Chapter 18: Oscillations

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You know, when you think about engineering a massive structure,

something like a huge suspension bridge or, I don't know, a commercial passenger jet,

you intuitively expect the physical challenge is to be, well, pretty monumental.

Right, like you're picturing.

It really is.

Yeah.

And when crash investigators scoured the wreckage, they weren't finding evidence of like a massive bird strike or some catastrophic engine explosion.

No, they found microscopic cracks.

Exactly.

Tiny, practically invisible metal fatigue located right at the high stress corners of their

almost square passenger windows.

And, you know, both of these completely different engineering disasters, a five million pound wobbly bridge and a tragic jet disintegration, they actually come down to the exact same underlying physical phenomenon.

Really?

Just the same basic physics?

Yep.

It wasn't the sheer size of the external forces that caused the failure.

It was, well, it was the rhythm, the fundamental rules of how things vibrate.

Okay, let's unpack this.

How do simple back and forth movements like things we take for granted every single day create such massive world altering engineering challenges?

It's a great question.

Welcome to the Deep Dive.

Today, we are grabbing the stack of notes you've shared with us on the physics of oscillations.

And hey, if you are prepping for your advanced physics studies right now, consider this your masterclass.

We're going to decode the physical principles, translate all that intimidating math into plain English,

and, you know, visualize the experiments you need to know.

So you can see exactly how these microscopic vibrations scale up to take down a bridge.

That's the mission.

And that is the perfect way to frame it.

Because to understand those massive structural failures, we first have to ground ourselves in the absolute fundamental rules of how objects vibrate.

We need to move away from just like looking at a swaying object and start understanding the precise predictable mathematical laws that govern its motion.

Right.

Before we can even calculate the forces tearing a bridge apart, we have to define what an oscillation actually is in a physical sense.

Broadly speaking, an object is oscillating when it moves back and forth repeatedly on either side of an equilibrium position.

Exactly.

The equilibrium position is key.

Yeah.

If you were to somehow strip away all its energy and just stop the object, it would rest perfectly at that exact equilibrium point.

And looking through chapter 8 tomb of your Cambridge textbook notes here, we categorize these movements into two distinct types, right?

Free oscillations and forced oscillations.

Right.

Let's look at free oscillations first.

If you displace an object, like you pull it out of its equilibrium, you just leave it to vibrate without interfering any further.

It's going to vibrate at what we call its natural frequency.

Okay.

So give me a visual for that.

Think of a simple pendulum, maybe a heavy mass hanging from like a two meter string attached to the ceiling.

You pull it to the side, you let it go and it swings.

Gravity and tension do the best.

That is a free oscillation.

So it's like plucking a guitar string or striking a tuning fork and just letting it ring out.

You give it that initial burst of energy and then it just does its own thing based on its physical properties.

Exactly.

But a forced oscillation is a totally different scenario.

That involves an object being continuously made to vibrate by some external driving force.

Okay.

Forced oscillation.

Like if you're sitting in the back of a bus that's idling at a stoplight and you suddenly feel your teeth rattling in your head.

Yes.

Perfect example.

Your jaw isn't vibrating at its natural frequency.

It's being forced to vibrate at the exact frequency of that idling bus engine.

It's super annoying, but it's a great physics example.

It really is.

And a fantastic way to visualize this in a laboratory setting is to use a signal generator hooked up to a loud speaker.

You can dial the generator to a very low frequency, let's say one Hertz.

And Hertz just means cycles per second.

Right.

So at one Hertz, you can physically watch the speaker cone slowly push out and pull back in exactly once every second.

That is a forced oscillation.

The speaker cone has no choice.

It is completely obeying the electrical signal that's driving it.

Okay.

So that's forced.

Now, if we want to observe a free oscillation up close, your notes detail this really elegant setup from the textbook.

You take a little wooden trolley, load it up with some extra weights, and you tailor it between two clamps on a desk using two identical horizontal springs.

