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Periodically Dampened Oscillator

  1. Sep 24, 2013 #1
    1. The problem statement, all variables and given/known data
    A body with mass m is connected to a spring in 1D and is at rest at X = A > 0. For the region X > 0, the only force acting on the mass is the restoring force of the spring. For the region X < 0, a viscous fluid introduces damping into the system.
    a) Find the speed of the particle at X = 0.
    b) Find the speed of the particle at X = 0, as it emerges from the X < 0 region.
    c) Find the maximal position D the body reaches in the positive region after having left the negative region

    2. Relevant equations
    F = -kx for x > 0
    F = -kx-bv for x < 0

    3. The attempt at a solution
    a) x(2) +k/m x = x(2) + ω02x = 0
    The characteristic equation for free oscillations:
    x = x0cos(ω0t + δ)
    where x0 = A and δ = 0
    x = Acosω0t
    From here its easy enough to find the the speed at x = 0 is Aω0
    b)x(2)+ 2βx(1)+ ω02x = 0 when x < 0
    The characteristic equation for this differential is:
    x = Ae-βtcos(ω1t) where ω1 is √(ω022)
    The period of oscillation is ∏/ω1

    My confusion at this point is how to incorporate the initial velocity into this equation.
    Once I know that I can easily plug in the period into the velocity equation and find the new velocity at x = 0
     
  2. jcsd
  3. Sep 24, 2013 #2

    TSny

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    Homework Helper
    Gold Member

    You need to reconsider how you wrote the solution here. If you redefine t = 0 to be the time the body enters the viscous region, then what should x equal when t = 0?

    Also, you can't assume that the constant factor "A" in the solution for x < 0 is the same A as in part (a).
     
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