Physics 325 Spring 2011 Problem Session 7 Solutions

Physics 325 Spring 2011 Problem Session 7 Solutions - 2 )...

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Physics 325 Spring 2011 Discussion 7 March 14, 2011 Consider an overdamped driven harmonic oscillator with driving force proportional to e - σt , for some constant σ > 0 . The equation of motion is therefore ¨ x + 2 β ˙ x + ω 2 0 x = F e - σt . (a) Find a particular solution of this differential equation. (b) Let e - λ 1 t and e - λ 2 t be the homogeneous solutions. Determine λ 1 and λ 2 in terms of β and ω 0 . (c) Suppose the initial conditions are such that the full solution is x ( t ) = x p ( t ) + B e - λ 1 t + C e - λ 2 t , where B and C are not zero. Here, x p ( t ) is the particular solution from part (a). If σ < λ 1 and σ < λ 2 , what is the behavior of x ( t ) as t → ∞ ? (d) Assume F is positive. For what range of σ will x p ( t ) be negative? Solutions : (a) We expect a particular solution of the same form as the driving force. Thus, let’s try x ( t ) = Ae - σt . Substituting this into the equation of motion gives ( σ 2 - 2 βσ + ω 2 0 ) Ae - σt = F e - σt A = F σ 2 - 2 βσ + ω 2 0 . (b) Now we substitute a trial solution e - λt into the homogeneous equation ¨ x + 2 β ˙ x + ω 2 0 x = 0 . ( λ 2 - 2 βλ + ω
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Unformatted text preview: 2 ) e-λt = 0 Using the quadratic formula, we find λ 1 = β + q β 2-ω 2 λ 2 = β-q β 2-ω 2 Since this is an overdamped system, both λ 1 and λ 2 are positive. 1 (c) The full solution is x ( t ) = Ae-σt + B e-λ 1 t + C e-λ 2 t x ( t ) = e-σt h A + B e-( λ 1-σ ) t + C e-( λ 2-σ ) t i Since both λ 1-σ and λ 2-σ are positive, the B and C terms become negligible for large t . Thus, x ( t ) ∼ Ae-σt as t → ∞ . (d) Note that because λ 1 and λ 2 are solutions to λ 2-2 βλ + ω 2 = 0 , we can rewrite the amplitude A from part (a) as A = F ( σ-λ 1 )( σ-λ 2 ) . If σ > λ 1 and σ > λ 2 , then A > . Similarly, if σ < λ 1 and σ < λ 2 , then A is also positive. Thus, we can only get A < if min( λ 1 ,λ 2 ) < σ < max( λ 1 ,λ 2 ) . 2...
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This note was uploaded on 10/06/2011 for the course PHYS 325 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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Physics 325 Spring 2011 Problem Session 7 Solutions - 2 )...

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