Acceleration amplitude of a damped harmonic oscillator

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1. Apr 21, 2017

TheBigDig

1. The problem statement, all variables and given/known data
The acceleration amplitude of a damped harmonic oscillator is given by
$$A_{acc}(\omega) = \frac{QF_o}{m} \frac{\omega}{\omega _o} \sqrt{\it{R}(\omega)}$$
Show that as $\lim_{\omega\to\infty}, A_{acc}(\omega) = \frac{F_o}{m}$

2. Relevant equations
$$\it{R}(\omega) = \frac{(\gamma \omega)^2}{(\omega _o ^2 - \omega^2)^2 +(\gamma \omega)^2}$$
$$Q = \frac{\omega _o}{\gamma}$$

3. The attempt at a solution
From substituting in the above equations into the formula and cancelling off, I've gotten this far
$$\frac{F_o}{m} \frac{\omega ^2}{\sqrt{(\omega _o ^2 - \omega^2)^2 +(\gamma \omega)^2}}$$
I'm fairly certain I haven't made any mistakes in my cancelling off. I don't see how the equation will tend to $\frac{F_o}{m}$ as surely $\frac{\omega ^2}{\sqrt{(\omega _o ^2 - \omega^2)^2 +(\gamma \omega)^2}}$ will go to zero.

2. Apr 21, 2017

kuruman

Look at the denominator. What does it become when $\omega$ becomes very large? In other words, what is its $\omega$ dependence?