F2001s - CORNELL UNIVERSITY ‘ Department of Physics ...

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Unformatted text preview: CORNELL UNIVERSITY ‘ Department of Physics « Physics 214 Prelim, I Fall 2001 NAME: 50407/0/1/5 SECTION: ’ Instructions —~ To receive credit, you must place your answers in the boxes provided whenever required. ' —— Closed book; no notes You may use a calculator. —- Check that you have all 16 pages (including cover page) The formula sheet is distributed separately. — Important note: Except for some Challenge problems, each part of this exam is designed to be answered Without the answers of previous parts. The parts within a given problem become more and more difficult If you get stuck on one part skip to the next problem and come back later if you have more time. Problem Score Grader .1. (28 pts) Total (100- pts) Page 2 of 16 PHYS 214 Prelim‘ 1 Contents 1 Problem 1: Damped Harmonic Oscillator [28 points] 3 (a)’ Equation of Motion (7 points) .............................. 3 (b) RealPartofAUpoints)...............................‘.. 4 (c) Imaginary Part of A (7 points) Challenge Problem! ................. 4 ((1) Rough Sketch of the Solution (7 points) ......................... 5 2 Problem 2: Wave Equation for a String [8 points] 6 3 Problem 3: Standing Waves in Sound Tubes [8 points] 8 4 Problem 4: Forces, Slopes and Curvature [14 points] 9 (a) Forces (7 points) . , . .‘ ................................. 9 (b) Acceleration and Curvature (7 points) Challenge Problem! ............. 11 5 Problem 5: Lab Experiment I [21 points] 12 (a) The Fundamental Mode (7 points) ........................... 12 (b) Higher Modes at Fixed Length (7 points) ........................ 13 (c) Higher Modes at Fixed Frequency (7 points) ...................... 13 6 Problem 6: Sound Waves versus Waves on a String [7 points] 14 7 Problem 7: Generalized Boundary Conditions for a Sound Tube [14 points] 15 (a) Air Pressure and Displacement of the Piston (7 points) ................ 15 (b) Boundary Condition at the Piston (7 points) Challenge Problem! ......... 16 List of Figures A mass-spring realization of a damped harmonic oscillator ............... A chunk of string with air resistance. .......................... Closed tubes filled with noble gases ........................... ‘. A chunk of Vibrating string. ........ - ....................... Lab Experiment I. ........................... y ......... 1 Generalized boundary conditions for a sound tube, .................. 1 ubb-DIOH mmcooomoo C740! PLEASE TURN PAGE ' Page 3 of 16 PHYS 214 Prelim 1 Figure 1: A mass—spring realization of a-damped harmonic oscillator. 1 Problem 1: Damped Harmonic Oscillator [28 points] A damped oscillator is modeled as a mass m with equilibrium position x 2 arm acted on by an ideal spring of spring constant k and a viscous drag force proportional to the velocity, Fdrag : —bm77. (See Figure 1.) (a) Equation of Motion (7 points) Show that the Equation of Motion for the damped harmonic oscillator is: l 5,42% ex resyz‘ 0 a F = , ) x) X misac Wax . 