Chapter 5: Problem 45
Consider a damped oscillator, with natural frequency \(\omega_{\mathrm{o}}\) and damping constant \(\beta\) both fixed, that is driven by a force \(F(t)=F_{0} \cos (\omega t) .\) (a) Find the rate \(P(t)\) at which \(F(t)\) does work and show that the average rate \(\langle P\rangle\) over any number of complete cycles is \(m \beta \omega^{2} A^{2} .\) (b) Verify that this is the same as the average rate at which energy is lost to the resistive force. (c) Show that as \(\omega\) is varied \(\langle P\rangle\) is maximum when \(\omega=\omega_{\mathrm{o}} ;\) that is, the resonance of the power occurs at \(\omega=\omega_{\mathrm{o}}\) (exactly).
Short Answer
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Key Concepts
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