Vector Spaces in Physics 8/6/15 Chapter 4. Practical Exaples. In this chapter we will discuss solutions to two physics probles where we ae use of techniques discussed in this boo. In both cases there are ultiple asses, coupled to each other so that their otions are not independent. This leads to coupled linear equations, which are naturally treated usin atrices. A. Siple haronic otion - a review. We are oin to discuss asses coupled by sprins and a copound pendulu. Let us start by reviewin the atheatical description of the oscillations of a sinle ass on a sprin or a siple pendulu. Fiure 4-1 shows the two siple systes which for the basis for the ore coplex systes to be studied. In each case there is a restorin force proportional to the displaceent: F displaceent (4-1) If we cobine this with Newton's law of otion, applied forces a (4-) we obtain d x acceleration soe constant / (4-3) dt or d x x dt You can easily show that for the ass on a sprin, (a) L x x, two faous relations. SHM (4-4) o x/l Fiure 4-1. (a) A ass on a sprin. (b) A siple pendulu. In both cases there is a restorin force proportional to the displaceent (for sall displaceents in the case of the pendulu). In the analysis of these systes we will inore vertical forces, which just cancel. (b) x L, and for the pendulu, 4-1
Vector Spaces in Physics 8/6/15 So, how do we find a function xt satisfyin equation (4-4)? Its raphical interpretation is the followin: the second derivative of a function ives the curvature, with a positive second derivative ain the function curve up, neative, down. So, equation (4-4) says that the function always curves bac towards the x = axis, as shown in fiure 4-. Loo lie a sine wave? x> curves down d x x(t) constant dt x t x< curves up Fiure 4-. The differential equation aes the curve x(t) eep curvin bac towards the axis, lie, for instance, a sine wave. The equation (4-4) cannot be siply interated to ive xt. Too bad. Second best is to do what physicists usually do - try to uess the solution. What failiar functions do we now which coe bac to the sae for after two derivatives? sin t sinh t cos t cosh t f '' f : f '' f : it t e e it t e e The first set of functions are the ones to use here, thouh they are closely related to the second set. The eneral solution to equation (4-4) can be written as cos x t C t (4-5) where C and are arbitrary constants. (Second-order differential equations in tie always leave two constants to be deterined fro initial conditions.) It is fairly easy to show (iven as a hoewor proble) that the followin fors are equivalent to that iven in equation (4-5). x t Asin t B cos t, A and B real constants it it * x t De Ee E D x t Fe F it ( ) Re, a coplex constant, coplex constants (4-6) 4 -
Vector Spaces in Physics 8/6/15 It turns out that the exponential fors are the easiest to wor with in any calculations, and the very easiest thin is to set it x( t) ae. (4-7) This loos strane, since observables in physics have to be real. But what we do is to use this for to solve any (linear) differential equation, and tae the real part afterwards. It wors. We will use this for for the eneral solution in the exaples to follow. B. Coupled oscillations - asses and sprins. Many coplex physical systes display the phenoenon of resonance, where all parts of the syste ove toether in periodic otion, with a frequency which depends on inertial and elastic properties of the syste. The siplest exaple is a sinle point ass connected to a sinle ideal sprin, as shown in fiure 4-1a. The ass has a sinusoidal displaceent with tie which can be described by the function iven in equation (4-7), with as the resonant frequency of the syste, and a a coplex aplitude. It is understood that the position of the ass is actually iven by the real part of the expression (4-7); thus the anitude of a ives the axiu displaceent of the ass fro its equilibriu position, and the phase of a deterines the phase of the sinusoidal oscillation. A syste of two asses. A soewhat ore coplicated syste is shown in fiure 4-3. Here two