(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider a damped oscillator, with natural frequency ω_naut and damping constant both fixed, that is driven by a force F(t)=F_naut*cos(ωt).

a) Find the rate P(t) at which F(t) does work and show that the average (P)_{avg}over any number of complete cycles is mβω^{2}A^{2}. b) Verify that this is the same as the average rate at which energy is lost to the resistive force. c) Show that as ω is varied, (P)_{avg}is maximum when ω=ω_naut; that is, the resonance of the power occurs at ω=ω_naut (exactly).

2. Relevant equations

(P)_{avg}= 1/τ ∫ Fv dt

F= F_naut*cos(ωt)

x(t)= A*cos(ωt)

v(t)= -Aωsin(ωt)

ω= √[(ω_naut)^{2}- β^{2}]

3. The attempt at a solution

I plugged in the equation for F(t) and v(t) in the integral ( P(t) = ∫Fv ) and used u substitution, making cos(ωt) the "u" and -ωsin(ωt)dt the "du".

So I then had ∫(F_naut)Au du

After integrating and plugging cos(ωt) back in for u, I had:

(P)_{avg}= (1/2τ)*(F_naut)A*cos^{2}(ωt), integrated from -τ/2 to τ/2.

This is where I hit a problem. My work doesn't seem like it will be equal to mβω^{2}A^{2}. Because I can't solve this, I can't figure out the next two parts either. Did I make a mathematical error in integrating, or did I set up the integral wrong? Thank you!

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# Homework Help: The rate at which a damped, driven oscillator does work

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