Resonance

0:00- [Instructor] Check out this demo. There's a speaker placed right next to a wine glass. As the speaker starts producing sound, the wine glass begins to vibrate, and eventually it shatters! This phenomenon is called resonance. But how can sound shatter a wine glass?

0:16What's going on over here? Let's find out. Since we're dealing with oscillations, let's start with a simple pendulum. If there are absolutely no resistive forces, like no air resistance or friction or anything, and if we pull it back and let it go, it'll keep oscillating forever, right?

0:34The amplitude will stay exactly the same. But that's not what happens in real life, because in real life, we have resistive forces. Here's my makeshift pendulum. Watch what happens when I pull it back and let it go. As time passes, the oscillations become smaller and smaller, because the resistive forces constantly sap mechanical energy from the pendulum.

0:54And if I fast-forward this, you'll see eventually the oscillations die out completely and it comes to a stop. Now, this ideal case where there are no resistive forces is called an undamped oscillation. And in contrast, the real-life case, which we just saw, where resistive forces are present, we call it damped oscillations, because the amplitude is getting damped.

1:13The oscillations are getting damped. Now, for the undamped case, we already know what the position-time graph looks like. It's pretty much a sinusoid, especially when we're dealing with small angles. But what does the graph look like for the damped oscillations?

1:27It'll be a great idea to pause the video and think about this. Well, since the amplitude keeps decreasing, we would probably expect something like this. The oscillations gradually die out because the amplitude decreases. And this is certainly one possibility, because the graph of damped oscillators can be way more interesting, because the behavior over here depends on both the nature of the resistive forces, like whether we're dealing with air resistance or friction or some other kind, and also the strength of the resistive force.

2:01In fact, let's look at a simulation. Right now we have absolutely no damping, so this is the ideal case, undamped oscillations. The amplitude stays the same. It'll oscillate forever. But now let's introduce some damping, some air resistance, some small amount of air resistance.

2:17And exactly as we saw, the amplitude reduces, the oscillations keep dying out. This is pretty similar to our real pendulum, right? It oscillates for a while and then eventually it'll come to a stop. If increase the strength of the air resistance, notice the oscillations die out quicker.

2:38But so far, it still oscillates for a while before coming to a stop, right? But now watch what happens if I make the strength very high. Look! Now it no longer oscillates. What's going on? Well, now the damping, the resistive forces are so high that when I let go of this pendulum, it just comes to the equilibrium position and stops.

3:04So notice this time the graph is so different. And this is what I meant earlier, that the graph of damped oscillators can be very, very interesting and they can be quite complex. But we don't have to worry too much about this, because in most everyday cases where we're dealing with oscillations in the air, the damping is pretty much what we saw in our demo.

3:24So you will see a lot of oscillations going by before it comes to a stop. So let's just stick to this weak damping case. Okay, so we know resistive forces dramatically affect the amplitude. But what about the frequency? Do they affect the frequency as well? I mean, we know in an ideal undamped case, a simple pendulum oscillates with an angular frequency given by this expression.

3:47Notice how the angular frequency is inversely related to the length of the pendulum. A shorter pendulum means the pendulum oscillates faster. A longer pendulum means it'll oscillate slower. We call this the natural frequency, because it's the frequency that a pendulum naturally oscillates with.

4:04Now, technically this is natural angular frequency because this is radians per second and it differs from ordinary frequency by a factor of two pi, but don't worry too much about that. We'll just simply call it natural frequency for the rest of the video. But this is assuming that there are no resistive forces, right?

4:20So now the question is, what if we do consider resistive forces? Will it still oscillate with that same frequency? Or would it be different? It'll be a great idea to pause the video and think about this. All right. Well, we know that resistive forces oppose pendulum's motion, so we would expect the pendulum to take a longer time to complete one full oscillation.

4:44In other words, you would expect it to have a longer period. A longer period means a lower frequency. So technically speaking, resistive forces reduce both the amplitude and the frequency. But again, how much does it affect the frequency? Let's go back to our simulation.

5:03So we have the undamped case again, no resistive forces. Now let's introduce resistive forces and increase its strength and let's see what happens. Here we go, here we go. We see the amplitude dramatically reduced, but look at the spacing between successive peaks.

5:17It's still pretty much the same. I mean, yeah, the period is slightly longer, but it would be hard to tell, right? Even with very high dumping, you can hardly tell that the spacing between the successive peaks have increased. Long story short, this means although the frequency does technically reduce, it's pretty close to what it was in the ideal case.

5:44So for most everyday situations, we can pretty much ignore this and we can say that the frequency under damped case is also pretty much the same as under the undamped case. So long story short, whether the oscillator is damped or not, the frequency, it's natural frequency, the frequency with which it naturally vibrates or oscillates when you disturb it and let it go, that is approximately the same, and it's given by this expression for a simple pendulum.

6:09And the same idea applies to any other oscillators, okay? Anyways, now comes a really interesting question, okay? We know that damping removes energy from the system, causing the oscillations to die out. So the question is, how can we stop that from happening?

6:23How can we make this pendulum, for example, keep oscillating for a long time? How do we do that? Well, if damping is constantly taking energy away, then the obvious thing we need to do is keep adding energy back into the pendulum system. I mean, over here, what that means is that I can just simply start moving my hand back and forth.

6:45That external force is what supplies energy to the pendulum. So I need to supply an external force, and that external force should also be periodic, moving back and forth. Only then it'll ensure that the pendulum keeps swinging. And for simplicity, we'll assume that this driving force varies sinusoidally, meaning it smoothly goes back and forth like a sine wave with a particular frequency.

7:10So notice, now we have a situation in which we are driving the pendulum's motion. Earlier, we just disturbed the pendulum and let it do its own thing, but now we are driving it by exerting an external force. Such oscillations are called driven oscillations, for obvious reasons, because it's driven by an external force.

