In this video, we’re going to learn about forced oscillations. We’ll see what effect these oscillations have on physical systems, and we’ll also learn about the phenomenon of resonance.
To get started, imagine that you and a friend set out to set a record for the highest elevation while on a swing. To set this record at your neighborhood playground, it will be helpful to know something about forced oscillations. Next time you’re around a piano, try out this experiment. Open the piano up, and with your head close to the strings, sing a quick loud note. You’ll notice if you do that the piano sings back to you. The energy in the sound waves of your voice interacts with the strings and gives them energy to oscillate themselves. This is an example of forced oscillation.
One thing you might also notice if you try this out is that not all the strings in the piano sound back to your voice. This leads us into the notion of natural frequency. An object’s natural frequency is the frequency at which it tends to move or vibrate. When it comes to the strings in the piano, some of the strings are designed to oscillate at the frequency at which you’ve sung; that is, the frequency of the note you’ve created. It’s these strings in particular that are set in motion and that create sound which resonates back to you.
Other strings though, say the very high or the very low ones that don’t match the frequency of the note you sang, have a different natural frequency. And it’s not just strings in a musical instrument that have a natural frequency. Nearly every physical object has one. Glasses have a natural frequency. If you put one in front of a loudspeaker and play the right sound at the right amplitude out of the speaker, it’s possible to shatter the glass. Diving boards have a natural frequency at which they want to bounce up and down. Even objects as large as suspension bridges have natural frequencies at which they prefer to oscillate.
This fact has been demonstrated by the failures of several bridges whose natural frequency has been matched by winds passing over the bridge. Not only do objects have a frequency at which they prefer to oscillate, but it’s possible to apply a rhythmic externally applied push to objects known as a forced oscillation. An example of a forced oscillation is the example of pushing your friend when they’re swinging on a swing. This oscillation is externally applied because you’re giving it, and it’s rhythmic because you only apply this push when your friend is within arm’s length.
In general, when we think about an object’s natural frequency and a forced oscillation that might be applied to it, there is no particular reason why these two frequencies, and they are both frequencies, should match. When we consider the natural frequency of your friend on this swing system, it’s completely possible for you to apply your pushes either too frequently or too infrequently in such a way that does not match the natural frequency of the swing system. When this happens, when you’re pushing frequency doesn’t match the natural frequency of the swing, your friend won’t move very far.
We could say that your friend’s amplitude of oscillation will never get very big because the pushes that are coming to your friend don’t match up with their natural frequency. But what if your pushes do match up? What if the natural frequency of the swing system and the forced oscillation you apply are both lined up at the same exact frequency? When that happens, when there is a match between the forced oscillations applied and the natural frequency of the system, we have a phenomenon known as resonance. And in this case, the amplitude of your friend’s oscillations on the swing will be very high.
Resonance is a very common physical phenomena. And many systems are designed to achieve some sort of resonance. For example if you play guitar, you can use a tuning fork to match the frequency of a particular string on that guitar. This is an example of resonance. Or when you’re scanning through radio frequencies, you’re adjusting the frequencies of oscillation of electrons in the antenna of your radio system. When these oscillating electrons find a match — that is, when they resonate — then you’re tuned in to a new channel. Or if you’re using a laser light or laser pointer, many lasers have a cavity which require optical resonance in order for laser light to be emitted.
These examples of resonance are all about a forced oscillation being matched to a natural frequency. Since resonance occurs when a natural frequency is matched, if we know the natural frequency of a system, we know it’s a resonant frequency. For example, we’ve seen that in a case of a simple pendulum, the natural frequency of the system is the square root of 𝑔 over the length of the pendulum’s arm. The natural frequency of the pendulum is also its resonant frequency; that is, the frequency at which it’s motion resonates. As with other objects, this would mean that the amplitude of the pendulum’s oscillations grow bigger and bigger if it’s driven at its resonant frequency.
Or consider the oscillating system of a mass on a spring. The angular frequency of this system is equal to the square root of the spring constant over the mass. So this is the natural frequency of the system and the frequency at which, if the system is driven, its amplitude of oscillations will increase dramatically. There’s such a close correspondence between an object being driven at its natural frequency and its oscillation amplitude increasing that we can actually use oscillation amplitude as a clue for whether the object is resonating.
Thinking back to our glass that we can shatter using sound waves, that shattering of the glass means its oscillation amplitudes have grown so big that the glass can’t sustain them. It also means that our sound source is now driving the glass at its natural frequency. Let’s look at an example that involves forced oscillations. An unloaded diving board oscillates at a frequency of 8.06 hertz. The board has an effective mass of 14.2 kilograms. What is the frequency of the oscillations of the board if a diver of mass 86.5 kilograms jumps up and down on it?
We can call this oscillation frequency we’re solving for 𝜔 and begin by recalling the relationship between 𝜔 and a spring system. We can recall that the natural frequency of a mass oscillating on a spring is equal to the square root of the spring constant 𝑘 divided by the mass amount 𝑚. In our case, we’re told that the board, when there’s no diver on it, has a natural frequency of 8.06 hertz and that the mass of the board is effectively 14.2 kilograms. We can write that 𝜔 sub 𝑖, the natural frequency of the board by itself, is equal to the square root of the spring constant of the board divided by 𝑚 sub 𝑏; or, rearranging to solve for 𝑘, it’s equal to the mass of the board times the board’s natural frequency squared.
We want to solve for the frequency of the oscillations of the board when a diver of mass 86.5 kilograms is on it. This frequency, which we’ve called 𝜔, will be equal to the square root of 𝑘 over the mass of the system: mass of the board plus the mass of the diver. Substituting in 𝑚 sub 𝑏 times 𝜔 sub 𝑖 squared for 𝑘, we now have an expression we’re ready to plug in for to solve for 𝜔. When we enter these values on our calculator, we find that 𝜔, to three significant figures, is 3.03 hertz. That’s the oscillation frequency of the board plus the diver.
Let’s summarize what we’ve learned so far about forced oscillations. We’ve seen that practically all objects have a natural frequency at which they tend to move or vibrate or oscillate. We’ve also seen that a forced oscillation is an externally applied rhythmic push. This forced oscillation is a source of energy from outside a system that applies this energy at a certain frequency. And finally, we saw that significantly increased oscillation amplitude happens when an object’s natural frequency is matched by an externally applied rhythmic push, a forced oscillation. This phenomenon is known as resonance.