Yep.

The tethered trolley.

Practical activity, 18 .1.

Right.

So you pull the trolley to one side, let it go, and watch it bounce back and forth.

And when you visualize that trolley moving, really pay attention to the changing velocity.

As it oscillates back and forth along the bench, where is it moving the fastest?

I mean, it's absolutely flying when it passes through the center, the equilibrium position.

Exactly.

And what happens at the extreme ends?

Well,

it decelerates, comes to a complete stop for a microscopic fraction of a second, reverses direction, and then accelerates back towards the center.

Right.

And what's fascinating here is that our eyes simply cannot process these back and forth oscillations if they happen faster than about five hertz, five cycles per second.

Oh, wow.

Really?

Just five.

Yeah.

Anything faster than that.

And our brain just interprets it as a blurry smear.

Which makes total sense.

And I guess it explains why physicists have to rely on these really slow -moving laboratory setups, like the weighted trolley or a really long pendulum.

Because if you wanted to study that loudspeaker cone we mentioned earlier, and it's vibrating at 100 hertz, you couldn't see a thing.

You'd just see a blur.

Right.

You'd have to bring in an electronic stroboscope, like a light flashing perfectly in sync with the speaker, at 100 flashes per second, just to create an optical illusion that freezes the motion so you can actually study the pattern.

Exactly.

Because once we can visually freeze or slow down the pattern, we need a reliable way to map it, to measure it mathematically.

Because our eyes can't track a trolley flying back and forth 10 times a second, we have to graph the motion.

This naturally leads us to the displacement time graph.

Okay, graphing.

So if you map that trolley's back and forth motion over time, you don't get a jagged mountain range of data points.

You get this incredibly elegant, perfectly smooth sine curve.

The motion is fundamentally signed to soil.

Right.

And to read that graph effectively, we have to lock down a few precise definitions for the listener here.

First is amplitude.

We represent this with an x, with a little subscript zero.

So x naught.

Amplitude is the maximum absolute displacement from the equilibrium position.

If you're looking at the sine wave, it's the absolute peak of the curve.

Got it.

And the period represented by a capital T is the exact amount of time it takes for one complete oscillation.

And complete is the crucial word there, right?

Absolutely.

It means the trolley goes from the far left, swings all the way to the right, and returns all the way back to the starting point on the left.

That's one period.

And then you have frequency, lowercase f, which is just a number of those complete oscillations that happen per unit of time, usually per second.

And the physical relationship there gives us a really simple equation.

Frequency equals one divided by the period.

F equals one over T.

Exactly.

But where it gets slightly more complex and incredibly important for understanding our structural failures later is the concept of phase.

Phase describes exactly where an oscillating mass is within its cycle at any given moment.

Like, are you at the peak?

Are you passing through the center?

Right.

And this is vital when you are comparing two different oscillating objects to see how they interact.

We look at their phase difference.

Okay.

Let's actually walk through a work example of this from the text, because I know this trips a lot of students up.

Imagine we are looking at a graph showing the motion of two identical pendulums.

They have the exact same period.

They take the same amount of time to swing, but they are out of step with each other.

Okay.

I'm visualizing it.

So the graph shows that the time gap between pendulum A reaching its peak and pendulum B reaching its peak is 17 milliseconds.

And we know the total period, the time for one full cycle is 60 milliseconds.

Right.

So we have a 17 millisecond delay in a 60 millisecond cycle.

To find the phase difference, you first express it as a simple fraction.

The delay is 17.

The whole cycle is 60.

So they are out of sync by 17 sixtieths of an oscillation.

Okay.

Simple enough.

But in physics, we don't just leave it as a fraction.

We need to express this in angles.

Well, thinking back to basic geometry,

one full cycle or one full circle is 360 degrees.

So if I just take that fraction, 17 divided by 60, and multiply it by 360 degrees, I get exactly 102 degrees.

That's their phase difference.

Right.

