Chapter 21: The Harmonic Oscillator

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Welcome back to the Deep Dive.

We had a request, a really good one actually, to tackle a cornerstone chapter from Feynman's lectures.

Ah, the harmonic oscillator, judge 21.

That's the look, this isn't just about, you know, fiddling with a spring and a weight.

Feynman makes it clear

this is maybe the most vital system to understand in all of physics.

He really does.

And the reason,

the core reason is the math, the equation we end up with for this super simple mechanical thing.

Yeah.

It's the exact same equation that governs AC circuits,

how molecules vibrate, sound waves, light.

It's astonishingly universal.

So the setup, the mass on a spring, it's almost like like a teaching model for a deeper mathematical truth.

Precisely.

Feynman calls it the simplest system governed by a linear differential equation with constant coefficients.

That sounds technical, but it's the key.

Okay, let's unpack this.

The goal here is getting our heads around this one equation because as you said, it unlocks so much else.

Exactly.

Master this and you'll see its reflection everywhere in physics and engineering.

So let's get into it.

Right.

Section 21 to 2, the setup.

It's simple enough to picture.

We've got a mass, let's call it a noller.

Sitting on a frictionless horizontal surface, maybe attached to a spring.

And it can oscillate back and forth around some middle point of the equilibrium position.

Let's say 6 to law.

Right.

And if you pull it away from $6, say you displace it by a distance dollar.

The spring pulls back.

It pulls back.

And Hooke's law tells us how it pulls back.

The force, $5, is proportional to displacement $6.

But in the opposite direction.

So $5 is x dollar.

Perfect.

$2 is the spring constant.

It tells you how stiff the spring is.

A bigger dollar means a stronger pull for the same $6.

Okay, that's the force.

Now physics connects force to motion using Newton's second law.

Always.

Okay.

Or since acceleration is the second derivative of position with respect to time.

Right.

Ax dt2 to x grid.

We can substitute that in.

So 8 times the second derivative of $6 equals the force, which is dd2.

And that, that's the equation.

That's the golden equation.

The equation of motion for the simple harmonic oscillator.

Everything flows from this.

And you mentioned it's linear.

What's the significance of that linear part?

The fact that it's 2x dm and not, say, 2x 0 to 3.

Oh, that's absolutely crucial.

Linearity means if you double the displacement $6, you double the force.

No funny business.

Okay.

This leads to predictable sinusoidal solutions.

If it were non -linear, say 6, 2 out of t or something, the math gets way harder.

And crucially, the period of oscillation would depend on how far you pulled it.

Ah, okay.

But here it doesn't.

We'll get to that.

Exactly.

Linearity also means superposition works.

You can add solutions together is why this form is so fundamental.

So let's rearrange that equation slightly.

Divide by dollar.

Fracti 2 xd.

And that combination of physical properties,

the stiffness divided by the mass, that gets a special symbol.

It does.

We define 22 xd meters.

So the equation becomes even simpler aesthetically.

Fracti 2 x me 2 of 2a 2.

Okay.

To make it not the natural angular frequency, what does that mean physically?

It's the system's own inherent frequency.

How fast it wants to oscillate if you just leave it alone, determined solely by its mass and the spring stiffness.

It's like the system's signature rhythm.

And it's an angular frequency.

So it's in radians per second to get the actual period, the time for one per cycle.

You use the relationship t dollars equal two mega dollars or substituting back t dollars equals two pet.

And here's where it gets really interesting.

That period t dollars depends only on other and others.

Right.

The physical property is the system.

It does not depend on how far you initially pull the mass out.

The amplitude doesn't affect the timing.

That's the defining characteristic of simple harmonic motion.

Pull it a little or pull it a lot, the time it takes to swing back and forth is exactly the same.

It's quite remarkable.

Yeah, it really is.

That's why pendulum clocks could work, roughly.

Small swing, big swing, same period.

Precisely.

Stemming directly from that five dollar instead of big linearity.

Okay, so we have the equation of motion.

What's the solution?

What function satisfies fractity obiv 1 and a sig se du 2?

Well, we need a function whose second derivative is proportional to the negative of itself.

Sines and cosines do that.

Exactly.

They're the functions tailor made for this.

The general solution can be written as f t f t plus delta.

Okay, let's break that down.

Eight is e b e is the amplitude, the maximum value of sexy tall, how far the mass swings out from the center and delta the delta.

