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Unformatted text preview: 2 ) eλt = 0 Using the quadratic formula, we ﬁnd λ 1 = β + q β 2ω 2 λ 2 = βq β 2ω 2 Since this is an overdamped system, both λ 1 and λ 2 are positive. 1 (c) The full solution is x ( t ) = Aeσt + B eλ 1 t + C eλ 2 t x ( t ) = eσt h A + B e( λ 1σ ) t + C e( λ 2σ ) t i Since both λ 1σ and λ 2σ are positive, the B and C terms become negligible for large t . Thus, x ( t ) ∼ Aeσt as t → ∞ . (d) Note that because λ 1 and λ 2 are solutions to λ 22 βλ + ω 2 = 0 , we can rewrite the amplitude A from part (a) as A = F ( σλ 1 )( σλ 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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 Spring '08
 Staff
 Physics, mechanics, Force, Quadratic equation

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