4.1 - Section :5 1 The Harmonic Oscillator 205 U - ’ 1 I...

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Unformatted text preview: Section :5 1 The Harmonic Oscillator 205 U - ’ 1 I I'w'é ":2 $353, a w ,. w . M ‘ ‘ _ ,' » up) ii: tfif 33213 . . _. -_3I?oim_ni__ations__of lilo diifomntial equation-s for-the moss—spring system from moohonios and tho LRCeci.rcuit from ' '._:'_o1¢ctri_czil £_ho_0ry._le'a'¢l gtoihe'_samo'_.l_inearsecond—oime modoloolled the harmoniooscillatoa .Soiutidns in tho .:.sundasnped.¢asearc sinusoidal. Vibrations-a = ' ' -' ' ' - - -' ' ' - - - '- -' - - " ' ' in rih an 4.1 Problems )cis . . of T he Undamped Osoiilator For Problems 15-8, find the simple Single~Wave Forms of Simple Harmonic Motion Rewriie harmonic motion described by the initial-value problem See Problems 15—18in the form A cos(wgr w 6) using the converw :6 also Problmm 213—30 and 32-39 sion equoiions ( 7) 6 egg 1. ix 2 0. .x(0) = 1, H0) = 0 15. cost ~§~ sin: 16. cos: —— sin: 2‘ +x 2 0, HO) m i, obi-{0) mi 17. w cos: + sin! 18. —cosi w sin: 3- + 9x = 0’ rm} m 1' HO) = 1 Component Form of' Simpie Harmonic Motion Rewrite " ‘ Problem 19—22 in rlzeform cl cos mm + C3 sin mg! usng the 4" x 4“ 4x W Di “0) = 1' 35(0) 2 "‘2 conversion equorions (8). Ir 5. +16): = 0, x(0) m —1, m) = o 19, 2cos(2t — n) 29. cos (i + E) 6. + Mr 2 0, .r{0) : 0. 0 a 4 < ) 21. 3 cos (i — 32. cos (3: — 7. +167r3x = o, x(0) 2 0, m3) 2 x I 1 ‘ Interpreting Oscillator Solutions For Problems 23-30, 3 8- 4.56. +751 m 0. $40) = I. 33(0) = 7? determine file amplitude, phase angle, and period of the motion {These are fire equations of Problems he? and : Graphing by Calculator For the combinations of‘sine and 32.39 ; » cosinefimctions in Problems 9—13, do the following 23" + x w; 0’ Km) 2 1’ H0) = 0 mm» (a) Use a graphing calculator or computer Io sketch the ; {Tent graph ofeochfimcrzonr 24‘ + x = 0’ Im) w is H0) m 1 stem (b) From yourgmphs, estimate the amplitude, period, and ; seen phase shift 6/590 oftlze rewiring oscillation 25m + 9x a 0, 37(0) = 1, MO) = 1 (c) Write each funcn‘on in the form A cosmor w 6). 25lx+4x=o x0 m1, i0 m—z 3 3p36~ 9.. x0} a cos! “i” sin: ( } ( ) mics a " —- — w ‘ = Camry 10“ I“) u 280“ + Sm! -7. x +163: m 0, ,x(0) _ 1, x{0) 0 1L = 560$ 3f + Sin 3’ 23A +161” 2 0, 110} “= 0, 29(0) m 4 12. x(r) m cos 3: + 531;] 3: 29. +167r1x = 0, x(0) = 0, km) 2 71* 13. W) 2 ~ cos 5! + 25in 5: 30. 4x + Hz): 2 0, .r(0} = 1, 32(0) m a: y M Alternate Ii‘ornis for Sinusoidal Oscillations To dexive 31” Relating Graphs For the oscillator DE +0251 20‘ m up the Comma“ “mamas (7) and (3}, use the idenmy Fig. 4 14 shown previousiy {inked solution graphs and mpthe cosm _" fl) = COSOICOSfi + Sinasinfi phase por‘uaii. Parts (:1), (b), (c), and (d) relate to that - figuro » I . . . “form 2:331} trigonlmeuy to Shaw maybe ffqmflx thszrg,usoldal (a) Manic on the phase portrait the starting points (where anic 1 atlorls comma; m ) can a wmten m t e orm I m D) Em, the trajectories Show“. I . C: €05 me! + 62 5111 war. (i3) Write expiicit solutions f'orxm and xo) Winn is the ,J Where c] z A cos 5 and c: w A sin 6. Value of “’0? (A) {B} FIG U RE 4 .1 . 7 Graphs of the solutions that match the IVPs in ProbEems 40-43 31. continued {c} Label the: axis in the solutions graphs of Fig 4.1 .4 with the appropriate values for t . HINT: Consider where the solution graphs cross the axis. (d) Given A as the amplitude of the solution with the largest osciilation, state the amplitudes of the other solutions shown. Phase Portraits For Problems 32—39, find Jim and then sketch the trajectory for the [VP in the .202 phase plane, with arrows showing the direction of motion (These are the equa» tions of Problems 1—8 and 23—30 ) Explain how and why your phase portraits dfierfi‘om each other and from Fig. 4.14. 32“ 5E 4-K m 0, 517(0) = l, = 0 33. + x = a, x(0) = 3, H0) 2 1 34. + 91' m 0, .x(0) r 1, HO) =1 35. If +4): 3 0, 175(0) z i, = —2 36. +16): = 0. x(0) = W}, 2 0 37" + 16.x m U, x{0) fl 0, t 4 38. +167r2x z 0, x(0) m 0, 55(0) : Jr 39» 455 + 3721 = 0, x(0) = l, 5H0) = 3’1" (C) (5} Matching Problems March the NP: in Problems 40—43 to the graphs in Fig 4.1.7. 40. + 4.): = O, x{0) z 0, 5.10) = l 41. + 4.2: = O, .110) m 1, MO) = O 42. + eh- m o, .r{0) = o. .J‘:{O) = 1 43. 