The entropy of the reservoir will stay the same. d“, W a; WWLQi] d. The entropy of the gas will decrease. The entropy of the reservoir will increase. F _ e. The entropy of the gas Will decrease. The entropy of the reservoir Will decrease. \q SJ—ic a 7L 3 'x k f. The entropy of the gas will decrease. The entropy of the reservoir will stay the same. Q 1)“) > g. The entropy of the gas Will stay the same. The entropy of the reservoir Will increase. Sm . h. The entropy of the gas will stay the same. The entropy of the reservoir will decrease. twig-w: laced” WW9 i. The entropy of the gas Will stay the same. The entropy of the reservoir will stay the same. (‘ngr JQLr (w lire A d“; J 1.13. Whenagas expands: WA; 1 SSC “’3 P 973 f» @ The gas does positive work on its surroundings. WM ‘1} 2 A V '2 «five '9?“ b. The surroundings do positive work on the gas. 7 (3 \ K ‘1‘ Mk 5? Iggy «’0 f c. Work is not necessarily done. _ i, V? Fm (My/S 1.14. If a gas undergoes a thermodynamic change whereby it somehow ends up in the same state it started in: a. The internal energy of the gas will be less than when it started. 1% larvon QALrj 1 :5 g b. The intenial energy of the gas Will be greater than when it started. 31mg Wm “3&ng C9 The internal energy of the gas will be the same as when it staited. d. The change in internal energy will depend on the direction of the change (clockwise vs. counter—clockwise). 1.15. Suppose you flip 13 coins simultaneously. How many different ways could the coins land to give you 8 heads and 5 tails? (1.e., What is the number of microstates in the 8H 5T inacrostate?) a. 32 i i c f. 8192 b. 256 (W) 2 «ii/t C if“? g. 13980 c. 520 % tr 5 - h. 432432 d. 688 i. 67108864 Q 1287 1.16. A “closed—’clgged” pipe and a “closed-open” pipe are the same length. Which will have the lower fundamental fi‘e uenc ? \n q y “We @\ W a. closed-closed . V b. closed-o en 9' p 1:9" )‘ ) 50 “MM”! same fimdamental frequency 1,17. In which case will there be no reflection from a wave on a string hitting a boundary? a. when the wave’s frequency is the same on both sides of the boundary b. when the wave’s speed is the same on both sides of the boundary 0. when the wave’s wavelength is the same on both sides of the boundary (t V «J CAM E 3 3 if V/ v.) t more than one of the above “mg (q) lac/lid 'ifot f {raga I g S Q , ‘ r A )Q q_l“V\- ’ \3 Sflflk/ ) thi =7: Phys 123 Final Exam — pg 3 f(x, t) plotted as f(x) for t=O . —3 1.18. Which wave function f(x,t) is represented by the two graphs displayed? The left—hand‘graph is the wave functiOn plotted for t=0; the right hand graph is the wave function plotted for t=1. V _ f(x,t) = 3 cos(2x - 2t) 6 0N1 bi/\9 NIL V» l”\ f. f(x,t) = 6 cos(2x - 2t) . f(x,t) = 3 cos(2x - 4t) L: L w: l H4 } g. f(x,t) = 6 cos(2x — 4t) 0. f(x,t) = 3 cos(3x — 3t) \ ' h. f(x,t) = 6 cos(3x - 3t) (1. f(x,t) = 3 cos(4x - 2t) i. f(x,t) = 6 cos(4x - 2t) e. f(x,t) = 3 cos(4x - 4t) j. f(x,t) = 6 .cos(4x - 4t) TV i A? 1.19. 7 Light going from a low index of refraction to a high index of refraction will always experience a 180° phase shift, regardless of the angle of the light ray relative to the boundary. 7A+ m, w/qu Mex-Wm Mi aka 1; kqmagi l ‘1' I a. true \ 'i’ r TL; \\§\ 1’ false (\t 0v [3% j " \WLSI. grfiudsk A mate WLS ‘ihL bow/Liij 1.20. The critical angle for total internal reflection exists on both sides of a material interface. Q31 N new 1422’“ n +0 low in 1.21. When you are designing a coating for a piece of glass that needs to minimize reflections for a given wavelength, there is only one coating thickness d that is allowed. _ \ k H _ Q4 4. l A a. \IQJK ‘9’} inoofhfiqj hm 1‘ e MKS/n — L) _( 3 (13 it KlQi’S‘sffiwsybfi W15/5"" Lot; £9055 My. Isl/.5. 