But your notes emphasize that we often need this in radians, not degrees.

Why is that?

That's a crucial point.

We use radians because they create a natural mathematical bridge between circular motion and this linear back and forth motion.

A full circle or a full oscillation cycle is exactly 2 pi radians.

Oh, okay.

It locks the geometry of a circle directly into the physics of the wave.

Exactly.

So to convert our phase difference, we take our fraction, 17 over 60, and multiply it by 2 pi.

That gives us about 1 .78 radians.

That single number tells a physicist exactly how out of sync those two objects are.

Okay.

Before we move on to the next section, I want to clarify something about these graphs.

We've been talking about mapping displacement over time and getting that smooth sine wave.

But what if I mapped the trolley's velocity over time instead?

Velocity instead of displacement.

Right.

Since it stops, turns around, and speeds up again, is that graph going to look like a jagged sawtooth pattern, like a bunch of sharp triangles?

No, it won't.

It will also be a perfectly smooth sine curve, just shifted slightly.

Really?

Still smooth.

And the physical reason is that the velocity changes gradually.

Remember, it decelerates smoothly as the spring stretches, comes to a momentary halt, and then smoothly accelerates back.

There are no sudden instantaneous jerks or infinite changes in speed.

Which brings us to the real heart of the matter for Chapter 18.

We see that these graphs, whether displacement or velocity, are these perfectly smooth, elegant sine waves.

But why?

What is the underlying physical mechanism forcing the trolley to move in this exact mathematically perfect shape?

And that leads us directly to a phenomenon called simple harmonic motion, or SHM.

Simple harmonic motion is truly everywhere in the universe.

We see it in the vibrating strings of a cello.

But it's also happening invisibly all around you.

When sound waves travel through the room right now, they are forcing the air molecules into simple harmonic motion.

Wow.

Alternating electrical currents are just electrons vibrating with SHM.

Even the atomic bonds inside a solid crystal lattice act like tiny springs vibrating with SHM.

But just because something wiggles back and forth doesn't mean it earns the title of simple harmonic motion, right?

Yeah.

The text is very clear that there are three incredibly strict physical requirements.

Yes.

Three rules.

Rule 1.

You need a mass that oscillates.

Rule 2.

There must be a fixed equilibrium position.

And rule 3.

And this is the absolute linchpin.

There must be a restoring force that is directly proportional to the displacement.

And that force must always be directed towards the equilibrium position.

Let's really dissect that third rule because it is the engine of SHM.

The restoring force is whatever physical push or pull is trying to get the mass back to the center.

Okay.

Directly proportional means there is a linear relationship.

If you pull the trolley one inch, the spring pulls back with a certain force.

If you pull it two inches twice as far, the spring pulls back exactly twice as hard.

That makes sense.

And directed towards equilibrium just means that if the mass is on the right, the force pulls left.

If the mass is on the left, the force pulls right.

It's always trying to correct the displacement.

Wait.

Okay.

Let me test this.

Okay.

I'm thinking about a kid jumping up and down on a backyard trampoline.

They are definitely oscillating back and forth past an equilibrium position.

Does that count as simple harmonic motion?

It's a great question, but no, it does not.

It doesn't.

Nope.

And the reason comes right back to that strict third requirement.

Think about the restoring force.

When the kid pushes down into the trampoline mat, the mat acts like a spring.

The deeper they go, the harder it pushes back.

Okay.

So far, so good.

But what happens when they bounce up into the air?

Oh, their feet leave the mat.

Exactly.

When they are airborne, the trampoline mat is no longer applying a force.

The only restoring force acting on them is gravity pulling them back down.

And near the earth's surface, gravity is a constant force.

It doesn't get stronger the higher they bounce.

Oh, I see.

Since the restoring force is no longer directly proportional to their displacement throughout the entire cycle, the strict rules of SHM are broken.

Precisely.

Gravity isn't a spring.