4.5: + n2): 2 o, .110) m 0, .i—(O) = 1 44. Changing Frequencies Consider the undamped har» mortic osciliator‘ defined by + m3): = 0 with initial conditions .x(0) = 4 and .i:(0) = O. (a) (b) For C00 = 0.5, l, and 2, the corresponding Lrajecto» ries are plotted in Fig 41861) on the same Lit—plane. In Fig. 41.80)) the corresponding trajectories are plotted on the .xic phase planer Make a trace of boih graphs and label each curve with the appropriate value Of 600, If an) is increased, we can see that the frequency of the oscillations in Fig. 4.1 8(a) increases (as expected, since an} is the natural (circular) frequency). De~ scribe what happens in the phase plane (Fig. 4. 1.803)) if we is increased. How is )3 affected if can is increased? (a) Sointions ([3) Phase ponraits FIGURE 4.1.8 Graphs off +ng =0{Probiem 44) 46¢ H 45. Detective Work Suppose that you have received the fol- lowing two graphs without their equations. Show how you can infer the equations from graphical information. (a) The graph in Fig 4 1 9(3) represents A cos(l — 5). Determine A and :5 from the graph and write the equa- tion of the curve in the form of equation (5) (b) The graph in Fig. 4 l 9{b) represents c1 cos 1+0; sin r Determine cl and Cg from the graph and write the equation of the curve in the form of (4). HINE: Find A and 6 first. 0 It 21: (:1) Graph olA cos (I w 8} 0 1: 2a: (b) Graph of c: cos M a: sin 1 FIGURE 4.1.9 Simple harmonic motions (Problem 45). 746a Pulling a Weight An object of mass 2 kg, resting on a frictionless table, is attached to the wall by a spring as in Fig. 4.1. 1. A force of 8 at is applied to the mass, stretching the spring and moving the mass 0.5 m from its equilibrium position. The object is then released. (It) Find the resulting motion of the object as a function of time (b) Determine the amplitude, period, and frequency of' the motion. (0) At what time does the mass first pass through the equi- librium position? What is its velocity at that time? Section 4 .1 47, 48. 49‘ 50, The Harmonic Oscillator 207 Finding the Differential Equation A mass 05500 gm is suspended from the ceiling by a frictionless Spring The mass stretches the spring 50 cm in coming to its equilib- rium position, where the mass acting down is balanced exact] y by the restoring force acting up. The object is then pulled down an additional 10 cm and released (a) hormulate the initial—value problem that describes the object’s motion, setting x equal to the downward dis- placement lrorn equilibrium. (i?) Solve for the motion of the ohject. {c} Find the amplitude, phase angle, frequency, and period of the motion M a 5 s - S p ri n g Watch the motion of the mass linked with the graphs of the position and velocity dis~ played in a time graph and a phase plane lnitial~Value Problems A ib-ib object is attached to the ceiling by a frictionless spring and stretches the spring 6 in. before coming to its equilibrium position. Formu‘ late the initial-value problem describing the motion of the object under each of the following sets of condim tions Set it equal to the downward displacement from equilibrium. (a) The object is pulled down 4 in. below its equilib— rium position and released with an upward velocity of 4 ft/sec. (b) The object is pushed op 2 in. and released with a downward velocity of l ft/see One More Weight A l2—ib object attached to the ceiling by a frictionless spring stretches the spring 6 in. as it comes to its equilibrium Find and solve the equation of motion if the object is initially pushed up 4 in. from its equilibrium and given an upward velocity of 2 ft/sec, Comparing Harmonic Motions An object on a table attached to spring and waiI as in Fig. 4 1.1 is pulled to the tight. stretching the spring, and released. The same object is then pulled twice as far and released What is the relationship betWeen the two simple harmonic motions? Will the period of the second be twice that of the first? What about the amplitudes and frequencies? Testing Your intuition Knowing {fmm Example I) what you now do about the damped harmonic oscillator equation mfc' + bi ~l~ kx : 0 and the meaning of the parameters :11, b, and k. consider Problems .51m56. How would you etpecr the solution of each equation to behave? Can you imagine a physical system being modeled by the equation? Who! would you expect for its long-term behavior? 