The angular frequency of the damped oscillato | vedclass

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(A) The angular frequency of an undamped oscillator is $\omega_0 = \sqrt{\frac{k}{m}}$.The angular frequency of a damped oscillator is $\omega = \sqrt

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increases by $1\%$

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(A) The angular frequency of an undamped oscillator is $\omega_0 = \sqrt{\frac{k}{m}}$.The angular frequency of a damped oscillator is $\omega = \sqrt{\frac{k}{m} - \frac{r^2}{4m^2}} = \omega_0 \sqrt{1 - \frac{r^2}{4mk}}$.Using the binomial approximation $(1-x)^n \approx 1-nx$ for small $x$,we get $\omega \approx \omega_0 (1 - \frac{r^2}{8mk})$.The time period $T = \frac{2\pi}{\omega}$,so $T \approx T_0 (1 - \frac{r^2}{8mk})^{-1} \approx T_0 (1 + \frac{r^2}{8mk})$.The fractional change in time period is $\frac{\Delta T}{T_0} = \frac{T - T_0}{T_0} = \frac{r^2}{8mk}$.Given $\frac{r^2}{mk} = 8\% = 0.08$,we have $\frac{\Delta T}{T_0} = \frac{0.08}{8} = 0.01 = 1\%$.Since the value is positive,the time period increases by $1\%$.

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