So if the force isn't always proportionally scaling with the displacement, it's not SHM.

That is such a clear distinction.

Now, you can actually prove these proportional forces in a lab setting, right?

Your notes mention using a motion sensor.

Yes.

It's a great setup.

You take that same trolley tethered between the springs, but you attach a piece of stiff cardboard to it.

Then you set up a motion sensor pointing at the card.

And what does the sensor do?

The sensor emits ultrasonic pulses, basically sonar that bounce off the moving card.

It feeds that positional data into a computer in real time.

And you can watch it plot that perfectly smooth sine wave on the screen, proving those proportional forces are at work.

And because we have proven that the restoring force is proportional to displacement,

we can do something very powerful.

We can translate that physical reality into mathematical equations.

Right.

Creating a mathematical engine.

Exactly.

An engine that allows us to predict the exact position, speed, and acceleration of an oscillating object at any given millisecond.

And to build that engine, we need to introduce angular frequency, represented by the Greek letter omega.

Omega.

We touched on this with phase earlier.

We did.

Since one complete cycle maps perfectly to a circle, which is 2 pi radians, angular frequency measures how many radians the object moves through per second.

So omega equals 2 pi times the frequency, or 2 pi divided by the period.

Got it.

Omega equals 2 pi f.

And that angular frequency, omega, is the cornerstone of the single most important equation in this topic.

The defining equation of simple harmonic motion.

Roll, roll, please.

The equation is a equals negative omega squared x.

Okay.

Let's translate that into physical reality.

A is acceleration.

X is displacement.

Right.

So this equation states mathematically what we just defined physically.

That an object's acceleration is directly proportional to how far it has been displaced.

The constant linking them is omega squared.

And we cannot ignore that minus sign.

That minus sign is the MVP of the equation.

The MVP.

I love that.

It's true.

It represents the directed towards equilibrium part.

It guarantees that if your displacement is positive, say you're pulled to the right, your acceleration will be negative, pulling you back to the left.

Precisely.

And if you were to graph this relationship, an acceleration displacement graph,

you would not see a curve.

No.

Because they're directly proportional, you get a perfectly straight line passing exactly through the origin.

But because of that minus sign, the line slopes downward.

The gradient or the slip of that straight line is exactly negative omega squared.

Perfect.

So that powerful little equation handles acceleration.

But what if we want to know exactly where the object is?

It's displacement at a specific time.

For that, we use trigonometric equations.

Either x equals x naught sine omega t or x equals x naught cosine omega t.

And picking between sine and cosine isn't random.

No.

It just depends on when you started your stopwatch.

If you started timing the exact moment the mass was flying through the center equilibrium point, you use the sine wave.

If you pulled the mass all the way out to its maximum amplitude, held it, and started your stopwatch the exact moment you let go, you use the cosine wave.

Exactly.

We also need to be able to determine the object's speed.

As we discussed, the maximum speed of the oscillator v naught happens precisely as it passes through the center.

And to calculate that maximum speed, you just multiply the angular frequency by the amplitude.

So v naught equals omega times x naught.

Right.

But if you need to know the velocity at some random point midway through the swing, the equation is a bit heavier.

It's v equals plus or minus omega times the square root of x naught squared minus x squared.

Okay.

If you're listening to this and your eyes just glazed over at that formula, let's look at what it's actually telling us physically.

It's essentially comparing two things.

Your absolute maximum possible swing, x naught.

And where you currently are right now, x.

That's a great way to look at it.

Right.

Because the closer your current position is to your maximum limit, the closer that term under the square root gets to zero, meaning your speed drops to zero.

It's just mathematically describing the trolley slowing down as it reaches the end of the spring.

Exactly.

It's just the math reflecting the physical reality.

I do have to throw a massive warning flag on the play here, though.

When you are actually plugging numbers into these displacement and velocity equations for your homework,

that quantity omega t inside the sine or cosine function is calculated in radians.

Oh, yes.

