fall2003solutions - Page 3 of 14 PHYS 214 Prelim 1 20693...

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Unformatted text preview: Page 3 of 14 PHYS 214 Prelim 1 20693 Figure 1: Wave patterns for different boundary conditions. 1 Problem 1: Standing Waves on Strings [16 points] (a) Sketches (12 points) On the axes which Figure 1 provides7 sketch the form of the lowest frequency mode for each of the three possible combinations of boundary conditions (fixed—fixed, fixed—free, free—free). (b) Comparing frequencies (4 points) Among the three possible combinations of boundary conditions from part ( a ), which allows the lowest possible standing-wave frequency for the same length L, tension ’1' and mass [M of the string? Provide your answer in the boa: below. PLEASE TURN PAGE Page 4 of 14 PHYS 214 Prelim 1 2 Problem 2: Lab Experiment I [20 points] As part of the first laboratory experiment in Phys 214, you measured the frequency of the lowest mode of an aluminum rod oflength L z 1 In. You should have found a frequency near f = 2x103 Hz. (For the purpose of this problem? take this to be the exact value.) Because you held the rod at its center while leaving both ends free, this gives the frequency corresponding to the freeefree combination of boundary conditions from Problem 3. Finally, further measurements on the actual rod you used Show it to have a cross sectional area A : 7rr2 : 500 mm2 : 500 X 10‘6 m2 and a total mass M = 1.35 kg. (a) Speed of sound in aluminum (8 points) Use the above information to compute the speed of sound in the rod. Give a numerical result in units of m/s. c: A? -.—. (at)? ~=~ it lwast s“ C 2 Lrooo mlg (Maia: mex Hit home.) an; 3% 35:11 57 K=QZL\ (b) Bulk modulus of aluminum (8 points) The actual speed of sound in Aluminum is c z 5100 m/s. (Note: You will not find this value from the data in part (a) because of the fake value given there for f!) Using the actual speed of sound c = 5100 m/s and the information given above, compute the bulk modulus B of aluminum. Give a numerical result in units of N/m2. a \ r ) B: gyoo Erik, (5/0?) wig) 2.7-0; “0‘1 1: 4m 1 PLEASE TURN PAGE "‘ Page 5 of 14 PHYS 214 Prelim 1 (c) Spring constant of the rod (4 points) Challenge Problem! The rod can be used as a very stiff spring. When the rod is compressed by a distance :r, it pushes back with a restoring force proportional to :13, F 2 km. Use the definition AV APz—B V0. to compute a numerical value for the proportionality constant k in units of N/m. Hint: For this problem, you may ignore any tendency for the radius of the rod to increase as the rod is compressed. The influence of this effect is small in aluminum. - 4,. AV 2;, Ear. AF-AE-A— Big-Pt SMELL. A r? 1;: BA : %.axaafifl§-soavio“mz L lm PLEASE TURN PAGE Page 6 of 14 PHYS 214 Prelim 1 Figure 2: A mass—spring realization of a harmonic oscillator with sliding friction. 3 Problem 3: Harmonic Oscillator with Friction[24 points] Consider a harmonic oscillator with mass m acted on by an ideal spring of spring constant k and equilibrium position a: : xeq : 0 and by a sliding friction force in the direction opposite to the motion and of magnitude f = amg where ,a is the coeflicient of sliding friction and g the acceleration of gravity. (See Figure 2.) (a) Equation of Motion (6 points) Show that, when the mass moves to the right, the Equation of Motion for the har- monic oscillator with friction is: (12:16 k Iii—2+,ug—tngzo, oni/E. (3.1) PLEASE TURN PAGE Page 7 of 14 ' PHYS 214 Prelim 1 (b) General solution (6 points) Show that the following is a general solution to the equation of motion (3.1): m) = i3 + §Re [Aewot] . (3.2) 2 tan Note: Be sure to list each requirement for a general solution as you Check it, Q¢ Eqvafl-L eel/ugé‘uc Fa%3 l [lg F 2224 00¢ 25w: Amataw, :3 Deal :1 Paw / L4. flmé mi “1’17““?“47 Pémlb =7 hwc Q P‘t‘fafi‘ PLEASE TURN PAGE Page 8 of 14 PHYS 214 Prelim 1 (c) Real Part of A (4 points) For parts (c)-(e)7 consider initial conditions where, at t = 0, the mass is at the equilibrium position $0 : areq = 0 and moving with a velocity 110 > 0 t0 the right. Find the real part 5R2 [4] given the above initial conditions and general solution Ea:- press your answer in terms of only 220, ,a, g, and mo. (Note: you may not need all of these.) 