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Please could someone explain the concept in the text below for me?

In a driven oscillator, the phase of the driver and the phase of the oscillator are not always the same. At low frequency ($\omega\ll\omega_0$) the oscillation is in phase with the driver. At high frequency ($\omega\gg\omega_0$) the oscillation and the driver are out of phase. The phase difference between the driver and the oscillator is given by: $$\phi(\omega)=\arctan\left( \frac{-\gamma\omega}{\omega_0^2-\omega^2}\right). \tag{1.6}$$

If the oscillator is in phase with the driver at low frequency then surely this would cause the oscillator to resonate, but there should only be one frequency(the natural frequency)that can achieve resonance? I think I am confusing things here...please help! Thanks in advance.

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    $\begingroup$ Hello! It is preferable to type out screenshots or images of text; for formulae, one can use MathJax. Thanks! $\endgroup$
    – jng224
    Nov 15, 2021 at 22:10

3 Answers 3

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starting with the equation of motion for a driven oscillator

$$\ddot x+\gamma\,\dot{x}+\omega_0^2\,x=F\,\cos(\omega\,t)$$

the particular solution is:

$$x(t)=A\,\cos(\omega\,t+\phi)$$

where
$$A=\frac{F}{\sqrt{\left(\omega^2-\omega_0^2\right)^2+\gamma^2\,\omega^2}}$$ and $$\phi=\arctan\left(\frac{\gamma\,\omega}{\omega^2-\omega_0^2}\right)= \arctan\left(\frac{\gamma/\omega}{\left(1-\left(\frac{\omega_0}{\omega}\right)^2\right)}\right)$$

hence

$$\frac{x(t)}{F\,\cos(\omega\,t)} =\frac{A\,\cos(\omega\,t+\phi)}{F\,\cos(\omega\,t)}$$

if $\omega >> \omega_0\quad\Rightarrow \phi~<<~$ the oscillator is in phase with the driver .

if $\omega << \omega_0\quad\Rightarrow \phi ~>>~$ the oscillator is out of phase with the driver .

if $~\omega=\omega_0\quad\Rightarrow \phi=\pi/2~$

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Resonance and phase difference between driver and oscillator are different concepts.

Lets start with the phase difference:

If you drive an oscillator really really slowly, it will follow you "instantaneously": when your driver is at maximum, so as the oscillator. When your driver is at minimum, so is the oscillator.

So far so good, but when you try to drive it really really really fast, it takes time to react to you: when you're pulling right it is still moving to the left.

The phase difference is related to how fast you're driving the oscillator. But you always need to ask yourself "fast and slow compared to what?". The answer is the "natural frequency" of the oscillator, the frequency it will oscillate with no driving force after perturbated.

This frequency is the resonant frequency of the system, and it is characterized by having a maximum "reaction" to a certain driving force. Meaning for the same driver amplitude, the oscillator will oscillate the most (biggest amplitude) when driven at its natural frequency, and that is the concept of resonance.

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Take a mug with a handle, and several elastic bands. Loop one band through the handle, and join a few more on to it, and put your middle finger through the final loop. The mug + bands acts as a resonant system and if you displace it (with your other hand) you'll see damped oscillations at the resonant frequency.

Hold your arm horizontal and move it up and down slowly in the best approximation to a sine wave you can manage. The mug will follow your hand: at low frequencies driver and oscillator are in phase, with a response factor of unity.

Now make your motion a bit faster. The mug will still follow you but the oscillations get bigger. That $(\omega - \omega_0)^2$ in the denominator starts to get busy.

Keep increasing the speed gently, and you'll hit resonance and the mug will jump around all over the place. You may have to pause the experiment at this point.

Move your hand rapidly, still trying to imitate a sine wave, and you're above the resonant frequency and the response gets smaller. The faster you can wobble your hand, the smaller the response. And the mug is now out of phase with the motion of the hand. You can see this and, what's more, you feel it: at high frequencies your hand is moving one way and the mug is moving the other.

Do try this at home! There's probably a video on YouTube, but nothing teaches you about forced oscillation phase and amplitude response like actually feeling it.

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