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Damped Oscillator equation  Energy 
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#1
Dec812, 09:06 PM

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the damped oscillator equation:
(m)y''(t) + (v)y'(t) +(k)y(t)=0 Show that the energy of the system given by E=(1/2)mx'² + (1/2)kx² satisfies: dE/dt = mvx' i have gone through this several time simply differentiating the expression for E wrt and i end up with dE/dt = x'(vx') im at a brick wall. Am i doing something wrong? Any help is much appreciated! Thanks 


#2
Dec812, 10:12 PM

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#3
Dec912, 11:39 AM

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PF Gold
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You can't just differentiate E the way you did and prove the theorem. You need to incorporate the basic diff eq representing a damped springmass system. The expression for E represents ANY springmass system, damped or not, linear or not, etc.
The only thing I can think of is to solve the diff eq (it's a simple 2nd order one with constant coeff). Apply an initial condition to x = x0. Derive the solution x(t) and then substitute in E, take dE/dt and there you are. (BTW why is y used in the diff eq and x in E?) Dick's comment is well taken! Not only his, but I noticed the dimensions don't make sense. dE/dt has dimensions of FLT^{1} whereas mvx' has dimensions of MF where M = mass F = force = MLT^{2} L = length T = time. Thus mvx' has the wrong dimensions to be dE/dt. 


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Dec912, 11:58 AM

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Damped Oscillator equation  Energy



#6
Dec912, 02:28 PM

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#7
Dec912, 03:37 PM

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PF Gold
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#8
Dec912, 03:39 PM

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