144 m9 o/z‘Aga 0ij ,[email protected];x =_ W‘Zx x/fl’ m 012%” aim)" PLEASE TURN PAGE Page 4 of 16 PHYS 214 Prelim 1 (b) Real Part of A (7 points) In the case b < 2% the expression _ w) = M + éRe [Ae(-b/2+w/)t] 7 w’ 5 mg _ 52/4 (1.2) is a general solution to the Equation of Motion (1.1). d2; Find the real part 3&2 [A], given the initial conditions x0 = x(t = 0) and 00: ch —( your answer using no quantities other than (co, v0, xeq, b, and w“. I??? 1%2. (hf/5.4% W51mm=4€fz¢= 0) 04L 231$ i W soflme/m (/ Z] are i; :L 56-:— :C(t= a): 7+?:Ee[Ae(€/ QQOJ=Zef+BZBJ => (0) Imnginary Part of A (7 points) Challenge Problem! t— — O) Empress Using the same initial conditions as in part (b), find the. imaginary part (311}[A] Eatpress your answer using no quantities other than 230, v0, xeq, b, and we. H__int: Write A— - A +2'41, where A, = Re [A] and Ai = Sm[A], and solve for A. I444 $144 fi/M’éuo‘fzw Wélébfl¢ofi o = ”(250) 014 1144 gaziagsowmf {2; M 077:4? [1: "FA +z/4 . gar/14.0)— 9+7 lager/“Wk =2€;[/4( g+zij0j7= @[él 7L:Aj/thw2/ =31? gfl ”MAM/35A +60%rj =~£fi[——w/lg => 4' = 2:; - €55 A” 57 AC: fiAJ=~28~x7~fi~¢9 4 _, _ / a); 5% 2 we 0522;»: Arm/1*] =Wtro-é/ww ’ , PLEASE TURN PAGE ' Page 6 of 16 PHYS 214 Prelim 1 5; i+A Figure 2: A chunk of string with air resistance. 2 Problem 2: Wave Equation for a String [8 points] Note: In this problem we use the same notation as in lecture. Consider a chunk of string of length A and mass per unit length a. The string is under tension 7'. The displacement in the vertical direction is given by the variable g which is a function of both position a: and time t. As opposed to the derivation 1n lecturywhere air resistance is neglected we now include it Note that air resistance on a chuck of string moving with velocity 1) creates a force opposing the direction of the velocity, 1. e with a y-component Fdrag,y = “bmchvy : (2.1) where b is a positive constant and inch is the mass of the chunki A free body diagram on Figure 2 indicates all the forces acting on the chunk. Note that the positive y direction is upward and that we ignore the (small) effects of gravity in this problem. Which of the following formulas best approximates the wave equation for small am- the next page.) (A) ”awn _ bays») _ Trim) Asagwazm 82 in > 62am) 30“?” “go/”22¢ ‘ (B) 83:2 ‘ M at? )jéras F W1 IE; :9wa we; (C)T Imam”) Maya” f 7m WM” 5* 17 3:102 at 8t2t ZLAC g/Zde 255% D 82y(a“, t) b 3mm) 32m (,xt) “9:7— ( ) 82:2 " " as =“ a2 0/ f/ic (9440M ”3%ch 8y(a:,t) 62y(ac,t) 8y(:t,t) (F) T — W = M F 9 8 3t2 at :2 plitude waves on the string with air resistance? (Provide your answer in the boa: on one? 8t _ at? 11% y 52 W t) / 321/,(32 t) 8y(ac,t) 82y($, t) 9 (1376.722 —b at :M atz IIZIJ =~ZV9%($)# PLEASE TURN PAGE Page 7 of 16 PHYS 214 Prelim 1 =m574 2% 6AM), m git M5297 Mm» .' / 72b fieépwofA—eOWu/fl E ’0), a/verLermsaffiLDon/é? (0;: %{£ij a), =%¢/x fjj/ we 0&2”;- r gig/W -— figgaw =flj—é—jgzz/zy aéléws ) 2%»? a3? fie Wax/e; fimflafu Gyfcwswer -. \ PLEASE TURN PAGE Page 8 0f 16 PHYS 214 Prelim 1 —1——1—-—1————1—1———1——1—> (A) P0 (Ne) (B) 2P0 (A1") (C) 4P0 (Kr) (D) 290 (Ar) Figure 3: Closed tubes filled with noble gases. 3 Problem 3: Standing Waves in Sound Tubes [8 points] Four tubes each with two closed ends are filled with different noble gases at atmospheric pressure and room temperature therefore, they have the same bulk modulus B but different mass densities p0, 2pc, etc (see Figure 3). The tubes also have different lengths, as shown on Figure 3 Which two of these (A 6'! B 01‘ B 85 D, etc.) tubes will have the same l___owest resonant frequency? Provide your answer in the boa: below. 