51. f6+x+x3fl0 52. .if+.x~.t3=0 208 Chapter 4 En Higher-Order Linear Differential Eqnations n _ ,_ l , (b) For t >- 0, use Kirchoff's voltage law to determine 53' x w" x = O 34“ ‘x + Y": + x = O the sum of the voltage drops around the circuit. Set it equal to zero to obtain a second—order differential SSt -t- (x2 -— l).i' +3: = 0 56. + rx 2 0 equation involving L, Q, anti C. What are the initial conditions? 57. [iii-Circuit 'Constder' the series LR»Circtnt shown in (C) Solve {he IV}, in {b} forum Charge Q on the capaciim th .1 10, in which a constant input voltage V0 has been Does your result agree with part (a)? supplied son] I = 0, when it 15 shut off. . , , , , (d) Obtain an expltcrt soiution {or L : 10 homes and C : EU” farads R = 40 ohms 59. A Pendulum Experiment A pendulum of length L is sus- pended from the ceiling so it can Swing freely; 6 denotes the anguiar‘ displacement, in radians, from the vertical, as shown in Fig 4 l 12. The motion is descrihed by the pendttl'mn equation, 9 + i: sinB = 0 I = 5 hemies _ _ Determine the period for small oscillations by using the “CU REN4'1'10 A“ L'R'C’mmt approximation sin 9 R 9 (the linear pendulum}. What is (Problem 37) the relationship between the period of the pendulum and g, the acceleration due to gravity? lithe sun is 400,000 times (a) Eefore carrying out the mathematicai analysis, de— more massive than the earth, how rnttch faster wouid the sctihe What you think Will happen to the Circuit pendniutn oscillate on the sun (provided it did not melt) (h) For r >« 0, use Kirchoff’s voltage iaw to determine than on [he with? the sum of the voltage drops around the circuit Set it equal to zero to obtain a firstworder differential equa« tion involving R, 1, L, and I What are the initial conditions? to} Soive the DE in (b) for current I. DOes year anSWet agree with (a)? fixpiain. (d) Use the values R = 40 ohms, L = 5 hertries, and Us = 10 volts to obtain an explicit soiution. 58, LC-Circuit Consider the series LC-circuit shown in Fig. 4. l .E l, in which, at! m 0, the current is 5 amps and there is no charge on the capacitor Voltage V0 is turned off at! = 0. C: i0"3 itnads L = 10 hent‘ies FEGURE 451.1! An LC~citcuit {Problem 58) {a} Before carrying out the mathematical analysis, de- scribe what you think wili happen to the charge on the capacitor FsinB FIGURE 4.1.12 Simple pendulum (Problem 59) Pendulums ' Yhis tool aliows you to compare the mo» tions of the linear, nonlinear, and thread pendulums with predictable and/or chaotic results Changing into Systems For Problems 50m64, consider the second—order nonhomogeneotts D153; ii’r‘r're them or a system offirsnorder DES m in (18) 60. 4.t——2.t~+3x 2» iiwcosr 61. Lr'j + R6] + chq : vo) when of me radio sprin 67. Met 18 i Fig. Section 41 The Harmonic Oscillator 2.09 mine 62» Sq" + 15¢ + l—‘Oq = 5 cos 3: pen‘od of vibration is found to be 2.7 sec. Find the weight Set a ‘ of the cyiinder. HINT: Archimedes’ Principle says that an 1mm 63. ta? + 4H + x = i sin 21 64. 41‘ + 16.x = 4 Sin 1‘ object submerged in water is buoyed tip by a force equal C the _' to the weight of' the water displaced, where weight is the I 65. Circular Motion A particie moves amend the circie product of voiume and density. The density of water is C“ r .x2 + y2 a r2 with a constant angniar velocity of we radi« 62.5 lh/f't3 0 ans oer unit time. Show that the projection of the particle _ _ . 0n the )6 His satisfies the equation + max = 0. 68. Los Angeles to Tokyo It can be shown that the force on 5 and .. an object inside a sphericai homogeneous mass is directed '- ' 66. Another Harmonic Motion The mass—springwpuiley towards the center of the sphere with a magnitude pro» system shown’in Fig. 4.1.13 satisfies the differential portionai to the distance from the center of the sphere 53%“ ' equation Using this principie, a train starting at rest and traveling “Pies ._ ' in a vacuum without friction on a straight line tunnel from “with '_ -. kRz Los Angeles to Tokyo experiences a force in its direction Byte " mm— ,20. ‘ ' —r- r r + (nle + I) x oi motion equai to In cosQ, where 9 r is the distance of the train from the center of the where x is the displacement from equilibrium of the object with of mass m In this equation, R and I are, respectively, the o .x is the distance of the train from the center of the rig the _ radius and moment of inertia of the pniley. and k is the tunnel, that: is it. : spring constant Determine the frequency of the motion. . 