- #1

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## Homework Statement

For a lightly damped harmonic oscillator and driving frequencies close to the natural frequency [itex]\omega \approx \omega_{0}[/itex], show that the power absorbed is approximately proportional to

[tex]

\frac{\gamma^{2}/4}{\left(\omega_{0}-\omega\right)^{2}+\gamma^{2}/4}

[/tex]

where [itex]\gamma[/itex] is the damping constant. This is the so called Lorentzian function.

## Homework Equations

[tex]

\text{Average power absorbed} = P_{avg} = \frac{F_{0}^{2} \omega_{0}}{2k Q} \frac{1}{\left(\frac{\omega_{0}}{\omega}-\frac{\omega}{\omega_{0}}\right)^{2}+\frac{1}{Q^{2}}} \\

\omega_{0} = \sqrt{\frac{k}{m}}\\

m = \frac{b}{\gamma}\\

\text{where $b$ is the damping constant and $m$ is the mass}\\

\Delta \omega = \frac{\gamma}{2}

[/tex]

## The Attempt at a Solution

The course of action that I took goes like:

1.Find [itex]k[/itex] and [itex]Q[/itex] in terms of [itex]\omega_{0}[/itex] and [itex]\gamma[/itex].

2. Chug through and do some algebra (and it is here that its very possible that a mistake was made, but I'll put my result not all the steps).

3. Expand a function about [itex]w_{0}[/itex] and make approximations so that [itex]\Delta \omega[/itex] is small.

(4) See the above equation fall out. This is the stage that I'm stuck at.

[tex]

k = b \frac{\omega_{0}^{2}}{\gamma}\\

Q = \frac{\omega_{0}}{\gamma}\\

2 \Delta \omega = \gamma

\\

P_{avg} = \text{plug in and do lots of algebra...}\\

P_{avg} = \frac{\frac{\omega^{2}\gamma^{2}}{(\omega+\omega_{0})^{2}}}{(\omega_{0}-\omega)^{2}+\frac{\omega^{2}\gamma^{2}}{(\omega+\omega_{0})^{2}}}

[/tex]

Then taylor expanding [itex]f(\omega) = \frac{\omega^{2}}{(\omega+\omega_{0})^{2}}[/itex] about [itex]\omega_{0}[/itex]....

Am I on the right try here? I'd like that taylor expansion to equal [itex]\frac{1}{4}[/itex] because then the equation would match the one described in the question but I'm trying it by hand and with mathematica and I'm not seeing them match.