7:31And what will be the frequency of this driven oscillations? Well, initially, as I start driving it, there will be chaos. The pendulum will try to oscillate with its own frequency, the natural frequency, whereas I'm trying to drive it with my own frequency, which we'll call the driving frequency, and so it'll be a mess.

7:49But once the dust settles, the pendulum eventually oscillates with the frequency with which it's being driven, with the frequency decided by the driver. But what will be interesting to look at is the amplitude of that oscillation. So let's, again, do a demo. We will do three demos.

8:10In the first, we'll drive this pendulum at a frequency much lower than its natural frequency, then in the second one, we'll drive it at a frequency much higher than its natural frequency, and in the third case, we'll try to drive it exactly at the natural frequency.

8:25And let's just see, let's just concentrate on the amplitude, okay? Here it goes. So let's first drive it with the frequency less, much less than the natural frequency. And you notice the pendulum now is oscillating with the frequency with which I am driving it, right?

8:44However, look at the amplitude. It's pretty low, right? I mean, it's pretty low compared to what we will see later. So this is the first case. We see very small amplitude. Now let's try to drive it at a frequency much higher than its natural frequency. You can see my hand is shaking much quicker.

9:02It's a much higher frequency. And what you notice is that the pendulum is still oscillating with that frequency, but you'll hardly call it as oscillations because the amplitude is very tiny again. All right, now the third case. Let's drive it at its natural frequency.

9:20Ready? Here goes. Here goes! Oh, wow! Look at that now. Oh, now it's definitely oscillating! And look at the amplitude. It's pretty high! So if we look at the three, all three together, we notice that when the driven frequency is much lower or much higher than the natural frequency, well, the pendulum doesn't oscillate with a very high amplitude.

9:46The amplitude is pretty low. But when you drive it at exactly its natural frequency, the amplitude is very high. The whole oscillation goes haywire. This phenomena is called resonance. It shows us that the most effective way to transfer energy is by matching the driving frequency to its natural frequency.

10:08So why does this happen? Well, think about it. Every time I disturb the pendulum, in all the three cases, whenever I'm disturbing the pendulum, the pendulum tends to oscillate with its own natural frequency. But when my disturbance, my external force, its frequency is much, much different from the natural frequency, there will always be times where the pendulum tends to move in one direction and my force would be in the opposite direction, and that sort of tends to cancel the motion.

10:37It opposes the motion, that slows it down, and that's why you see it doesn't, you know, gain a lot of speed and therefore it doesn't oscillate much. Over here, our frequencies are in sync. So when the pendulum is moving to the right, my force is also towards the right.

10:54When the pendulum is moving towards the left, my force or the external force is also towards the left, and therefore it nicely amplifies the motion and that's why we get a huge amplitude. That's why we get a resonance. And now we can go back to our speaker and the wine glass example.

11:12Before the speaker is turned on, we can imagine the air to be completely undisturbed and we can model the air particles to be at rest. Of course, the neutral molecules are moving randomly because of the thermal motion, but there is no overall motion, so we can just model them to be at rest.

11:27Okay, now let's switch the speaker on. The speaker cone starts moving back and forth, and that motion makes the air molecules, it pushes and pulls on the air molecules. And the air molecules over here in turn pushes and pulls on their neighboring air molecules and so on and so forth, and so look, the disturbance propagates outwards.

11:47We call this a sound wave. What's important now is that if you look at any air molecule going back and forth, its frequency of oscillation is pretty much the same as the frequency at which the speaker cone is oscillating at. If the speaker cone oscillates at a lower frequency, look, all the air molecules now oscillate at a lower frequency.

12:06If it oscillates with a higher frequency, then all the air molecules over here also oscillate with a higher frequency. Now, of course, in real speakers producing music, the motion is usually far more complicated, but we don't have to worry about that. Let's just place a wine glass now.

12:23What's going to happen? Well, again, because of reflection and all the sound waves are gonna get much more complicated, but again, let's not worry about those things. What's important is that if you look closely over here, the air molecules are now going to hit the walls of the wine glass.

12:43They're gonna push and pull on the walls of the wine glass, and as a result, the wine glass will also start oscillating. So notice, the air molecules over here act like the driver. They are driving, they're forcing the wine glass to oscillate. And since that frequency is set by the speaker, the speaker now becomes the driving frequency.

13:08The frequency of the speaker becomes the driving frequency of the oscillation. And just like with the case of pendulum, a wine glass has its own natural frequency that it likes to oscillate with, and if you want to hear that frequency, you just disturb it. You just hit the wine glass with a spoon, for example, and you can hear that frequency.

13:26That is the natural frequency with which it tends to oscillate. That's our omega 0. Again, a small detail over here, a wine glass is not the same thing as a simple pendulum, so it's far more complicated. So turns out that there are more than one single natural frequency, but again, that's not important.

13:41What's important is that there is a natural frequency with which the wineglass tends to oscillate. And just like before, when the driving frequency is much higher than the natural frequency, we will see that the energy transfer is not very effective. And so although the wine glass does vibrate, the amplitude is very small.

14:04The same thing happens if the driving frequency is much, much smaller than the natural frequency of the wine glass. But what if you could somehow match the driving frequency to the natural frequency of the wine glass? Then we get resonance, just like before, and the amplitude will increase.

14:23And if we have enough energy, if the sound is loud enough, then the amplitude could be so high that the wine glass shatters. That's exactly what we saw earlier, resonance. And what's even cooler is that some skilled singers can do the same thing with their voice.

14:43They simply tap the wine glass to hear its natural frequency and match the pitch of their voice to it and then sing loud enough to shatter it. Resonance is pretty cool.