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Acceleration amplitude of a damped harmonic oscillator

  1. Apr 21, 2017 #1
    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. jcsd
  3. Apr 21, 2017 #2

    kuruman

    User Avatar
    Homework Helper
    Gold Member

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