That's the phase angle or phase shift.

It basically sets the starting point of the oscillation at time two dollars.

Does it start at maximum displacement or passing through the center or somewhere in between?

So, and delta are constants, but they depend on how you start the motion.

Right.

We'll come back to how to determine them from initial conditions.

But first, Feynman introduces this brilliant visualization.

Ah yes, the circle analogy.

Section 21 to 3.

Why does he do this?

We have the solution, the cosine function.

Because pure math can feel abstract.

The circle makes it instantly intuitive.

It connects this oscillation to something we can physically picture.

Uniform circular motion.

How does that work?

Describe the analogy.

Imagine a point p moving steadily around a circle of radius r.

Let's say it moves with a constant angular velocity omega.

Okay, going round and round.

Now imagine a light shining down from above, casting the shadow of point p onto the horizontal diameter of the circle.

Right.

So the shadow just moves back and forth along that line.

Exactly.

And the position of that shadow along the diameter, I'll call it six, is given by r cos theta, where if it is the angle the point p makes with the horizontal axis.

And since the point moves at a constant angular velocity omega, the angle theta is just a pic of t.

So the shadow's position is six x r a t t t.

Hang on, that looks exactly like our solution.

Six x is infill each a t p t plus delta.

If we set the radius three dollars to be the amplitude of a t p and omega to be omega and maybe adjust the starting angle for delta.

Precisely.

The back and forth motion of the shadow is simple harmonic motion.

It's not just an analogy.

It's a geometric representation of the same mathematical form.

Wow.

So understanding the simple oscillator is like understanding the projection of smooth circular motion.

Yes.

And it makes understanding velocity and acceleration much easier too, visually.

The velocity of the point p on the circle is tangent to the circle.

Its projection gives the velocity of the shadow.

Which is maximum when the shadow passes through center and zero at the end.

Right.

And the acceleration of p in uniform circular motion, acceleration always points towards the center of the circle.

Okay.

Its magnitude is constant v two dollars or omega two dollars.

When you project that acceleration vector onto the diameter.

It points opposite to the displacement six dollars and it's maximum when six dollars is maximum at the ends of the shadow's path.

Exactly.

The projection gives an acceleration x daily x omega two by omega two.

Which is just omega two by.

It geometrically confirms the original differential equation.

Chickadee d tooth has x d two by.

That is genuinely elegant.

It turns calculus into geometry.

You have a classic find and move.

Finding a way to see the physics, not just calculate it.

The maximum acceleration is omega two tally two.

Which you see directly from the circle's properties.

Okay.

So we have the general form of the motion.

Now, section 21 to four.

Fixing those constants l and delta.

You said they depend on how it starts.

Right.

Initial conditions for a second order differential equation like ours.

You need two pieces of information at time.

Two dollars.

Typically the initial position, let's call it six dollars and the initial velocity v dollars.

Exactly.

Where is the mass and how fast is it moving right at the beginning?

Once you know six and V s dollars, the entire future motion is fixed.

A delta are determined.

How do we calculate them?

Freinman uses an alternative form of the solution.

Sometimes six x c omega t plus d sin omega t is that easier.

It can be for finding the constants at t sin one and sin is just su dollar.

Ah, so sig it is just dollar.

Simple.

And you need the velocity.

Differentiate six deal with respect to time.

V ten c omega t plus d omega t.

So omega t.

Okay.

Now evaluate that at two dollars.

So v omega t and omega two.

Meaning v dollars is z dollars less d omega t.

Meaning v dollars or the dollars of v omega t.

There you go.

If you know the initial position, six dollars and initial velocity v dollars, you instantly find the motion is solved.

You can relate the dollar back to the amplitude dollar stays delta too, right?

Through trigonometry.

Yes.

One always score t c two plus d two and j delta t two.

It's all consistent.

The point is two initial conditions fix the two constants of integration.

Makes sense.

Now let's talk energy.

If there's no friction, no air resistance.

An idealized system.

Yes,

then total mechanical energy should be conserved.

What forms does the energy take?

Two forms that trade back and forth.

Potential energy baller stored in the spring when it's stretched or compressed.

And the formula for that is one dollar freq one two k by two two.

Correct.

It depends on the square of the displacement.

Maximum potential energy happens when six dollars is largest at the amplitude points six dollars of p .m.

