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class9 - 2 3 Overdamping In the case where o < 2 the roots...

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3.) Overdamping: In the case where ω 2 o < γ 2 , the roots of the equation are real. We can define ω 2 = γ 2 - ω 2 0 (44) So that the general solution becomes x ( t ) = e - γ t A 1 e ω 2 t + A 2 e - ω 2 t (45) The motion is not oscillatory. The displacement goes to zero asymptotically, but not as fast as a critically damped system. The velocity is ˙ x ( t ) = - γ x + e - γ t ω 2 A 1 e ω 2 t - ω 2 A 2 e - ω 2 t (46) For all illustrated paths on the phase diagram, the asymptotic paths at long times are along the dashed curve ˙ x = - ( γ - ω 2 ) x . On in a special case (see assignment 3) does the phase path asymptote at the other dashed curve. 6
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(2) x x . (1) Figure 3: Schematic phase dia- gram for an overdampled oscilla- tor. Line (1) is ˙ x = - ( γ - ω 2 ) x , and line (2) is ˙ x = - ( γ + ω 2 ) x . Precise paths depend on initial conditions x 0 and ˙ x 0 . example: Consider a pendulum of length l with a mass m at the end, moving through oil with decreasing θ . There is a resistive force present of F = 2 m g/l ( l ˙ θ ). Evaluate the type of damping in this system.
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