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Can someone explain to me what my professor did?

  1. Nov 6, 2015 #1
    1. The problem statement, all variables and given/known data
    upload_2015-11-6_17-35-30.png


    Two things I dont understand; how did he get that omega is sqrt(k/m-(b/2m)^2)
    And second; why is it that x(t) = e^(-bt/2m)* cos (omega*t+phi)
    shouldnt it rather be; x(t) = c1*cos(ωt) + c2*sin(ωt)


    2. Relevant equations


    3. The attempt at a solution
     
  2. jcsd
  3. Nov 6, 2015 #2
    I figured out why omega is what it is, but still dont understand how he got the equation for x(t)
     
  4. Nov 6, 2015 #3

    gneill

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    Staff: Mentor

    I think he's assuming that the system is underdamped, so will be a damped sinusoid. The general solution can be molded into that shape:

    ##x(t) = c_1 cos(ωt) + c_2 sin(ωt)##
    ##~~~= \sqrt{c_1^2 + c_2^2}\left( \frac{c_1}{\sqrt{c_1^2 + c_2^2}} cos(ωt) + \frac{c_2}{\sqrt{c_1^2 + c_2^2}} sin(ωt) \right) ##
    ##~~~= \sqrt{c_1^2 + c_2^2}\left(cos(\phi) cos(ωt) + sin(\phi) sin(ωt) \right) ##
    ##~~~= \sqrt{c_1^2 + c_2^2} cos(ωt - \phi)## where: ##~~~~\phi = tan^{-1}\left( \frac{c_2}{c_1} \right)##

    You can fudge the sign of the angle ##\phi## by negating it and interpreting it appropriately.
     
  5. Nov 6, 2015 #4
    Aren't c1 and c2 constants? If so, I see a sinusoid, not a damped sinusoid.
     
  6. Nov 6, 2015 #5

    gneill

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    Staff: Mentor

    Ah, right. I left out the damping term in the general solution o:) So:
    ##x(t) = e^{-\alpha t}(c_1 cos(ωt) + c_2 sin(ωt))##
    Go from there. The roots of the auxiliary equation will be complex conjugates of the form ##\alpha ± \omega##, where ##\omega## can be further broken down as ##\omega = \sqrt{\alpha^2 - \omega_o^2}##
     
  7. Nov 6, 2015 #6
    This is exactly the point where I dont know where to go from :P. Where does the c2sin(ωt) part go?
     
  8. Nov 6, 2015 #7

    gneill

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    Staff: Mentor

    See post #3. The sin and cos terms can be amalgamated into a single cos (or sin) term with a constant phase shift.
     
  9. Nov 6, 2015 #8
    Look at the undamped case:

    ##x(t)=c_1 \cos (\omega t)+c_2 \sin (\omega t)##

    and

    ##x(t)=A \cos (\omega t+ \phi)##

    are equivalent. Note that each contains two constants of integration.
     
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