56$}? fMfi/QZ %@ 7130776214th ffp 19:5 wax/€— f/Lj Off SZL 757mg]; War/,1, a, 24% 3 e7 W (on/WW ”5226044 5200” smce we we Aéoézlg J4“ fie Lon/557‘ 71220» Wwa are %cu/e age 542 wave 51 00W€SfWJ 0,_ fie w:o:£2‘7'w :fawoé/ wau/e ale; We thus see that Wag W Weft W50!” PLEASE TURN PAGEW Page 9 of 16 PHYS 214 Prelim 1 Figure 4: A chunk of vibrating string. 4 Problem 4: Forces, Slopes and Curvature [14 points] A chunk of vibrating string (tension 7‘ z 1000 N and linear mass density a = 0.6 kg/m) is shown on the graph below: Neglecting air resistance and gravity, and using the graphical information on Figure 4, ’complete the tasks below: (a) Forces (7 points) For each of the parts below choose the value, (A), (B), (C), (D), (E), or (F), which is closest to the answer of the question: (Note that your answer may not match any of these exactly.) (A)60N, (B)6N, (C)20N,I (D)15N, (E)1000N, (F)53N. Provide your answers in the boxes below each question. . What is the magnitude of the y- component of the force on the right hand and of the chunk? F;- = $%(3-5m): 7000/1/x( (002:)N N 66?” => 72:; c405557’ aura/er as (8)6/1/ PLEASE TURN PAGE Page 10 of 16 , PHYS 214 Prelim 1 o What is the magnitude of the y-component of force on the left hand end of the chunk? 22624547 254a Sé/De % tie/1704) we 74pm! 57 = ‘ 2%f0-W */000Nx(W_6;o_y => 5r Z‘Ae MAeA/I'TUDE //Z / we. (”KM 24; 0100.96 Mover (A) 60M, J o What is the magnitude of the net force on the chunk? 72.1 A/Er Face: 0% 2%? GM fob/1f: fie g’axdg/ simce fie x—WW c f(Fm2jx= @x’Léjm :3 +Z"ZV=0). flaw/4n; we Fw=5¢ =F+5 =677V—60/1/z—533/l/ e7 ‘27 Q“ -~ ‘ ' => W22 CLOSE-'57— aurwer 21% #AGM/‘Tl/DE Eat 1'5 (F) 53 M 74V / / PLEASE TURN PAGE Page 11 of 16 PHYS 214 Prelim 1 (b) Acceleration and Curvature (\7 points) Challenge Problem! Using-your answer from part (a we won’t penalize propagati of your errors): 44462, rite —cc3/ 0W0 5f; 0 Find the acceleration gag of 't eflrclhunk. (Write out ‘your numerical answer in- cluding units. This part is got multiple choice!) ——_s_ fig? 5634:7415: , f4 M) “Z M ‘. mm W :3 «2-3 A/ 72%;, /’£ A (26%”)xzub'm 2t 35 3 m S 2 0 Using the wave equation, estimate the curvature [3—3 of the chunk. (m ggyyour numerical answer including units. This part is n_ot multiple choice!) FI’DM Mt; wax/e ”rod/L /" 244$ WMIé we 7%. 7“ fi ‘ .2 Z 2 4 . 4 7%:fl:&% => //§&24{2/= JZé'é/Cclf/ 7:“: a: #160 ”71235379752- 2:- PLEASE TURN PAGE Page 12 of 16 PHYS 214 Prelim 1 frictionless rod """ (has --------- (freq f) . <—> adjustable massless ring Figure 5: Lab Experiment I. 5 Problem 5: Lab Experiment I ' [21 points] In your first laboratory experiment, you studied a vibrating string With two fixed ends. Consider now a. modification of the experiment in which the right hand end of the string is attached to a small massless ring which slides up and down a horizontal (in and out of the page) post without friction. The post is attached through additional strings to a pulley on the other side of which spring scales (like the ones in lab) provide constant tension (see Figure 5). As in lecture, L is the length of the string between the rod and the shaker, f is the frequency of the shaker, T is the string tension and ,u is the mass per unit length of the string. Assume ,u is constant throughout the problem. The shaker'vibrates the string in and out of the page. Assume that the left hand end is fixed at the shaker While the horizontal motion (along-the y—direction) of the right hand end has no constraint, i.e., the right hand end is free. (a) The Fundamental Mode (7 points) Sketch the