# 震动与波浪 Vibrations & Waves PHYS125001

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$$A=\frac{F}{2 \pi \sqrt{4 \pi^{2} m^{2}\left(f^{2}-f_{0}^{2}\right)^{2}+b^{2} f^{2}}}$$
Show that the maximum occurs not at $f_{\mathrm{a}}$ but rather at the frequency
$$f_{\text {res }}=\sqrt{f_{o}^{2}-\frac{b^{2}}{8 \pi^{2} m^{2}}}=\sqrt{f_{o}^{2}-\frac{1}{2} \mathrm{FWHM}^{2}}$$

## PHYS125001COURSE NOTES ：

\begin{aligned} a &=F / m \ &=4 T h / \mu w^{2} . \end{aligned}
The time required to move a distance $b$ under constant acceleration $a$ is found by solving $h=\frac{1}{2} a t^{2}$ to yield
\begin{aligned} t &=\sqrt{2 b / a} \ &=w \sqrt{\frac{\mu}{2 T}} . \end{aligned}
Our final result for the velocity of the pulses is
\begin{aligned} |v| &=w / t \ &=\sqrt{\frac{2 T}{\mu}} . \end{aligned}

# 震动与波浪 Vibrations & Waves PHYS10302T

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The speed of the rotating mass is the dramference of the circle divided by the period, $v=2 \pi A / T$, its acceleration (which is directly inward) is $a=v^{2} / r$, and Newton’s second law gives $a=F / m=(k A+F) / m$. We write $f_{\text {ra }}$ for $(1 / 2 \pi) \sqrt{k / m}$. Straightforward algebra yields
$$\frac{F_{r}}{F_{s}}=\frac{2 \pi m}{b f}\left(f^{2}-f_{r s}^{2}\right)$$
This is the ratio of the wasted force to the useful force, and we see that it becomes zero when the system is driven at resonance.
The amplitude of the vibrations can be found by attacking the equation $|F|=b v=2 \pi b A f$, which gives
$$A=\frac{\left|F_{\mathrm{t}}\right|}{2 \pi b f} .$$
However, we wish to know the amplitude in terms of $|F|$, not $|F|$. From now on, let’s drop the cumbersome magnitude symbols. With the Pythagorean theorem, it is easily proven that
$$F_{s}=\frac{F}{\sqrt{1+\left(\frac{F_{r}}{F_{s}}\right)^{2}}},$$

## PHYS10302T COURSE NOTES ：

\begin{aligned} a &=F m \ &=4 T h / \mu w^{2} . \end{aligned}
The time required to move a distance $b$ under constant acceleration $a$ is found by solving $h=\frac{1}{2} a t^{2}$ to yield
\begin{aligned} t &=\sqrt{2 h / a} \ &=w \sqrt{\frac{\mu}{2 T}} . \end{aligned}
Our final result for the velocity of the pulses is
\begin{aligned} |v| &=w / t \ &=\sqrt{\frac{2 T}{\mu}} . \end{aligned}