PHY053 F06 Exam 3A Sol - Physics53 Fall 06 Exam 3 A Name M6...

Unformatted text preview: Physics53 Fall 06 Exam 3 A Name: M6 V Section: I certify that] have abitled by the rules and the spirit of the Duke Community Standard. “WW PLEASE READ CAREFULLY BEFORE YOU START “*i‘ Do not forget to write your name and section number above. You must do all multiple choice Questions and two of the longer problems. If you attempt to do more than two of the longer problems, please, indicate below which two problems you want be counted. Remember that credit will only be given for solutions that show your work. Do not just write down your result, but show how you got it. No work shown = no credit. ll‘ numbers are required, use g = 10 111/52. Record your MC answers (by letter A - E) here. No credit will be given, ifyou do not record your answer here! mafia MC3_§Q MC4§ MCSQ Check here, which two of the long problems should be counted: MCI P1 P2 P3 P4 [To be filled in by graderz] Grading scores: MC P1 P2 P3 P4 Total: M ulti le Choice Problems: 6 Points Each 1. The staff of an excursion boat in distress on a small lake lowers the lifeboat together with several frightened passengers into the water. They succeed without dropping a passenger or letting any water into the lifeboat. Does the water level of the lake change in the process? :jAgNo, the water level remains unchanged. B) Yes, the water level rises. C) Yes, the water level falls. Consider two identical oscillators x1(t) and x20). The maximum displacement of the first oscillator, A1, is twice as large as that of the second oscillator, A2. The first oscillator is released at rest from its position of maximum displacement, the second oscillator is released at the same moment with positive velocity from the position x220. If the period of both oscillators is T = 3.003, when do the two oscillators first have the same displacement (x1=x2)? [Assume that all motion is frictionless] v?- A l , 1338:3212: t“ “ 2.5. if” ”i 5! my. ~———2> tie“ at " Z— n (10.375 S. reg-Ir- a an wi‘ i as, m: nit m» ’érlffriq‘wg @0529 s. E) The two oscillators never have the same displacement at the same time. A transverse wave propagates along a string, which has a gradually decreasing linear mass density. As it propagates toward the thin end of the string, its 3?” ~ w A) frequency increases. ”if: ....:t. A = :7 f: aim, Waltz B frequency decreases. f i’ / wavelength increases. > J;— D) wavelength decreases. M779 ?\ M f“ - E) both its frequency and wavelength change. 1% Consider two vessels of equal volume. One is filled with air, the other with hydrogen gas. Both gasses have the same pressure and temperature. Which vessel contains more gas molecules? A) The one filled with hydrogen. B) The one filled with air. :5 W @Both contain the same number of molecules. 5. An organ pipe (Closed at one end), which was designed for a room with a temperature T = 20°C, was produced outside of specifications. The organist determines that the pipe plays with the correct pitch, if the room is heated to 30°C. The velocity of sound in air is given by the expression v = (331 + 0.6T) m/s, where T is measured in degrees Celsius. By how much was the pipe too long, if it was intended to play the note E (330 Hz) at 20°C? [Neglect the thermal expansion of the pipe itself] A) 0.227 cm. _ z %% ’13” 0.303 cm. @0455 cm. % S if Mg 5 2 i D) 0909 cm. g “T sf? E) The pipe was produced too short. The following problem is worth 4+6+5+5 = 20 points: P1. A pendulum is made of a uniform disk of mass M and radius R pivoted around a point at its edge, as shown. Consider small oscillations (small 0) of this disk about its stable equilibrium position. a) Draw the free—body force diagram of the disk for the situation shown, in which the center of mass (CM) is displaced by x. b) For a small displacement x of the disk, corresponding to a small rotation 0 about the pivot point, what is the torque of the disk about the pivot point in terms ofM, R, and x? c) Use the result of (b) to derive an equation involving x and its second derivative, dzx dt2 d) What is the period of oscillation ifR : 0.500 m and M = 1.20 kg 7 which has the form + cozx = 0 . % a P a a,” M fifiémg igw gig,” e53 Egg; “fi‘erm fig, @fihg gmfimg “We ta wetligx i , a, 5» fi‘ig a: an &% 2%, 6i gem-g ml 15%;. ”3 gwre W? gee» a are» semmwéfflaeffia %%[email protected]%rfiiag @ éwfiéafipé” “if? e a “3 153% as”. ' W. 5%: a ” ”ragga @5%& fig% “3% g??? E; gag a: Q a. 2:. 