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Math Help - Damped Oscillations

  1. #1
    Cobra150
    Guest

    Damped Oscillations

    Hey,

    I'm a bit stuck on these questions about damped oscillations. It's actually from a physics course but the question is very maths based.



    I'm basically stuck (well can't do) sections a) b) and c). Normally I can do differentiation but this one is a real beast. I've tried doing the basic stuff in part a) (produce rule and all that) but i'm getting no where and end up with a huge equation which i don't know if its right or not. I must be doing something wrong.

    Any help would be greatly appreciated!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Cobra150 View Post
    Hey,

    I'm a bit stuck on these questions about damped oscillations. It's actually from a physics course but the question is very maths based.



    I'm basically stuck (well can't do) sections a) b) and c). Normally I can do differentiation but this one is a real beast. I've tried doing the basic stuff in part a) (produce rule and all that) but i'm getting no where and end up with a huge equation which i don't know if its right or not. I must be doing something wrong.

    Any help would be greatly appreciated!
    (a) find \dot{x}

    <br />
x(t)=C e^{-\gamma t/2} \sin(w_f t+\phi)<br />

    Using the product rule:

    <br />
\dot{x}(t)=C (-\gamma/2) e^{-\gamma t/2} \sin(w_f t+\phi)+C~ w_f~e^{-\gamma t/2} \cos(w_f t+\phi)<br />
<br />
= -\gamma ~x(t)/2 +C~ w_f~e^{-\gamma t/2} \cos(w_f t+\phi)<br />

    Hence (again using the product rule):

    <br />
\ddot{x}(t)<br />
= -\gamma ~\dot{x}(t)/2 +C~(-\gamma/2) w_f~e^{-\gamma t/2} \cos(w_f t+\phi) - C~ (w_f)^2~e^{-\gamma t/2} \sin(w_f t+\phi)<br />

    ............. <br />
=-\gamma ~\dot{x} (t)/2 -(w_f)^2 \dot{x}(t)+C~(-\gamma/2) w_f~e^{-\gamma t/2} \cos(w_f t+\phi)

    ............. =-\gamma ~\dot{x} (t)/2 -(w_f)^2 \dot{x}(t)+(-\gamma/2)[\dot{x}(t)+\gamma ~x(t)/2]

    Now check this (there is probably a mistake somewhere) and simplify.

    RonL
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