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Critically damped oscillator: Classical mechanics help!

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  • #1
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Homework Statement



A critically damped oscillator with natural frequency [tex]\omega[/tex] starts out at position [tex]x_0>0[/tex]. What is the maximum initial speed (directed towards the origin) it can have and not cross the origin?

Homework Equations


For the case of critical damping,
[tex]x(t)=e^{(-\gammat)}(A+Bt) where \gamma=\omega=\sqrt{k/m}[/tex]


The Attempt at a Solution


Well first I derived the above equation (and verified it with my textbook). I then evaluated the initial position:
[tex]x(0)=A[/tex].
I then took the derivative of the position function to get velocity:
[tex]v(t)=(e^{-\gamma t}) ( B-Bt\gamma -A\gamma)[/tex]
Setting t=0 I obtained
[tex]v(0)=B-A\gamma[/tex] recalling that x0=A, [tex]v(0)= B-x_0\gamma[/tex]. I then solved for Beta: [tex] B=v_0 +\gamma x_0. [/tex]
I then tried substituting this back into the position equation, and solving for the initial velocity, the program is there is always time dependency that i cant get rid of.... what am i doing wrong? any advice would be much appreciated!
Thanks
 
Last edited:

Answers and Replies

  • #2
237
5
umm its supposed to be e-tgamma(B-Bt(gamma)-A(gamma)) for the velocity and v(0)=B-A(gamma)=B-x_0(gamma) for the initial velocity
and x(t)=e-tgamma(A+Bt) for the position but for some reason I all the sudden fail at latex
 
  • #3
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I think I solved it... you're on the right track. Solve for x(t) in terms of gamma, w, Xo and Vo, then make x(t)=0 and solve for Vo. Next, derive V(t) by taking dx/dt (remember to include initial conditions!). Take the limit as t goes to inf of V(t), and substitute terms so that you're left with w and Xo, and you should get Vmax.
 

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