6b_sample_mt2soln

6b_sample_mt2soln - Problem #1: Part 1: Traveiing waves...

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Unformatted text preview: Problem #1: Part 1: Traveiing waves move along five strings as shown below. All five strings have the same finear mass density M, but are not necessarily at the same tension. fen-“fl”? e. Rank the five strings according to their tension, from highest to lowest (7 ' 5 4t? __________ 4 Part 2: Two cylinders, A and B, each initially contain 4 cubic meters of (diatomic) nitrogen gas at identicai pressures and temperatures. Cylinder A then expands isothermaily to a final volume.attiredb‘ic;_._rtjetersr, while cylinder I B expands adiabatically to the same final volume of 8'.cu't§§te}'-mete s‘i';':._: _ __ In each square of the followrng table, write ” l; " If the quaint expansion, ” ” if it decreases, and " - ” it it stays the same-ii Cylinder A (isothermal) Temperature of the gas H— Entropy of the gas Mass of a single gas molecule Mean free ath Of ‘ as molecules 4? awv _ w Average see I = v _ ‘ 0e ules i" j ‘I-' ? Extra credit: next to each quantity that you marked as increasing or decreasing above, indicate the factor by which it changed. (1e, write "x 4” if the quantity became four times bigger, or “x 1/3“ if it became three times smaller.) Problem #2: True/False Mark each statement as true or false. 7 if a statement is false, then correct it, by adding, deleting, or changing a few words, so that it becomes true. M" ‘9 7 1) An irreversible process always causes the total gy of the universe to dé‘CI‘ease, while a reversible process does not change the universe‘s Mam inn/mac - 09%? m gall“ 922406 2) High frequency sound waves travel 3 er )hfi low-frequency sound waves in the same medium. 3) At the same temperature, the molecules of a diatomic ideal gas have a higher avera - nergy than the molecules of a monatomic gas. (more aggro? mf- I 4) Two otherwise identical guitar strings are at different tensions. If waves travel . nine times faster on string A than on string B, then the tension in string A must be t etimess er. 2g w 'W W” J?) 5) The fundamental difference between the past and future directions in time, is that eyKQy increases as we move toward the future. - 6) if an ideal gas contracts isobarically, then the entropy of the gas ingeés. i _ decider->305 (Le/(mug, Lwéat‘i‘ (aw; (Dz/FD ' Problem #3: Sanity Check As always on sanity check problems, do not try to solve this problem directly (unless you want to try for extra credit at the end.) Instead, think about the physical situation, examine the five equations given as possible answers, and try to reason out why at least four of the equations give impossible or physically absurd answers. The Situation: - An air-filled cube of sidelength L floats on the surface of a lake. The water in the lake has density pw, while the air-filled cube can be considered to have zero density in comparison. (Thus, it floats on the surface of the water with none of its volume submerged.) A diver then ties a string to the cube; on the other end of the string is a metal sphere of density p and radius R. The sphere's weight pulls the cube downward until they come to a new equilibrium, with the bottom of the sphere a distance 2: below the water's surface. ' The possible answers: -iil Z; gfiéfl—fi ’- a) What units should the answer have? Min/5 Which equation(s), if any, does this eliminate? Eli} 69M} 0m¥d§Mm9EmQ795> b) If the density of the sphere were equal to the density of water, what would be the value of z? Which equation(s), it any, does this eliminate? Briefly justify your answer. W Wat/kl; h€h+fitl£9 g0 H” wot/iii “0+ C‘wa “V‘J’lfwfi‘kf What/l, Tim/L9, ha Prior,” 2 ’3 a Tin; elim‘mmle9 (xvi/nth dorm} +hqiL i9 TF9» Z a c) If you were to replace the sphere with a larger sphere of the same density, would .2 increase, decrease, or remain the same? Which equation(s), if any, does this eliminate? Briefly justify your answer. 