7 1 (d) Imaginary Part of A (4 points) Find the imaginary part $111M] under the same conditions Empress your answer in terms of only 110, n, g, and we. (:Note you may not need all of these ) we > 6%? Palms/“3°1— wear PLEASE TURN PAGE Page 9 of 14 PHYS 214 Prelim 1 (e) Maximum distance (4 points) Challenge Problem! What mamimum position rmax will the mass reach before stopping and turning around? Empress your answer in terms of only 110, a, g, and mo. (Note: you may not need all of these.) PLEASE TURN PAGE Page 10 of 14 PHYS 214 Prelim 1 y FJiSCfiL’ + A,t) wascm, t) Figure 3: A chunk of fluid experiencing viscous forces 4 Problem 4: Equation of transverse motion in a fluid [12 points] Consider transverse motion along the y direction of a fluid with density ,00 and bulk modulus B. For plane waves propagating along x, the constitutive relation for such motion is that the driving force per unit area on a chunk at :1: due to the adjacent fluid is FJi5“(:r,t) : c 82y(x,t) A U 8x525 7 (4.1) with the direction of the force as in Figure 3. Here7 Cu is a constant known as the coefficient of viscosity (a constant which is large for c‘,‘t.hick” fluids like honey). Which of the following formulas best represents the equation for transverse mo- tion in a fluid? (Provide your answer in the boa: on the newt page.) 63y(x,t) 82y(;c,t) (A) CU 8$28t :pO (9t? 212(WB: Fog—£1 62y(x,t)_ 82y(ar,t) (B) 6” 8x623 ‘p" 6t? B62 y,(17 t) 82y(:1:,t) C\I gngt >Pb3 9-62. 8x2 2’” at? (D) airing—MM=p0___62y<$’t> 521% 2:“ - «Ms C” am at? 83:2 at? ,_ ’ (0)3 833/06 t) 021/03, t) 323106; t) (E) Cu 6?th +BW:PO 5752 AC 3133 C2¥zgi) - AC 2245:) A g V u : . 9130+”? '3‘ (7N 9‘ a? __ 34"...) 22 (‘9‘;3-‘(3 “* C” A #526 PLEASE TURN PAGE Page 11 of 14 ‘ PHYS 214 Prelim 1 Marz $/§W/)/f Flka-fé‘)—— Eéxlf) in m. 32 g y dz 272122, goAA- : Walt, w»,/ £21- a: _L_ r (A; 175.. ,7} jg“: .221: yr“! 90 91”, A z A A—-> o 5 I) 9 j ( j 92 ¢ ,5 9%; kg- “ ¥ .11... gohé A 2X CV 9X 23ng 2f 71 [Waggf Q(fle¥:’,€91/Z 21 33 Z . 6’ J r M (A) 02464 v 22,? 222: PLEASE TURN PAGE Page 12 of 14 PHYS 214 Prelim 1 5 Problem 5: Sound Waves versus Waves on a String [12 points] The following equation is true for sound waves: 8:1: 81’ 8t (9t 6 [:13 1359—5—35] +9— [mpoas]: 0. (5.1) (a) Equation for strings (8 points) What is the analogous equation for strings? (b) Another conservation law (4 pointS) Challenge Problem! Conservation of what physical quantity does your equation for strings represent? Hint: The densitz of the conserved quantity appears inside the 6—t[ ] and the driving ‘force” for the conserved quantity appears inside thea 8—1 [ ] ma. $9 "Oéekéryyfl “*9/41 ={£#?-/5L 171ng J'EJ ‘19, 2/4315: 2 P3 :(fifisz‘j uér OR MM '~ ‘ n) a. —- -> rn5.0439;2 3 is with 3 = 2’73 ‘3 ’DKZU’RF) 1%“: mJ @ereJTZI/Cé 67‘“ qa’ MM PLEASE TURN PAGE Page 13 of 14 PHYS 214 Prelim 1 mass ess piston Figure 4: Generalized boundary conditions for a sound tube. 6 Problem 6: Generalized Boundary Conditions for a Sound Tube [16 points] A tube of length L is filled with air of bulk modulus B and mass density {)0 at atmospheric pressure P0. One end of the tube (r : L) is closed. At the other end (at z 0) a massless piston of area A can slide freely (without friction) along the tube. The piston is attJached to a shock absorber which generates a horizontal external force on the piston flex” 2 ‘bfl, where i7 is the velocity of the piston. (See Figure 4.) We denote the sound displacement inside the tube as 5(zr7t) and the pressure inside the tube as P(:I:, t). (a) Air Pressure and Displacement of the Piston (6 points) Assuming that the piston is moving to the right with velocity v : 85(zr : 0)/(9t7 draw a free body diagram for the piston, indicating the directions and the magnitudes of all the forces using only P(a: = 0,t), 88(x = 07t)/8t, b, A, p0, L, and P0. (Note: you may not need all of these.) . PLEASE TURN PAGE Page 14 of 14 PHYS 214 Prelim 1 (b) Equation of Motion for the piston (6 points) Assuming Small amplitude waves, so that you can take the location of the piston to be LU % 0, write the Equation of Motion for the piston in terms of only the constants describing the problem (6, A, B7 pg, L, and P0), and s(:r,t) and its derivatives evaluated at CE = 0. (Note: you may not need all of these) 0 Z Felt ‘ fig ‘ x ' X : O Ukslfiflrf) M legs/gm W): Fo— eons/c f2 hat/011 (c) Standing wave solutions (4 points) Challenge Problem! In the limit b ——> 007 what is the frequency of the lowest frequency standing sound wave in the tube in terms of L, p0, and B? Hg 1: a 6'0] TC; piston amt may: + 0543, ”(L a ale/gel canal Er (JosaL 4/wa 84.) Lug,- l’kw'e. KM M1“ 1, Hiat— % = L. ms, p; 5:71: 5:— [52? L‘ 311 END OF EXAM ...
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