identical asses are connected to each other and to riid walls by three identical sprins. The otions of asses 1 and are described by their respective displaceents x1(t) and x(t) fro their equilibriu positions. The anitude of the force exerted by each sprin is equal to ties the chane in lenth of the sprin fro the equilibriu position of the syste, where it is assued that the sprins are unstretched. For instance, the force exerted by the sprin in the iddle is equal to (x - x1). Tain the positive direction to be to the riht, its force on 1 would be equal to + (x - x1), and its force on would be equal to -(x - x1). Newton's second law for the two asses 1 and then leads to the two equations x 1 x Fiure 4-3. Syste of two coupled oscillators. 4-3
Vector Spaces in Physics 8/6/15 x1 x1 ( x x1 ) x ( x x1 ) x, (4-8) or, in full atrix notation, F x Kx (4-8a) (eneralized Hooe's law). where 1 K (4-8b) 1 In the absence of external forces the asses will vibrate bac and forth in soe coplicated way. A ode of vibration where both asses ove at the sae frequency, in soe fixed phase relation, is called a noral ode, and the associated frequencies are referred to as the resonant frequencies of the syste. Such a otion is described by it x1() t a1e, (4-9) it x() t ae or i t x() t ae, (4-9a) Note that the frequency is the sae for both asses, but the aplitude and phase, deterined by a 1 or a, is in eneral different for each ass. d it it Substitutin (4-9a) into (4-8a), and usin the fact that e e, we obtain two dt coupled linear equations for the two undeterined constants of the otion a 1 or a : a1 a a1, (4-1) a1 a a or Ka a, (4-1a) Here we have introduced a diensionless constant, (4-11) where is the anular frequency of this ode of oscillation, and (4-1) is a constant characteristic of the syste, with the diensions of an anular frequency. Note that is not necessarily the actual frequency of any of the noral odes of the syste; the frequency of a iven noral ode will be iven by 1/. Equation (4-1a) is the eienvalue equation for the atrix K, and the eienvalues are deterined by re-writin (4-13) as K I a. (4-16) 4-4
Vector Spaces in Physics 8/6/15 This syste of linear equations will have solutions when the deterinant of the atrix K I is equal to zero. This leads to the characteristic equation: 1 K I 1 1 4 3 ( 1)( 3). (4-17) (1) There are thus two values of for which equation (4-16) has a solution: 1 and () (1) () 3, correspondin to frequencies of oscillation and 3. We will investiate the nature of the oscillation for each of these resonant frequencies. (1) Case 1.. This is the sae frequency as for the sinle ass-on-a-sprin of fiure 4-1a. How can the interconnected asses resonate at this sae frequency? A ood uess is that they will ove with a1 = a, so that the distance between 1 and is always equal to the equilibriu distance, and the sprin connectin 1 and exerts no force on either ass. To verify this, we substitute = 1 into equation (4-16) and solve for a1 and a: 1 K I a a 1. (4-18) 1 1 a 1 1 ivin two equations for two unnowns: a1a a1 a. (4-19) Both equations tell us the sae thin: a1 a. (4-) Both asses have the sae displaceent at any iven tie, so the sprin joinin the never influences their otion, and their resonant frequency is the sae as if the central sprin was not there. () Case. 3. This frequency is hiher than for the sinle ass-on-a-sprin of fiure 4-1a, so the iddle sprin ust be stretched in such a way as to reinforce the effect of the outer sprins. We iht uess that the two asses are ovin in opposite directions. Then as they separated, the iddle sprin would pull the both bac towards the center, while the outside sprins pushed the bac towards the center. The acceleration would be reater and the vibration faster. We can see if this is riht by substitutin = 3 into equation (4-16) and solve for a1 and a: 4-5
