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So far, we analyzed...

We calculated the periods of lots of oscillators: pendulums, springs, rulers, hula hoops.

We gave them a kick, moved them off equilibrium, and then they were oscillating at their own preferred frequency.

Today, I want to discuss with you what happens if I force upon a system a frequency of my own.

So we call that forced oscillations.

I can take a spring system, as we have before.

This is x equals zero, this is x, and we have the spring force, very familiar, minus kx.

But now this object here, which is mass m, I'm going to add a force to it, F zero, which is the amplitude of the force, times the cosine of omega t.

So I'm going to force it in a sinusoidal fashion with a frequency that I choose.

This frequency is not the frequency with which the system wants to oscillate.

It is the one that I choose, and I can vary that.

And the question, now, is what will the object do? Well, we have Newton's Second Law —

ma equals minus kx plus that force, F zero cosine omega t.

a is x double dot, so I get x double dot, plus — I bring this in —

k over m times x equals F zero divided by m times the cosine omega t.

Now, the question is what is the solution to this differential equation? It's very different from what we saw before, because before, we had a zero here.

Now we have here a driving force.

It's clear that if you wait long enough that sooner or later that system will have to start oscillating at that frequency.

In the beginning, it may be a little different.

In the beginning, it may want to do its own thing, but ultimately, if I take you by your arms and I shake you back and forth, in the beginning you may object, but sooner or later, you will have to go with the frequency that I force myself upon you.

And when we reach that stage, we call that the steady state as opposed to the beginning, when things are a little bit confused, which we call the transient phase.

So in the steady state, the object somehow must have a frequency which is the same as the driver, and it has some amplitude A.

And I want to evaluate with you that amplitude A.

So this is my trial function that I'm going to put into this differential equation.

x dot equals minus A omega sine omega t.

x double dot equals minus A omega squared cosine omega t.

And so now I'm going to substitute that in here, so I'm going to get minus A omega squared cosine omega t plus k over m times A cosine omega t, and that equals F zero divided by m times the cosine of omega t.

And that must always hold.

So therefore I can divide out my cosine omega t.

I can bring the A's together, so I get A times k over m minus omega squared equals F zero divided by m.

Now, this k over m is something that we are familiar with.

If we let the system do its own thing —

we bring it away from equilibrium and we don't drive it —

then we know that omega squared, which I will give the zero, equals k over m.

This is the frequency that we have dealt with before.

This is the driving frequency —

it's very different.

And so I'm going to substitute in here for k over m omega zero squared, and so I find, then, that the amplitude of this object here at the end of the spring will be F zero divided by m divided by omega zero squared minus omega squared.

And this amplitude has very remarkable characteristics.

We calculated the periods of lots of oscillators: pendulums, springs, rulers, hula hoops.

We gave them a kick, moved them off equilibrium, and then they were oscillating at their own preferred frequency.

Today, I want to discuss with you what happens if I force upon a system a frequency of my own.

So we call that forced oscillations.

I can take a spring system, as we have before.

This is x equals zero, this is x, and we have the spring force, very familiar, minus kx.

But now this object here, which is mass m, I'm going to add a force to it, F zero, which is the amplitude of the force, times the cosine of omega t.

So I'm going to force it in a sinusoidal fashion with a frequency that I choose.

This frequency is not the frequency with which the system wants to oscillate.

It is the one that I choose, and I can vary that.

And the question, now, is what will the object do? Well, we have Newton's Second Law —

ma equals minus kx plus that force, F zero cosine omega t.

a is x double dot, so I get x double dot, plus — I bring this in —

k over m times x equals F zero divided by m times the cosine omega t.

Now, the question is what is the solution to this differential equation? It's very different from what we saw before, because before, we had a zero here.

Now we have here a driving force.

It's clear that if you wait long enough that sooner or later that system will have to start oscillating at that frequency.

In the beginning, it may be a little different.

In the beginning, it may want to do its own thing, but ultimately, if I take you by your arms and I shake you back and forth, in the beginning you may object, but sooner or later, you will have to go with the frequency that I force myself upon you.

And when we reach that stage, we call that the steady state as opposed to the beginning, when things are a little bit confused, which we call the transient phase.

So in the steady state, the object somehow must have a frequency which is the same as the driver, and it has some amplitude A.

And I want to evaluate with you that amplitude A.

So this is my trial function that I'm going to put into this differential equation.

x dot equals minus A omega sine omega t.

x double dot equals minus A omega squared cosine omega t.

And so now I'm going to substitute that in here, so I'm going to get minus A omega squared cosine omega t plus k over m times A cosine omega t, and that equals F zero divided by m times the cosine of omega t.

And that must always hold.

So therefore I can divide out my cosine omega t.

I can bring the A's together, so I get A times k over m minus omega squared equals F zero divided by m.

Now, this k over m is something that we are familiar with.

If we let the system do its own thing —

we bring it away from equilibrium and we don't drive it —

then we know that omega squared, which I will give the zero, equals k over m.

This is the frequency that we have dealt with before.

This is the driving frequency —

it's very different.

And so I'm going to substitute in here for k over m omega zero squared, and so I find, then, that the amplitude of this object here at the end of the spring will be F zero divided by m divided by omega zero squared minus omega squared.

And this amplitude has very remarkable characteristics.

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