Please listen to this.

You absolutely must have your scientific calculator set to radian mode.

If it's in degree mode, your answers will be complete nonsense.

Whenever you see pi involved in an angular frequency calculation, that should be your mental alarm bell ringing.

Switch to radians.

Such an important tip.

And you know, if we connect all these equations to the bigger picture, this is exactly why physicists love simple harmonic motion.

The concept has incredible universality.

What do you mean by universality?

Well, once a structural engineer or a physicist solves this math for a simple wooden trolley bouncing on a spring, they don't have to invent entirely new fields of mathematics to solve other problems.

They take this exact same mathematical engine and apply it to alternating electrical currents or seismic waves hitting a building.

Oh, I see.

Yeah.

The math of SHM is basically a master key for decoding the universe.

I absolutely love that perspective.

But you know, as perfect and elegant as these mathematical models are, we live in the real world and the real world is messy.

It is very messy.

In reality, pendulums eventually stop swinging and bridges.

Well, sometimes they wobble violently out of control.

To understand why, we have to look at how energy moves through these systems.

Let's start with an ideal, theoretically perfect system.

In a vacuum with perfect springs, energy is perfectly preserved.

It just swaps back and forth between two distinct forms, kinetic energy and elastic potential energy.

Okay.

Think about the extremes of the swing.

When the trolley is pulled to the very end and stops for that microsecond, its kinetic energy is absolutely zero.

But the spring is fully stretched, meaning the elastic potential energy is at its absolute maximum.

Then it snaps back.

As it flies through the center equilibrium point, the spring is totally relaxed, the potential energy drops to zero.

But the trolley is moving at its maximum speeds, so kinetic energy peaks.

It just trades this energy back and forth 100 % efficiently.

The total energy of the entire system remains constant.

And mathematically, that total energy depends on the mass, the square of the angular frequency, and the square of the amplitude.

E equals 1 half m omega squared x naught squared.

Right.

Think about it.

A heavier mass oscillating faster with a wider swing contains massively more energy.

But in the real world, we have air resistance.

We have internal friction in the springs.

These non -conservative forces constantly steal kinetic energy away from the system, usually dissipating it as heat.

And we call this phenomenon damping.

Your notes highlight a brilliant experiment from the book to visualize damping.

You take a springy hacksaw blade, clamp one end firmly to a workbench, and attach a mass to the free end so you can twang it back and forth.

Yep, the hacksaw oscillator.

Then you attach a flat piece of cardboard to the mass.

That card acts like a sail, dramatically increasing the air resistance.

If you plot the amplitude of that blade over time, the peaks don't just shrink in a straight linear downward slope.

They form a very specific curve.

That curve is called an exponential decay envelope.

And because the amplitude decays exponentially, it behaves mathematically a lot like radioactive decay.

Wait, really?

Like radiation?

Yeah.

You can actually measure the half -life of the amplitude, the exact amount of time it takes for the swing to shrink to half of its previous size.

That is so cool.

But here is the critical detail, and it might sound totally counterintuitive at first.

Even as that amplitude shrinks down to almost nothing, the frequency of the oscillation does not change.

Not at all.

Think about a child on a playground swing.

It might take them two seconds to complete a massive high swing.

As friction slows them down, it still takes exactly two seconds to do a tiny prophetic little swing at the very end.

The amplitude dies, but the frequency survives.

That is a vital point for exams.

The natural frequency remains constant, unless, of course, you start feeding external energy back into the system.

Which brings us to the phenomenon of resonance.

Resonance.

The bridge destroyer.

Exactly.

We've established that every object has a natural frequency.

If you apply a forced oscillation to that object, and the forcing frequency exactly matches the object's natural frequency, the object absorbs that energy incredibly rapidly.

The amplitude builds and builds to massive, sometimes destructive levels.

The classic physics demonstration for this is Barton's Pendulums.

Oh, I love this one.

It's so good.

