Mech6Sols

Mech6Sols - Physics 105B Problem Set 6 May 13, 2008 Jeff...

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Physics 105B Problem Set 6 May 13, 2008 Jef Schonert: schonert (at) physics.ucsb.edu Taylor 11.27 a) Working in cartesian coordinates, the Lagrangian is L = 1 2 m x 2 1 x 2 2 ) - 1 2 k ( x 1 - x 2 ) 2 (1) = 1 2 m x 2 1 x 2 2 ) - 1 2 k ( x 2 1 + x 2 2 - x 1 x 2 - x 2 x 1 ) (2) From this, the mass and spring constant matrices can be read of M = ± m 0 0 m ² , K = ± k - k - kk ² To ±nd the normal ²requencies, we need to calculate det( K - ω 2 M ) = 0, which is m 2 ω 2 ( ω 2 - 2 ω 2 0 )=0 , (3) where we de±ned 2 0 = k . So the ²requency is either ω = 0 or ω = 2 ω 0 . b) For the mode with ²requency ω = 2 ω 0 , we must solve ( K - ω 2 M ) a =0 , which is equivalent to ± - 2 0 - 2 0 - 2 0 - 2 0 ²± a 1 a 2 ² In order to satis²y this equation, a 1 and a 2 must be related by a 1 = - a 2 , meaning that the masses oscillate with equal amplitude but out o² phase. c) For the mode with ²requency ω = 0, we have ± 2 0 - 2 0 - 2 0 2 0 a 1 a 2 ² , which requires that a 1 = a 2 . As the problem suggest, try a solution o² the ²orm x ( t )= a f ( t ), and plug it into the equation M ¨ x = - Kx . In order to satis²y the equation o² motion, f ( t ) must satis²y ¨ f = 0, i.e. f ( t bt + c . This mode describes two carts whose separation is ±xed, but whose center o² mass moves with constant speed. Taylor 11.35 a) Recall that the equations o² motion are ¨ φ 1 = - ω 2 0 φ 1 +( k/m )( φ 2 - φ 1 ) (4) ¨ φ 2 = - ω 2 0 φ 2 - ( k/m )( φ 2 - φ 1 ) (5) 1
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It is natural to defne ξ 1 = φ 1 + φ 2 and ξ 2 = φ 1 - φ 2 , as can be seen by adding and subtracting the two Euler-Lagrange equations. IF we add them, we get ¨ φ 1 + ¨ φ 2 = - ω 2 0 ( φ 1 + φ 2 ) , which can be rewritten as ¨ ξ 1 = - ω 2 0 ξ 1 . Similarly, iF we subtract the second equation From the frst, we get ¨ φ 1 - ¨ φ 2 = - ω 2 0 ( φ 1 - φ 2 ) + (2 k/m )( φ 2 - φ 1 ) This can be rewritten as ¨ ξ 2 = - (2 k/m + ω 2 0 ) ξ 2 Both oF these equations are uncoupled.
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This note was uploaded on 07/15/2008 for the course PHYS 105 taught by Professor Martinis during the Spring '08 term at UCSB.

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Mech6Sols - Physics 105B Problem Set 6 May 13, 2008 Jeff...

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