1.22. When light diffiacts through two wide slits, the resulting diffiaction pattern will be the same as the pattern from a sinfiwide slit, times the pattern of two infinitely narrow slits. . a: gig W cm mm w W 1 . a V,falwfttcj ’i‘jA’IL.,.. gr ain'tgmy 51(qu has A , false 0 am am”) true false 1.23. <Stiransparent glass, which travels faster: red light (7t=630 Inn) or green light (7t=500 nm)? red light b. green light c. both travel at the same speed flame (U;”:‘ ) Phys 123 Final Exam — pg 4 W, \é/ ‘3 ‘ Zfl‘f (10 pts) Problem 2,Di/sgfis/te/d:vith their final exam, the Physics 123 students gang up and ship Dr. Colton out from Earth on a fast rocket traveling aw? to live the remainder of his days in isolation from the rest of humanity. He lives for 60 more years (in his flame of reference), a the while wishing he had been kinder to the students. How far is he irons the Earth when he dies (in the Earth’s frame of reference)? ‘ Jr (53?“ So 3cm; 9V (V g) 40 Lima J3 AVA fian’Lfii I fl N K A“ g . N I [Fl W {if-"h" can! fv mar. ‘ 11 F bkéquyc L‘{ :5 ’flvvxg-éuiq Q)‘ C;X J’$0iu{.\& “WHQ 4445'“ (0Q f P \/ drgd‘cls {my (:00 r; 91:5 fl; Hm. UJMLQ 44w Wang La Mm analyze.) : ’15?“ "l Phys 123 Final Exam — pg 5 (13 pts) Problem 3. Lee is traveling at +0.70 relative to John. Cathy :is traveling at +0.90 relative to Lee. ;-1 i . .l (a) How fast is Cathy traveling relative to John? - :' . . 7 Ci WWW (17 C e A” l} L! 5 9 JV ‘ 7— "w d me (5 ~— L . cm .» . C 3 l ’* (it, fit: I "l (WC?) (b) John and Cathy both fire laser guns at Lee, and coincidentally both of their shots (traveling at the speed of light) arrive at Lee at the same time. Draw the situation on three different space-time diagrams: from John’s perspective, from Lee’s perspective, and from Cathy’s perspective. On each diagram label the world-lines from each of the three people, the two laser beams, and label these three events: 1 = John fires laser gun 2 = Cathy fires laser gun 3 = Lee gets hit by both guns all A) at ; his”? Jo (q A NC“)?th . L 5M J C (c) In J ohn’s frame of reference, who fires first? In Lee’s frame? In Cathy’s frame? (You should be able to use your pictures from part (b) to answer this; no equations needed.) F / . Sal/had ‘ifirum‘ i AMA '(lril "lr‘a—‘L l {(9 < {@w m ‘ J 1!" (NVKL " £3 s, ' . .lél '{v' k’7 ("k-"‘é (:‘\ \ Phys 123 Final Exam — pg 6 (12 pts) Problem 4. (a) How fast would a 1 kg block need to be traveling in order to have as much momentum as a 0.999 electron? L \Y; 22,364. 0 mchr (etch/m, nm/quhwgvk (b) When you burn a match, about 1000 J of energy are released. If all of that energy goes into accelerating an electron (that is, it all gets turned into kinetic energy), how fast would that electron be traveling? Note: because your answer will be very close to the speed of light, please write your answer as, for example, (1 — 7.7x10‘“)c, instead of 09999999999230 (or whatever the correct answer turns out to be). Hint: remember that (1+x)n w 1+nx when x = small. \(E —; (Y‘ (\mc 2‘ ‘ loOO » fl ,(‘2 " ~ ‘Lt “— ‘t few“. 9, — 2.22040 K ' \4 $2”; ‘ (94m amaze“) Phys 123 Final Exam — pg 7 (a) 5% pH ' I \1 ) t (14 pts) Problem 5. ( S ( 7 ~ (a) A U-tube, open on both ends, is filled with water. The right end is then shielded while air is blown across the left end. This creates a decrease in pressure by the :left end,:which “sucks” the water'up; How fast must'the air be blown in‘order for the water in the left—hand section to end up 10 cm higher than the water in the right-hand section. (Densities of air and water can be found on page 1.) i J V 2‘ ‘\& \ oar/N .., . 1‘“ v I" 2 , 4 L" 7’ QMmmAk ‘ )v J \ I"? l P7— F f ' qurfiw i 1, I l tr/"$v°“‘5”l’ «"73 I: /’ l (b) An air bubble has a volume of 1 cm3 when it is released by a diver 50 in below the surface of a lake. What is the volume of the bubble when it reaches the surface? Assume that the temperature and the number of air molecules in the bubble remain constant during the ascent. fl V V:14g\ ’1 63‘1ng 0\ V\ t 07’ v L MW a Q Q @Vfivfi (‘Qfix ’ k If QI7\‘Q\\D ){ssv "L v, Ir V2” A q, (of Q r I \‘t\.