324% at M an a E??? a lead” m t {A} 5"?“ «fig figg’g? £73» Z°€© «o “Wig a. m2 522%. « v;s«n .rj g at 9%“ gr? at m a? {at an a The following problem is worth 6+5+5+4 = 20 points: P2. Two identical cylindrical liquid containers of radius R and height h are connected by a narrow pipe of radius r, as shown in the figure. The left container (A) is initially filled with water; the container on the right (B) is initially empty. The pipe, which leads from the bottom of container A to the middle of B, is controlled by a valve, which is initially closed and then suddenly opened to allow water to flow from A to B eventually equalizing the water levels in both containers. The pipe has a diameter of 11.00 cm. [Assume that water is an ideal fluid] 3 a) Find the speed with which water is flowing through the pipe as a function of the remaining water level in container A. b) Use the result obtained in (a) to obtain an equation for the amount (mass) of water remaining above mid—level in container A as a function of the speed of flow in the connecting pipe. c) What is the rate (mass per unit time) of water flowing through the pipe, when the container B is filled to one quarter of capacity? Use R 2 10cm, 11 = 50cm, and r = 0.5cm to obtain a number for the flow rate. d) What is the total work done by gravity on the water when the water level has been equalized and both containers are half full, for the numbers given in section c. The following problem is worth 5+5+5+5 = 20 points: 1’3. Consider two pipes of the same length L = 0.520m, each closed at one end. One of them (pipe A) is equipped with a movable piston. which can be used to reduce the length of the available air column in the pipe. The other pipe (B) is fixed. [Assume that the air temperature is 20°C.] a) What is the fundamental frequency of pipe B, and what are its first two higher harmonics? b) At which positions of the piston (measured relative to the open end of the pipe) will the fundamental of pipe A have the same frequency as the first two higher harmonics of pipe B? c) How long would a string need to be to have the same fundamental frequency as pipe B, if the speed of wave propagation on it would be 265 m/s? d) What is the wavelength of sound emitted by the string described in (c), when it vibrates in its second harmonic mode? B D L ., at? ”if m» r gee esteem egg,” {gamiw it at are” 3’1 afieflfi the , w .. a . e a. fa My? (“M p, 5 a w ‘7‘ g Fé’ w , be 6%” , fig'iagg fl 6“ g {g} rate. gratin», Manatee at; gaiwweeéi, a» We greet” t’t 5?? a it ta ; a; m “ h ’7 a ateggfi if??? %% Emerge? Qéawwmfi my“ “fig 5 7 ‘ «2‘9” 19 I {£ng '3 f” iflflf iiaz gfiéfigfltiigwe §Q° ”% fig; j ($25??? a ”t? t if tie i m ’3» . (fl; .2 f If" g; 55* "1:: mg” ‘3 if) giving: ifiemammewgw mafia Aafi 39%, «g meg .. ”at? Eé§wfs me i=3 v Magm— a; W. Mfr?” " if g €123 eta 3% ‘ng fifesr’ig “I: - }‘ Vim, $3; 5 {i “t W I 5;; 6 W23» Wm”? 51?; M $319} Q” “2;; 5M3» I ”if T Lag \ , t a a?“ éeff m gigogw‘agjfi a: tg‘ m : ZL The following problem is worth 10+5+5 = 20 points: P4. Two loudspeakers standing 1.60 m apart emit sound of the same frequency (526 Hz) and loudness (65 dB measured at 1.00 m distance from the speaker, when only one speaker is turned on). Assume that the loudspeakers are in phase a) Find all the locations on the dashed line shown in the illustration (directly in front of one of the speakers) where the two sound waves interfere most constructively. b) What is the loudness measured at a point 4.50 m in from of the center of the baseline of the speakers (indicated by P in the illustration) when both speakers are turned on? c) What would the loudness be at point P if the two speakers were 180° out of phase? as {a} is: amiss m mamasa’gsaaes smears} usesaamafwstg ”Essa astsaas’i‘a Wstaiiafif or 3 f} I til «{st (sass tag; asaaasaa’; assess“? saLafi-%aa gas ésaasas Qje spit . t9 MMWWW WWW/hm...” Qina‘éas‘ae’ mafia- str W axisaajrsaasah was 3t :5 5%: e W% a flags? ss. 3?” gagié 7 s' ”L at (a g“ fibwaw was Kassléi“ Ra's {Kat/h sfmhfig‘ We? A 5:- ”m 2;in W as as: rasat ) aaassgsaaasa as saw astaasa {wean . , a 4 fl... 1 ,. ,- W was =artaio§ig§gaags 2: salsa; a. armaasamm i. s sax ,. 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