9 is the an gl 8 betweefl r and at, an , . mm; 0 2d ts the length of the tunnel between L .A and Tokyo, aid the 0 R is the radius of the earth (4.000 triiies), it {me ) as shown in Fig. 4 1.15. :.Cente_r_ I of'tunnel t A. - 'i'j‘ Tokyo mg FIG URE 4.1 «13 Mass-spring~ pulley system {Problem 66). 7 Motion of Buoy A cylindrical buoy With diameter F] CURE 4“ 1‘1 5 Tunnel from Los 18 in, floats in water wrth its axas vertical, as shown in An ales m Tokyo (Pmmem 68) Fig 4.1.14. When depressed slightiy and released, its g ' (a) Show that the train position .r can be modeled by the initiaI-vaiue problem W J~ ' = ._d M 3 ? mfit' + kx u U, :c{0) = d, ,iCD) m 0, ic [ ' 1 I - i where x is the distance of the train to the center of the I ' . , '. ' ' wzmé earth and R is the radius of the eanh . , e I : Skierstizifl l ' w (b) How iong does it take the train to go from Los Angeles “3’ : _ _; toTokyo? ‘ ; E bugymicy force, ((3) Show that if a train starts at any point on earth and " goes to another point on earth in this science fiction FIGURE 4.1.14 Motion of a scenario, the time wiil be the same as calcuiated in haoy (Problem 67). part (a)! 210 69“ Facioring Out Friction The damped oscillator equa— tion (2) can be solved by a change of variable that “factors out the damping" Specificnliy, iei (a) Show that X0) satisfies 4.2 Characteristic Equation Chapter4 Higher- Orde: Linear Diiferential Equations } (b) Assuming that k — [22/4111 0, solve equasion (.21) for X f r); ihen show that the solution of equation (2) is .J:{I) 2 AK‘b/M' coslwoi m 8), (22) where we = v4mk w bZ/Zm. 70, Suggested journal Entry With the help of equation (22) from Problem 69, ciescribe the different shori— and long- lerm behaviors of solutions oi the mass-spring system in the clamped and undamaged cases illustrate with sketches xm m e'(b/3"‘}’.X{I). (21} Real Characteristic Roots S YN OPS i5: For the linear second—order homogeneous differential equation with constant coefficients, we obtain a two—dimensional vector space of explicit solu- tions iii/hen the roots of the characteristic equation are real, examples include over- o’ampco’ and critically damped cases of the harmonic oscillator, with applications to mass—spring systems and LRCvcircuits We generalize our results ai [he end of this section We begin by solving a very straightforward sci of DES: linear homogeneous DES with constant coefficients, After we have gained insight from this experience, we will examine more general linear DES and prove the existence of bases for the solution spaces of {he DES. SohflnglfionfiantCoeffidentSecond-Order Lineai DES In the special case of a linear second—order homogeneous equation with constant coefficients, the custom is to write the DB in the form WWWM+W=. m where a, b, and c are real constants and a 9—"- O. The first—order linear examples in Sec 23, which can be written y’ —— r.)I 0, suggest that we try an exponential solution.1 Ifwe let y z e” as atrial solution, then )r’ = re” and y” = r38”, so (1) becomes ai'ge” + br'e” -l- cc" = finial": + bi" + C) = 0' ! Because :2" is ncvex zero, (I) will be satisfied precisely when or2 + l)!" + c = 0. This quadratic equaiion, called the characteristic equation of the DE, is the key to finding solutions that f'ossn a basis for {he solution space. By the quadratic EEiqioncntiai solutions were his! iriccl by Leonhmd Euler {l 707—1783}, one of the greatest mathe- mnlicinns of all time and surely the mosiprolilic. I-ie contributed in neariy every binnch oflhe subject. and his amazing producsivily continued even ulier he became blind in 1768 ...
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This note was uploaded on 04/07/2008 for the course MATH 5A taught by Professor Rickrugangye during the Fall '07 term at UCSB.

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4.1 - Section :5 1 The Harmonic Oscillator 205 U - ’ 1 I...

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