\~\sf ’t \3°°'Ct'%v50> ‘ \Un ‘ ' W" , v Q Q \[2 *’ i‘lcwt 2) {\ [,ol 40‘ '1, \/L: 3.95 CM Phys 123 Final Exam — pg 8 (12 pts) Problem 6. A11 insulated chest has a surface area of 1.2 m2 and a wall thickness of 3 cm. The temperature of the inner surface is 4°C, and the temperature of the outside surface is 28°C. The chest is made out of a material that has a thermal conductivity of 0.06 in standard SI units. (a) What are the standard SI units for thermal conductivity? Eqm‘. P: kits/V N) \L: 371‘ Siqfi \" aL ;; OT gaw‘K,¢3 ‘0\ (GT \u 53" Eh \ mg. Qicvh-‘\V“VQ“~ gfi’KVQ‘LVQ—l l9-“Q. 3mm!“ \NI big—3"") W): Hi5 L" 3 at ;. “if/55 t; /* m Phys 123 Final Exam — pg 9 t it - Kai/g] CV v stag I ate a V K I. .u (16 pts) Problem 7. An engine using 0.12 moles of a diatomic ideal gas is driven by this cycle: starting fiom "state A, the gas is cOmpresseduadiabaticallyuntil it reaches state B, Then, the gas is heated at constant pr‘essUre untilit reaches state C. Finally, the gas is cooled at constant volume back to the original state. Various pressures, volumes, and temperature are given in the table. ' Pr . - II W:- nM «a T = A Emma was; Nj 05M T :zmn. TL‘LA ‘6‘ \“P‘QCNV- kcubl POM w 2 \fh‘sikfi ‘3th {a {CA "\W‘AJKQ- (b) Find the heat added to the gas during each of the three legs. arm” fl Alohoiraficv a Q‘QcQ ( 32:9 eel-54 9— (Sta nCPth 412%; “‘HLW'S‘K) 2 .-—-- i 6.477: T 4245‘”; HQ? J a?” C-A cox/Q” V ' OZ ”’ “Cu AT 1 L®\1)(”2; g y) (7)00» Mo?) K” ’— ,. TL:ZVNVL 2 £155? - My: a W9 62> % How much net work is done by the gas each cycle? 17" F — ‘3): [EU thck, : O _—5) KQM’CMK \ LON) , ’- iaglt \N c» G‘ictol‘5143 : lli‘lbj “W ‘Lu'u ‘91 :‘vwfiv’ub (a) W What is the efficiencyfl of the engine? A .3} \’L ‘0 .a a we b.) a ,3.— : @220 mm \\’\ MW»? (e) fidj What is the maximum theoretical efficiency for an engine operating between the same minimum and maximum temperatures? ’ , TR ’ 2313 a f‘ [0 QM“ “x :(‘u t \‘ M0? \ Phys 123 Final Exam ~ pg 10 ("12 pts) lgroblem 8. While attempting to tune the note C at 523.3 Hz, a piano tuner hears 236mm between a reference oscillator (at 523.3 Hz) and the string. 1,9 {3 = €113 t. a (a) When she tightens the string slightly, the beat frequency she hears rises smoothly to 3.5 beats/s. What is the frequency of the string now? \A P 3 «Fm/WV} vak kfj LAW M 3 I Tumkgwg—e "P (am An f Big H? (b) By what percentage should the piano tuner now change the tension in the string to bring it into tune? at“ WA? M 2 W “mas—«WT to, [\L!7’: ifs.) 'L TL Q (at) ’(K ~\‘(:\ .2. T , 1 52w) , r 1 M (52ch Phys 123 Final Exam — pg ll p I (14 ptS) Problem 9. As a car drives paSt Steve, the driver sounds the hom. Steve has perfect pitch, and notices that as {the car approached, the horn sounded the A above middle C (440.0 Hz). After the car passed, the pitch dropped dowri'three half steps to an F-sharp. How fast was the car going? W0 mutt/31cm '. moo” 4&7, on: gm 3 3g??? I” \/:\/a 1 0W4” i \ f «m t ‘ Vt 0 v. Vol/ma (11/ ’fVWWéiS : 199W) : F‘F v-\/5 ' m .‘C 0W”) W7 " if we ,‘Ffing‘ii K Vivi V+\/§ "\ v t. “f +qv—m7 fifi’vvhg \f-fl‘rs VS A E x\/ W Jr : *‘ A w ,1 m 2 W X W3 E “(is , atQPxfiii W, m Phys 123 Final Exam — pg 12 (l0 pts) P‘roblem 10. You book lists this as the “lens-maker’s equation”: i =(,7_1)(_1___L] f R1 R2 However, when I consulted Wikipedia, htt ://en.wiki edia.or /wiki/Lens o tics , I found a slightly different version of that equation: .‘1 d i:(n—1) i._._1.