CDIM12 - CHAPTER 12 Coupled Oscillations 12-1. m1 = M k1 x1...

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CHAPTER 12 Coupled Oscillations 12-1. m 1 = M k 1 x 1 k 12 k 2 m 2 = M x 2 The equations of motion are ( ) () 11 1 2 1 1 2 2 22 1 2 2 1 2 1 0 0 Mx x x Mx x x κκ κ ++ = = ±± (1) We attempt a solution of the form it xt Be ω = = (2) Substitution of (2) into (1) yields ( ) ( ) 2 2 2 2 2 12 1 2 12 2 0 0 MB B BM B +− = −+ + = (3) In order for a non-trivial solution to exist, the determinant of coefficients of and must vanish. This yields 1 B 2 B ( ) ( ) 1 12 2 12 12 MM = 2   (4) from which we obtain 2 12 1 2 1 2 2 1 4 + (5) This result reduces to ( ) 2 12 12 M ωκ =+ ± for the case = = (compare Eq. (12.7)]. 397
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398 CHAPTER 12 If were held fixed, the frequency of oscillation of m would be 2 m 1 ( 2 01 1 12 1 M ) ω κκ =+ (6) while in the reverse case, would oscillate with the frequency 2 m ( 2 02 2 12 1 M ) (7) Comparing (6) and (7) with the two frequencies, + and , given by (5), we find () 2 22 12 1 2 1 2 1 24 2 M κ +  + + +   ωκ 2 1 2 12 11 2 0 2 2 MM + + = + = 1 >+ (8) so that 01 + > (9) Similarly, 2 1 2 1 2 1 2 M + + 2 1 21 2 0 2 2 + = + = 2 <+ (10) so that 02 < (11) If , then the ordering of the frequencies is 1 > 2 01 02 ωω + >>> (12) 12-2. From the preceding problem we find that for 12 1 2 , ± 2 2 ; ++ ≅≅ (1) If we use 1 01 02 ; 2 == (2) then the frequencies in (1) can be expressed as
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COUPLED OSCILLATIONS 399 () 12 10 1 0 1 1 1 12 20 2 0 2 2 2 11 κ ωω ω ε =+ ≅+ (3) where 12 12 12 ; 22 == (4) For the initial conditions [Eq. 12.22)], ( ) ( ) ( ) ( ) 1 212 0 , 00 x x x === ±± , = xD (5) the solution for ( ) 1 xt is just Eq. (12.24): 1 cos cos D t t +  =   (6) Using (3), we can write ( ) ( ) 1 2 01 02 1 01 2 02 ++ += + + + ≡Ω+ (7) ( ) ( ) 1 2 01 02 1 01 2 02 −− =−+ (8) Then, ( ) ( ) ( ) 1 cos cos xt D t t t t =Ω + + (9) Similarly, ( 2 sin sin sin sin t t Dt t t +− ) t = + + (10) Expanding the cosine and sine functions in (9) and (10) and taking account of the fact that + and are small quantities, we find, to first order in the ’s, ( ) 1 cos cos sin cos cos sin D t t t t t t t t + − ++ − −+ ≅ΩΩ −ΩΩ   (11) ( ) 2 sin sin cos sin sin cos D t t t t t t t t ++− + ΩΩ +ΩΩ (12) When either ( ) 1 or ( ) 2 reaches a maximum, the other is at a minimum which is greater than zero. Thus, the energy is never transferred completely to one of the oscillators.
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400 CHAPTER 12 12-3. The equations of motion are 2 12 0 1 2 22 0 2 0 0 m xxx M m M ω ++= ±± (1) We try solutions of the form ( ) ( ) 11 ; it xt Be == (2) We require a non-trivial solution (i.e., the determinant of the coefficients of B and equal to zero), and obtain 1 2 B ( ) 2 2 4 0 0 m M ωω    = (3) so that 2 0 m M −= ± (4) and then 2 2 0 1 m M = ± (5) Therefore, the frequencies of the normal modes are 2 0 1 2 0 2 1 1 m M m M = + = (6) where 1 corresponds to the symmetric mode and 2 to the antisymmetric mode. By inspection, one can see that the normal coordinates for this problem are the same as those for the example of Section 12.2 [i.e., Eq. (12.11)].
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CDIM12 - CHAPTER 12 Coupled Oscillations 12-1. m1 = M k1 x1...

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