Vector Spaces in Physics 8/6/15 ivin the equations confirin that () () 1 K I a a () 1 1 1 a 1 1 1. (4-1) a1 a a a, (4-) a a. (4-3) 1 Thus we have the followin eienvalues and eienvectors for the atrix K : 1 1 1 a 1 1 1 1. (4-4) a 1 1 3 The equations above only deterined the ratios of coponents of a ; I have added the factor of 1/ to noralize the vectors to a anitude of 1. Three interconnected asses. With three asses instead of two, at positions x1, x and x 1 x x 3 Fiure 4-4. Syste of three coupled oscillators. x3, the three coupled equations still have the for of equation (4-13), with 1 K 1 1 1 and characteristic equation 1 1 (4-5) K I 1 1. (4-6) It will be left to the probles to find the three noral-ode frequencies and to deterine the way the asses ove in each case. Systes of any coupled asses. A lon chain of asses coupled with sprins is a coonly used odel of vibrations in solids and in lon olecules. It would not be too 4-6
Vector Spaces in Physics 8/6/15 hard to write down the atrix K correspondin to such a lon chain. However, analyzin the solutions requires ore advanced ethods which we have not yet developed. C. The triple pendulu There is an interestin proble which illustrates the power (and weanesses) of the trained physicist. Consider three balls, suspended fro a fixed point, as shown in fiure (a) (b) (c) 1 L 1 1 T x 1 L x x L 3 T 3 3 x 3 3 3 Fiure 4-5. Three balls, forin a copound pendulu. (a) Hanin fro the ceilin, at rest. (b) Oscillatin in the first noral ode. (c) Free-body diara for ball. 4-5a. If the balls are displaced fro equilibriu and released, they can ove in rather coplicated ways. A further ausin proble is to iaine ain the point of support ove bac and forth, or in a circle. We ay not et quite this far, for lac of tie. To ae a tractable proble, tae the usual scandalous physics approach of siplifyin the proble, as follows: 1. Consider only otion in a plane, consistin of the vertical direction and a transverse direction.. Consider only sall displaceents. The idea is to be able to ae the sall-anle approxiation to trionoetric functions. 3. Tae all three asses to be equal, iven by, and tae the three strin lenths to be equal, iven by L. Now the proble loos lie fiure 4-5b. The three variables of the proble are the transverse positions of the three balls. The forces on the three balls are not too hard to 4-7
Vector Spaces in Physics 8/6/15 calculate. For instance, the free-body diara for ball is shown in Fiure 4-5c. In the sall-anle approxiation, sin x x1 / L. (4-7) sin x x / L 3 3 3 Also, reasonin that the strin tensions ainly just hold the balls up, they are iven by T 3 T 1 T3 The vertical forces autoatically cancel. For forces in the horizontal direction, Newton's second law for this ball then ives " a F " x T sin T sin 3 3. (4-8) 1 3 L L x x x x x x x x 1 3. (4-9) Here we have used the fact that a siple pendulu consistin of a ass on a strin of lenth L oscillates with an anular frequency of o. (4-3) L Siilar reasonin for the other two asses leads to the three coupled equations in three unnowns, x 3 x x x 1 1 1 x x x x x 1 3 x x x 3 3. (4-31) We now loo for noral odes, where i t x ae. (4-3) Substitutin into equation (4-31) ives a factor of on the left-hand side, suestin that we define a diensionless variable as before,. (4-33) ivin with K Ia Ka. (4-34) 5 3 1. 1 1 (4-35) 4-8
Vector Spaces in Physics 8/6/15 This is the classic eienvector-eienvalue equation, Ka a. (4-36) (You iht want to fill in the steps yourself leadin fro equation (4-31) to this point.) In this way, the physical concept of a search for stationary patterns of relative displaceents of the asses translates into the atheatical idea of findin the eienvectors of the atrix K. As with the coupled asses, we write this equation in the for 5 K I a 3 1 a 1 1. (4-37) Solutions will exist if and only if the deterinant of the atrix K I vanishes, leadin to the "characteristic equation" for the eienvalues, 5 I K 3 1 1 1 5 3 1 1 4 1. 3 9 18 6 (4-38) This is a cubic, with three roots, and is hard to solve analytically. There is in principle a closed-for solution, but it is pretty hairy. Here is how Matheatica does it: NSolve[x^3-9x^+18x-6,x] {{x.415775},{x.948},{x6.8995}} Another pretty ood way, however, is just to calculate values usin Excel until you et close. In the spreadsheet to the riht you can see that the cubic oes throuh zero soewhere near =.4, and aain near =.. You can easily ae the step saller and pin down the values, as well as findin the third root. The values are iven in Table I. labda dlabda.1 labda equation -6.1-4.89. -.75.3-1.383.4 -.176.5.875.6 1.776.7.533.8 3.15.9 3.639 1 4 1.1 4.41 1. 4.368 1.3 4.387 1.4 4.34 1.5 4.15 1.6 3.856 1.7 3.53 1.8 3.7 1.9.569.1 1.371..688.3 -.43.4 -.816.5-1.65.6 -.464 4-9