shape of the standing wave corresponding to the lowest (fundamental) mode. Find an ewpressz'on for the lowest (fundamental) frequency f1 in terms of L, 7' and ,u. 579m #5; 5% we see, Mai“ #42. for flu. FUNDAMENTAL MODE: 1 zi=L=> 99:41. PLEASE TURN PAGE Page 13 of 16 PHYS 214 Prelim 1 (b) Higher Modes at Fixed Length (7 points) Sketch the shape of standing wave on the string corresponding to the second mode. What should the frequency f2 of the shaker be for this standing wave mode to be formed? (Express f2 in terms of LT and M.) From 2‘4 3% mesa, fimffw the SECOND mode: 3 fl; = L = ._ 4 > AZ - 3-1. (c) Higher Modes at Fixed Frequency (7 points) Now suppose that you can vary the length of the the string. Find the string length t at which the 2nd standing wave mode will be formedxife efrequency of the shaker is fixed at f1 and the tension 7' remains constant. (Express t in terms 0 riginai length L.) Szmce we ans mfwsz‘eo/m Me SECOA/D moat/a We cm 4/575 all/re 2:4?" SMaZ/WZ @224 240(5):! igw PLEASE TURN PAGE Page 14 of 16 ' PHYS 214 Prelim 1 6 Problem 6: Sound Waves versus Waves on a String [7 points] ' i The following equation is true for strings: 6 am; 81 32121 3_y2_ ax[5ifi]+ 67L (5%) +2”(8t) —0‘ (6'1) It expresses conservation of energy for waves propagating On a string. What equation expresses conservation of energy for sound waves? Hint: Use the unified picture for string and sound waves presented in class and take a good guess! lD/Fefimngm wayeswagiiu Z3,__ if; wit-fa 3% flew Mei in MWLCJ 322% w :7? fifiefij in; 5* fl; 1’12: W3 iii? 3 77/015 Wye/V 20% o/%% malt/o WAV PLEASE TURN PAGE Page 15 of 16 PHYS 214 Prelim 1 massless piston Figure 6: Generalized boundaryr'conditions for a sound tube. 7 Problem 7: Generalized Boundary Conditions for a Sound Tube [14 points] A tube of length L is filled with air of bulk modulus B and mass density p0 at atmospheric pressure. One end of the tube (at = L) is closed while the other end (a: = 0) is attached to a massless piston of area A that can slide freely (without friction) along the tube. (We consider low amplitude waves so that you can take L m const.) The piston is attached to an ideal spring of spring constant k that exerts a horizontal force on the piston; the spring is relaxed when the piston is at x = 0. (See Figure 6.) We denote the sound displacement inside the tube as S(x, t) and the pressure inside the tube as P(:c, t). (a) Air Pressure and Displacement of the Piston (7 points) Draw a free body diagram for‘the piston, indicatingthe directions and the magnitudes of all the forces, using no quantities other than P(a: = 0,15), S($ = 0,t), k, A, pg, L, and PO. A PLEASE TURN PAGE Page 16 of 16 PHYS 214 Prelim 1 (b) Boundary Condition at the Piston (7 points) Challenge Prob- lem! _ Write the Equation of Motion for the piston in terms of the degrees of freedom, i.e., the displacement function 3(x, t) and its derivatives evaluated at x ; 07 using no quantities other than P(x= (Lt), 5(93 20,0, k, L, A, B, p0, and P0. —_...> 1% 1/5 while 555': MP5; 74‘ %e/¢:S‘f0% a“? W fl 50g; 0&er m/OWzLflL/5 it, gate/xewri 2% — A W W = ”i am 525mg 777/0 -—->0 [wayséyg /0/32:m,)) we ZWCJ *4 sfmw A2 —/4P/x=o, 29 = 0 5(resrzév 91/ W 2"» 254M778 0 fig 0.0F 5/295) ail Z‘ZQ/Zé/G/Zéll/es/ Magicifioé Qéo _ fim% Z’fiqz‘j: 2-3%[17192 we 0gZLaZ¢zJ _£S&=o,tj f%~%+/IB%§-(x=%tj = 0 / END OF EXAM ...
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