+(n ) f 7 R1 R2 anR2 In that equation, d is the thickness of the lens. Assuming the Wikipedia equation is correct (which I believe it is), apparently the textbook’s equation is making an approximation that the lens is infinitely thin. But, of course, lenses are not infinitely thin. If I have a bi-convex lens (the normal lens shape, curved outward on both sides), with both radii of curvature equal to 10 cm in magnitude, and my lens is 0.6 cm thick, what is the difference in the focal lengths predicted by the two equations? (The index of refraction of the material is 1.55.) s i . ‘( q i ch hug; (Mill) i : “ KY:— > 1 [i V% Cw rah“ { 3 gm? ea“ " (A \ i L 4 =[ oss’ (fim] : toil cm 1“ K; v?! {\(flu‘fiwfiy 5.J [gafh‘kwuu QM < 4 f 9 if" "ix '+\ [I m {a V‘\ )N I is, V) i. o v J :9 £3 .9 Phys 123 Final Exam — pg 13 ' k (14 pts) Problem 11. An object is placed 50 cm to the left of lens 1 (diverging, f = —8-0 cm). Lens 1 is placed 30 cm to the left of lens 2 (diverging, f = —40 cm). How far (in magnitude) from lens 2 will the final image be formed? Will the image be to the left or the right of lens 2? Will it be real or virtual? What will be the total magnifiCation? You do not have to provide any ray diagrams for this problem. HSQ 7 H I W o —_\ 0‘ ’9 V“ \1 pl: Kfi'%l)~\ : ~30 V 90 Vii—F F. _ 23:17”) A M: 35.. t a) 1 + £93 I --—-\ \vxg 1 'v a?) L U»; \ A‘- byjuxk'oi-(LM A C kW L . Kb 3 1L“ “lib'fi 2 p = + edit“! el ‘ ) )4 “W ‘ i “'L "' «Tia/DEC:th Vlflwli Oil ‘ ’ch F2, r W djénlp talk “New 2._ z , ,— » ft; “GEE—L4 ? M2“ 6” ’ (90.19“? «3 9' }\!\%;Mi(”\/\‘Z : (ham?) a mm A? lCll : AL cm (away from lens 2) Final image will be to the [LL ‘i’ (left/right) of lens 2 real vs. virtual: \/ (“71 "i total magnification: A: Phys 123 Final Exam ~ pg 14 / i pts, no» partial credit) Extra Credit. You may pick one of the following extra credit problems to do. (If you work more than one, only the first one will be graded.) (a) Use the Lorentz transformations to prove the relativistic velocity addition formula. Hint: Pick an arbitrary velocity in reference frame 1, find a point on the world-line corresponding to that velocity, do the Lorentz transformation to reference frame 2, then find the slope (and hence velocity) of the world-line in that reference fi‘ame. l L—‘32/6%ei4* I L 3 (1732‘. , A' / / SKA/I QILKLQ‘ oh MW\&\t\‘\SL 3 , l W M “l at) Sig? 3' TWA guvfi Us Chorcfigfi off-f 04 of iii/NC} : 3(091 : Cf (Wrap {53‘ 2 K" Xfi/fifl, X [331 The; cwth ’ figzfilz+l (b) Suppose I developed the ability to travel faster than the speed of light. That means (in the Earth frame of reference) I could travel one light year in 0.5 years. This would cause a tremendous problem for causality! Namely, the endpoint of my trip and the start of my trip will have a “spacelike” relationship to each other instead of a “timelike” relationship. And, as I claimed in class, if two points on a space-time diagram have a spacelike relationship, then one can always find a reference frame where one point happened before the other, and another reference frame where it’s the other way around. Therefore, there is some reference frame where my trip’s endpoint happened before my trip’s start! In that frame, cause and effect are reversed! In fact, there are an infinite number of such reference flames. Find the speed of the specific reference frame where I arrived 0.5 years before I left. Hint: that reference frame will be traveling in the same direction that I’m traveling. WWII“ -, , Mimi! 7‘ l “E‘faw‘k E1” 14" 1"“sz ram/‘1. t" . . § . F f-:> ~ E T’i ‘ \ i r x i a. L I I ‘ 5 ) i] 1 \- A / IIII ~~Wfli : \ “A ‘k ”\ (n '_ \S “th rcpt, 1 L0 or ,Zblfi"f C) P—ES «>3 2 N /// 0* 7 FL (I7 (‘5 + 6) ¥/{// 23,103“ (9 I Lr Phys 123 Final Exam —pg 15 f g, ((33 ...
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final exam - solutions - ms 123 $6.9. waxlvzw “71qu ac......

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