0155miminfi 'l’laat‘ 7fcl/ 0' [0593f waif? WOVJCQPV‘H w CMEC/ pnr’f'wif MV‘J@VM#VI Tia, am; 2 5ch will, Er‘w‘una 9 culmle DION“? “Fl/“71+ 0‘9 2i. d) If the cube were replaced by one with a larger side length L (but sill zero density), would z increase, decrease, or remain the same? Which equation(s), if any, does this eliminate? Briefly justify your answer. I]? L Mug \afggr/ -]’Le (LAM Cat/1U 670‘s” 0! SLer-DI/ 433mm 0M4 G16” W “KL gap-e, VOlmw-e §MEMV§e4 0W; WW W 964% L“030’”+—€7M' 60 01‘9 Ll] ZSMWMLL Tbfié @/ VVXMPOk claim/x9 Z hail" clean/ml on Lori all. e) Which etion(s), if any, might be the correct answer to this problem? Extra C redit: Derive the answer to this problem " or real ", from first principles. (Try using logic similar to the damn dam problem fiom your homework, with a few important details changed.) Does your answer match the eqaation(s), ifany,,_that you selected in part (e)? Problem #4: A cylinder of radius Ft = '10 cm contains a movable piston as shown below. Initially, the piston is a distance x1: 2 meters from the (closed) left end of the cylinder. The (airtight) section to the left of the piston contains a quantity n == 3 moles of helium gas, at unknown temperature T1 and pressure P1. The section of the cylinder to the right of the piston is open to the outside, and will remain at pressure Patm throughout the problem. ' A string is attached to the piston, passes over a pulley, and supports a weight of . mass M: 60 kg. a) Calculate the initial pressure P1 of the helium gas. ‘ Page/“ball; JFajl’aM Pig'ka ' 4: 77—22 b) Calculate the initial temperature T1 of the helium gas. Heat is now gradually added to the cylinder, and the piston moves slowly to the right until it is a distance X2 = 3 meters from the left end. This process takes place very slowly ("quasi—statically"), so that you can assume the piston's acceleration is essentially zero at ali times. ‘ “gm”? [E 0) During this expansion, does the gas pressure P increase, decrease, or remain the same? Briefly justify your answer. hemning ‘H‘mfl, 90ivaj w ‘Furcaé W‘o/ COWGI'JQVeCé \lv‘ Patl+ Oi WWW mgr stall 10:41am in may! “ll/“\ch News Ovif SOKW‘t—BOA fiw P! Lolclfig Tkm‘fi/ ‘19 OWL {go‘OCtrt‘C Prof/€49, d) Calculate the final temperature T2 of the helium gas. «—.—~ ,. W; of HA V r 77rle I __ 2 _ N k WAR V; ‘ Ei-Vt [91% (52:9 Nokwm MOM; {feat/wee) l’x’ Ctrfi‘l’le éawe a9 Tn Fat—FL) Thug] 7:: ’3')“, : 2061c ' (014% Q-Lgo’ til/Lyle RAUL? HUME” Eta-$2; TJQQK?Q§ (OW/ q9¢sz 9) Calculate the work done by the gas on the piston during the expansion ‘hmé 76 01 Camsltfiw‘l‘ ’ fin‘fiéw/a Precesg $0 W 2“ 'PA r I : (gllocvo POO (Va-V.) : (at 000 Pa) (0.094256. 062%?) f) Calculate the amount of heat 0 which was added to the gas. M AUSQWW Q:\A/+AL)' VIE \K‘mam/ W alrtmfé. term; DU! \LQZ’ RMJFVWA Qvélffiy oL-q WVWHGW‘TC 765? U:?—;NW w Edi“ 7°‘Q:w+au 3- 2§4OT+ 3910 T “an-AU 3 - M) : gamma; at) -' ' __-—“ » 39l‘93 Problem #5: An inventor of mass m = 60 kg wants to rescue a cat who is trapped in a tree, at a height h = 5 meters above the ground. She can‘t remember where she put her ladder, so instead she builds a perfectly reversible engine which extracts heat from a furnace at temperature Tr: 1127 Celsius. The engine turns a wheel which can be used to do work on the environment (and thus to lift the inventor up the tree.) ' The engine also dumps a certain amount of heat into its coolant lines, which contain water at temperature To = 77 Celsius. ‘ She fires the engine up for a test run. In a time tfest 20 seconds, the engine extracts Of: 400 Joules of heat from the furnace. a) What was the change in the furnace's entropy during this twenty-second interval? b) How much heat 00 was dumped into the coolant lines during this time? (Remember: the engine is perfectly reversible.) aim-CZ Me Pfage§5 TS five/9mg A SC Jr A :o c) How much work did the'engine do during this interval? Eyérfiy QWSQVVOL‘Flrf/‘Vl '1 Call 3 W + Q0 W ‘3’ 0H "— QC} Satisfied with the engine's performance, the inventor now hooks the engine to a pulley, and uses it to lift herself from the ground, moving upward at constant speed. d) How much time does it take the inventor to reach the stranded cat? w (Mei/«e; can clcr 2003" o(" Work ’in 2C9 Semi/[{9} “Pl/MS} 1+ (p69 :9; (or [EM/aha mang if: W (Free) _ a - ‘ Defiant “two a K _ ‘ . i0 olmbd’lx ‘lwse/ “he Fereon MM+94IH grownla‘iccnql emu W Mal1 1—. (got—Q {row/51%;”) 270 ' ‘ K PC If The ‘hvae {or ‘ike €“97re‘lv CLQLMK We (fin/95 . - e [5600?]? : mam, or eta-ww- ...
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6b_sample_mt2soln - Problem #1: Part 1: Traveiing waves...

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