Picture a horizontal string pulled tight across a room.

Hanging down from this horizontal string are several light paper cone pendulums, all of varying lengths.

Because the length of a pendulum determines its period, they all have different natural frequencies.

Right.

You hang one very heavy driver pendulum from that same horizontal string.

You pull the heavy driver back and let it swing.

As that heavy driver pendulum swings back and forth, it sends a forcing frequency traveling right through the horizontal string, disturbing all the other little paper pendulums.

And most of the little pendulums just jiggle around chaotically, right?

They don't absorb much energy because their natural frequencies don't match the driver.

Exactly.

But one of those little paper pendulums happens to be cut to the exact same length as the heavy driver.

Which means it has the exact same natural frequency.

Bingo.

Only that specific pendulum is in sync.

It matches the forcing frequency perfectly, so with every single swing, it absorbs more kinetic energy.

Its amplitude builds and builds into a massive swing, while the others barely move.

That is the visual definition of resonance right there.

It is.

So bringing this all full circle, what does this mean for our Millennium Bridge?

How does a modern multi -million pound steel suspension bridge fail in two days?

The Millennium Bridge was essentially a gigantic catastrophic version of Barton's pendulums.

The physical structure of the bridge itself was the pendulum.

It had its own natural frequency of lateral sway.

Okay, so what was the heavy driver pendulum?

The thousands of pedestrians walking across it, they acted as the driving frequency.

And the average person walks at a pace of roughly two steps per second.

Right.

When the frequency of the pedestrian's footsteps happened to align with the natural sway frequency of the bridge, resonance began.

The bridge started to sway just a tiny bit laterally.

Okay, a tiny sway doesn't seem so bad.

But here's where human biomechanics made a bad situation worse.

When the ground beneath you sways, you automatically unconsciously widen your stance and adjust your stride to keep your balance.

Oh wow.

They synchronized.

Yes.

Without even consciously communicating, hundreds of pedestrians locked their walking rhythm into the exact same swaying motion, perfectly matching the bridge's natural frequency.

So they became a massive, unified driving force, pumping massive amounts of kinetic energy into the structure with every step.

The amplitude of the sway increased exponentially until it became dangerous.

It was a perfect storm of forced oscillation and resonance.

And the exact same physical principles apply to the de Havilland Comet Jet we talked about.

The metal fatigue.

Right.

The immense vibration of the jet engines and the aerodynamic turbulence provided a constant forcing frequency.

The rigid square window frames had their own natural frequencies.

And windows matched.

Microscopic resonant vibrations concentrated at the sharp corners of the windows, tearing the metal apart at the atomic level over time.

It is genuinely humbling.

A simple concept, things wiggling back and forth, and a straightforward mathematical equation, a equals negative omega squared x, ultimately dictate the life and death of massive suspension bridges and jet airliners.

It really is profound.

Before we wrap up, I want to leave you with one final thought to mull over.

We focused entirely on mechanical systems today.

Springs, bridges, pendulums, airplanes.

But your physics notes briefly mention medical MRI machines, magnetic resonance imaging.

Wait, I see where this is going.

Magnetic resonance.

Exactly.

Your body is mostly water, which means it is full of hydrogen atomic nuclei protons.

And because of their quantum properties, those protons have their own natural frequency.

No way.

Oh, yes.

What do you think happens when a medical machine applies a targeted radio frequency that perfectly matches the natural frequency of the protons inside your own brain?

It's resonance.

The exact same fundamental physics of resonance that caused a steel bridge to wobble is what allows doctors to peer inside the human body without making a single incision.

It's all connected.

That is absolutely incredible.

The mechanics of disaster are the exact same mechanics of life -saving medicine.

Well, thank you for joining us for this deep dive into the physics of oscillations.

On behalf of the Last Minute Lecture Team, we wish you the absolute best of luck with your physics studies.

You've got the tools to master this now.

And remember, keep those calculators in radian mode.

See you next time.