The other form is kinetic energy.

The energy of motion t where t around one he sees mat v two two depends on the square of the velocity.

Maximum kinetic energy happened when the mass is moving fastest, which is when it passes through the equilibrium point six dollars.

Exactly.

At six dollars, the potential energy is zero at six p .m.

A .D.

The velocity is momentarily zero.

So the kinetic energy is zero.

So the total energy e e t plus u frac one two v plus rack one two plus frac one two k by two two should be constant throughout the entire cycle.

It is.

As six dollars gets bigger, six gets smaller and vice versa in just the right way to keep A .D.

the same.

How can we find the value of A .D.?

The easiest way is to look at the moment the mass reaches its maximum displacement six degrees.

At that point, why dollars?

So kinetic energy t t dollar.

The total energy e d dollar is purely potential energy at that instant.

E d dollars frac one two k a two.

So the total energy is fixed entirely by the spring constant dollar and the square of the amplitude dollar.

That's it.

Pull it out further increases.

You give the system more total energy and that energy just sloshes back and forth between kinetic and potential form forever.

Ideally, Feynman points out something neat about the averages too.

Yes.

Yeah.

If you average the kinetic energy over one full period of oscillation and you average the potential energy over the same period, they turn out to be equal.

Exactly equal.

And since their sum is always the constant total energy upright, each average must be t two Eddie dolls.

So on average, the energy is split 50 50 between kinetic and potential.

Yep.

Another elegant consequence of the pure sinusoidal motion.

Okay.

All of this describes the three oscillator just doing its own thing.

But what happens if we like push on it?

Right.

Section 21 of five introducing external forces.

This moves us towards more realistic scenarios.

What if there's some other force acting on the mass besides the spring?

Call it flighty to like maybe we're periodically pushing and pulling it or maybe there's fiction though.

He treats damping later for now, just the general external force.

How does that change our fundamental equation?

Well, Newton's law includes all forces.

So we just add fi t two to the spring for exactly the net force is now kx plus ft two.

So the equation of motion becomes 42 xt two plus ft two.

That looks more complicated.

It is often it's rewritten to put all the sexto terms on one side.

42 two plus kx ft two.

This is the equation for a forced oscillator.

Conceptually, what's the big shift here?

Now, the motion isn't just determined by the system's internal properties, giving mega mags.

It's also influenced by the external force fine collar, which might have its own frequency.

So you have the system's natural frequency omega dollars competing or interacting with the driving frequency from five dollars.

Precisely.

And that is where things get really interesting.

This sets the stage for understanding resonance.

What happens when the driving frequency matches the natural frequency, which finding doesn't fully dive into in this chapter, but he lays the groundwork by showing how the equation changes structure.

Exactly.

He establishes the mathematical form for forced oscillations, which is the next big step.

Okay.

So what does this all mean?

We've gone from the basic setup to the governing equation, its solution, the brilliant circle analogy, energy conservation, and a peak at external forces.

I think the three crucial takeaways are one, the linearity of five dollars is why the system is so simple and why its math is universal.

Remember that got it.

Linearity is key to the solution is sinusoidal sex x equals a a omega t plus delta.

And the circle analogy makes this intuitively clear.

Motion is the projection of rotation, amplitude a dollar natural frequency, a mega dollars phase delta and three energy is conserved in the ideal case with the total energy is frac one two K a two determined solely by the amplitude energy swaps between kinetic and potential averaging out to 50 50 a very neat package.

Before we wrap up that independence of period from amplitude, let's just hit that one more time.

It's worth reflecting on.

Imagine if the period did depend on amplitude.

If five are born, wasn't linear pendulum clocks wouldn't work.

Grandfather clocks would run faster or slower depending on how far the pendulum swings.

Musical instruments based on vibrating strings or air columns wouldn't produce stable pitches.

Our ability to define and use specific frequencies relies fundamentally on this property of simple harmonic motion.

That the rhythm doesn't care about the size of the motion.

Exactly.

That mathematical quirk born from linearity is what gives us stable frequencies in the universe from mechanics to light itself.

It's profound derived from just watching a mass bobbing on a spring.

Amazing.

Well, thank you for walking us through this foundational chapter.

My pleasure.

It really is central to so much physics.

And thank you for joining us on the deep dive.

Hopefully you now feel equipped to spot the harmonic oscillator wherever it appears because it will appear everywhere.