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Forced oscillations without damping

  1. Apr 17, 2013 #1
    1. The problem statement, all variables and given/known data
    We are to develop the equations of motion for an undamped horizontal spring system, the mass of which is being driven by a periodic force: F=F0 cos wt. I know how to do it but my teacher has defined an odd term, the meaning of which I want to be clarified.

    2. Relevant equations

    The potential energy of the system is V = 1/2 kx^2. So the force is -kx.
    The driving force is F0 cos wt.
    The natural frequency of the system is w0^2 = k/m

    3. The attempt at a solution
    So, here's my solution:
    [itex]m\ddot{x}+kx = F_{0} cos(ωt)[/itex]

    My teacher has the same thing. But what he does next is that he says
    [itex]a = \frac{F_{0}}{k}[/itex]

    And then:
    [itex]\ddot{x}+ω^{2}_{0}x = ω^{2}_{0} a cos(ωt)[/itex]

    Which is all fine...but what the heck is [itex]a[/itex]? is it just some random thing or is it something of physical significance?

    Thanks a lot guys!
     
  2. jcsd
  3. Apr 17, 2013 #2

    haruspex

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    Suppose that the applied force were constant, at its max value F0. What would F0/k represent?
     
  4. Apr 17, 2013 #3
    By constant, do you mean it's not periodic? That means it has sort infinite time period. So it's like moving it in a straight line. Logically I'd think that F0/k would be the extension of the spring due to the force but I'm not sure if that's correct...
     
  5. Apr 17, 2013 #4
    sort of*
     
  6. Apr 17, 2013 #5
    I think my reasoning for this is correct.

    Consider first that the frequency of the driving force goes to zero.

    [itex]lim_{ω\rightarrow0}\:F_{0}cos(ωt) = .... [/itex]

    The driving force then goes to it's max value of [itex]F_{0}[/itex]
    This can now be described by Hooke's Law (considering only the magnitude, not direction):

    [itex]F_{0} = kx → x = F_{0}/k [/itex]

    So essentially [itex] a [/itex] can be thought of a non-periodic driving force amplitude, where the spring constant effectively determines the value of it.

    My textbook literally reads: "At very low frequencies the amplitude of oscillations tends to the value of the amplitude a of the point of suspension. Under these conditions the motion is governed by the spring constant of the spring."
     
  7. Apr 17, 2013 #6

    haruspex

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    Quite so.
     
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