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Mass spring system. Am I on the right track?

  1. Sep 23, 2011 #1
    1. The problem statement, all variables and given/known data
    A mass at the end of a spring with natural frequency [itex]\omega[/itex] is released from rest at position [itex]x_0[/itex]. The experiment is repeated byut now with the system immersed in a fluid that causes the motion to be overdamped with damping coefficient [itex]\gamma[/itex]. Find the ratio of the maximum speed in the former case to that in the latter. What is the ratio in the limit of strong damping? [itex]\gamma >>\omega[/itex] In the limit of critical damping?

    2. The attempt at a solution
    Right now I am in need with the first part as after I get this I think I should be fine with the other two parts. I just need to know if I am on the right track with my thought process.

    So first I solve the undamped case (Which is frictionless).

    [tex]-kx=x''m[/tex]
    This leads to the solution..
    [tex]x(t)=x_0 cos(\omega t)[/tex]
    Taking the derivative and setting sine equal to 1, this has a max velocity of..
    [tex]v_{max}=x_0 \omega[/tex]

    This next part is the part I am unsure about. In the damped case, I solve the following differential equation..
    [tex]x''m+bx'+kx=0[/tex]
    And obtain a solution of..
    [tex]x(t)=x_0 e^{-\gamma t}cos(\omega_0 t)+\frac{\gamma x_0}{\omega_0}e^{-\gamma t}sin(\omega_0 t)[/tex]
    Where..
    [tex]\gamma=\frac{b}{2m}[/tex]
    and
    [tex]\omega_0=\sqrt{\gamma^2+\omega^2}[/tex]

    So my guess as to what I do next is to find the maximum of this velocity by differentiating x(t) twice and setting it equal to zero and getting a value of t. Then plug that value in and get a value of v. Doing all this I get a velocity of..
    [tex]\frac{\gamma x_0 (\omega_0-\gamma)}{\sqrt{2\gamma^2 +\omega^2}}[/tex]

    So the ratio would just be this value over the value I got in the frictionless case would it not? Something is telling me this isn't correct.

    I apologize for not showing all my work, as it was a lot of work and would take forever for me to type up the code. If need by ill type it up in my Math Type program and post the image.
     
    Last edited: Sep 23, 2011
  2. jcsd
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