Vector Spaces in Physics 8/6/15 otion eienvalue noralized frequency (sinle ball) 1. ode 1.4158.64487 ode.943 1.5147 ode 3 6.899.58 Table I. Eienvalues for the three noral odes of the three-ball syste, and the correspondin frequency, iven in ters of the frequency for a sinle ball on a strin of lenth L. Next, for each of the three eienvalues, we ust deterine the correspondin eienvector. This aounts to solvin the syste of three hooeneous linear equations, i 5 i i 3 1 a i 1 1. (4-39) Here (i) i and a are the i-th eienvalue and eienvector, respectively. For instance, for the first eienvalue iven above, this ives 4.584 1.584 1 a. (4-4) 1.584 The anitude of the eienvector is not deterined, since any ultiple of the eienvector would still be an eienvector, with the sae eienvalue. So, let's tae the first coponent of a to be equal to 1. The we can find the ratios a/a1 and a3/a1 fro 4.584 1.584 1 a. (4-41) 1.584 a 3 For instance, the equation fro the first line of the atrix is 4.584*1 * a. (4-4) ivin a.91. (4-43) Next, ultiply the third line in the atrix by.584 and add it to the second line, to ive 4-1
Vector Spaces in Physics 8/6/15 4.584 1.597 a 1.584 a 3. (4-44) The equation fro the second line is.597 a 3. (4-45) ivin a3 3.94. (4-46) Or, a 1 1.91 3.94. (4-47) for the first eienvector! In this ode, the coordinates of three balls are iven by x1( t) cos1t 1 i1 t x( t) x( t) a e.91 cos1t. (4-48) x3( t) 3.94 cos1t Note that the balls all ove in the sae direction, in this ode. The other eienvectors can be found in a siilar way. The exact values are left to the probles. But fiure 4-6 shows the displaceents of the balls in the three odes. The hiher the ode (and the hiher the frequency), the ore the balls ove in opposite directions. 4-11
Vector Spaces in Physics 8/6/15 Fiure 4-6. The three noral odes for the triple pendulu. The balls are shown at axiu displaceent, when they are all (oentarily) at rest. PROBLEMS Proble 4-1. (a) Usin identities fro Appendix A, show that C cos t Asin t Bcos t and find A and B in ters of C and. (b) Usin identities fro Appendix A, show that it it C cos t De Ee and find D and E in ters of C and. (Here C is taen to be real.) Proble 4-. Find the noral-ode frequencies i, i 1,3 for the proble described in the text (see fi. 4-4) of three identical asses connected by identical sprins. Express the frequencies in ters of, where. Proble 4-3. Find the noral odes for the proble described in the text (see fiure 4-4) of three asses connected by sprins. Proble 4-4. Consider a syste of two asses and three sprins, connected as shown in fiure 4-3, but with the iddle sprin of sprin constant equal to. 4-1
Vector Spaces in Physics 8/6/15 (a) Try and uess what the noral odes will be - directions of otion of the asses and frequencies. (b) Write the equations of otion, find the characteristic equation, and solve it, and so deterine the frequencies of the two noral odes. Copare with your uesses in part (a). Proble 4-5. Find the eienvectors a and a 3 for the triple pendulu correspondin to the second and third eienvalues, and. Give a qualitative interpretation, in ters of the co- or counter-otion of the balls, with respect to the first one. Proble 4-6. Repeat the analysis of the ultiple pendulu in the text, but for two balls, rather than three. You should deterine the two noral-ode frequencies i and the noral-ode eienvectors a i In this case it should be possible to find the eienvalues exactly, without havin to resort to nuerical ethods. Discuss the solution. 4-13