p2214_p1_solutions_10_07_08 - x mm '7‘“ // Paw? 5 5...

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Unformatted text preview: x mm '7‘“ // Paw? 5 5 (#5:, M .— <3 .25 U r ‘3] (117,43 Short answer questions 1—3 (18 points — 6 points each) (Show your work in the spaces below the questions.) 1. A particle of mass m = 0.8 kg is constrained to move along the x-axis. The particle is acted upon by a force given by Fx(x) =x2—x—6 where x is in meters and Fx is in newtons. What is/are the possible angular frequencies for small-amplitude oscillations of this particle? FSOS‘; (y7»Xr—6§ E O cafe-w 2’3 (y—~3\(xe¢2> :0 -"—" :2 fl —’ i 9% / Effigy/c 'f I.“ ht, (K U»! K /’l C M P rot-“fl *5 Fe 13”?» afaw~£X e 9 U? ' “'2' ‘“"”\~ a QC) ‘X’ I: m '2 e ’zx‘/§ LS? 1.. ___k -2 *0: ‘ V ‘“ 25 2. A thin organ pipe is closed at one end (x = 0) and open\at the other (x = L). Sketch graphs showing the pressure amplitude as a function of x for the first three standing wave patterns, in order starting with the lowest frequency. Lowest frequency Page 2 of 11 3. A string fixed at both ends vibrates in a standing wave with no nodes (other than at the ends) at frequency f when the tension is To. To what should the tension be changed so that the string has two additional nodes (in addition to the ends) when it vibrates at the same frequency f? Express your answer in terms of To. ,I/ /‘" Page3 ofll Multiple choice questions 4-7 (20 points — 5 points each) 4. At resonance (maximum power), which quantities are i90° out of phase with one another? (List all correct choices.) A. Driving force and position B. Damping force and position C. Driving force and velocity D. Damping force and acceleration E. Spring force and velocity 5. The graph is for a transverse wave pulse moving to the right (+x direction) without changing shape. Which of the points at X = l m, 1.5 m, and 2 m has a negative velocity (—y direction)? (A) x = 1 In only (B) x = 1.5 m only (C) x = 2 m only (D) bothx=1mandx= 2 m (E) all three x (m) 6. In a standing wave, what is the distance between a node and D the closest antinode? (A) )b (B) 27» (C) M2 (D) M4 (E) 4)» 7. A sound wave in a thin tube travels in a fluid with bulk modulus BO and equilibrium density p0 toward an abrupt change in medium at x = 0. The new medium has bulk modulus BI and equilibrium density p1. In which case will the transmitted wave have a larger displacement amplitude than the incident wave? (A) Bo/po>Bi/P1 (B) B0/100 <Bi/P1 (E),00>,01 (F)P0<Pl Page 4 of 11 8. (20 points) Consider the equation of motion of a mass (of value m) displaced along a single direction (i x) and subject to friction proportional to its velocity, a restoring spring force and an external sinusoidal driving force of amplitude F0 (a real number) and angular frequency a) (also real). The other parameters will be defined by their appearance in this equation of motion: 2 d x dx - . m— = —b —— — kx + F0 elwtwhere following convention we assume that 1n order dtz dt to get physical quantities we take the real part of this equation. a) Suppose that we have waited long enough that steady state response to the driving force is achieved, no matter what the initial conditions are. What characteristic time must we wait for this to happen? Don’t derive this answer; rather just give an expression in terms of the parameters of the equation of motion. Provide brief reasoning for your answer. WC WQ$‘Q‘ Wei—l >> 2—: “(Ag [éa.fafl.‘@¢¢%{i g “(9470 7 I’M) r/lim‘fi «(gir— U M Maw) o 8:36 [{cfin‘ G'V'A 5: 2 M b) Being in steady state means that we can express the displacement as follows: x(t) = Aeupezmwhere the amplitude A, the phase shift q) (relative to the driving force) and the driving force amplitude F0 are all real. Recalling the requirement that the amplitude of the sinusoidal force is real, derive a condition (simplified as much as possible) that 915 must obey (involving m, b, k, and a) as needed). Remember to show your work. [Continue on p. 6.] PageS ofll Shite: I‘M ex, talk-cam? Fa W M L“ $4. Answer: c) From what you found in part b), argue that as the driving frequency is reduced toward zero, the displacement of the mass does a better and better job of moving in sync with the oscillating force. ' Page 6 ofll 9. (22 points) Consider a string of linear mass density (,u ) with tension (17) that experiences transverse waves. It is terminated at x = 0 with a dashpot with a damping coefficient (1)) that produces damping force proportional to the velocity of the string in the y direction. The tension is assured by means of a massless slip ring on a pole as shown. The other end of the string is far off atx = +00. a) What boundary condition on y must be met at the damped end of the string? wk g 23.x. 'Z a”. X39 {a 2)? 2 "441—3 54:5 2: O ‘ act“: Wu st :0 Page7of11 b) By considering an incident wave of arbitrary shape traveling toward the dashpot in the — xdirection, find the reflection coefficient (the ratio of the reflected to incident amplitude) as a function of b and any other needed quantities. Simplify your result as much as possible. W“ [m Winfrey, eme +$< 3 Wm-~~~—«::~> we {be ~ 3-) Page 8 ofll c) For what value of b does your result demonstrate the idea of impedance matching? We 44-1 I‘M/9214‘“er wastrL-X. 544. 5:: iv .= gz/Lfl’i —_ fl ’4“, Answer: d) In the limit that the damping coefficient goes to zero, give the boundary condition that results and briefly say why this is to be expected. A ~> C3 / Ml {X/aec/‘g @[AM Qapg government? (Ow/top? #6230“ “792,3 amt—«A Page9ofll 10. (20 points) Consider a metal rod (of linear mass density y) clamped down on both ends (at x = 0 and x = L). It is under no tension, but has a bending elasticity coefficient Fthat allows for transverse bending obeying the following wave equation: 02_y__ <94_y_ arz 6x4 7 L.» a ; / f . zit-:0 5:0»th V‘ooL a) Assuming that we might have standing waves of the following form (either one or both may be valid), y(x,t)=Acos(kx) cos(a)t), y(x,t) = A sin(kx) cos(cot) indicate which of these form 3) is tare) valid and find the allowed values for k. L252. Sci/L >693 ‘éQ 5“ cl§€7 %.€. >< ‘3’ 6 M 1/16ch kLih—r M.:/,7/3;..‘._ 55 411.255 5% CAL):@ $0 $o.‘fi-3'Es'(7 Ef’, 45'6" KEL_ Page 10 ofll b) Use your answer to a) to find the lowest three frequencies for such standing waves. jog-Lot‘s weak WCNQ “Cytfl‘f‘lqv‘ Sam K’WQD do; c) Find the net force on a segment of the rod from x to x + Ax . Do not assume that Ax is small. [Hintz what would you have to do to derive the given wave equation from the forces?] 322 : Was-.5. “11% Page 11 ofll ...
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p2214_p1_solutions_10_07_08 - x mm '7‘“ // Paw? 5 5...

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