Simple Harmonic Motion: Crash Course Physics #16

Summary

In June 2001, the Millennium Bridge in London had to be closed almost immediately after opening due to severe swaying caused by pedestrians' footsteps. This swaying was a result of oscillations and resonance, a phenomenon where force applied at the right frequency increases amplitude. Engineers initially overlooked horizontal swaying, leading to dangerous resonance. By analyzing simple harmonic motion and comparing it to uniform circular motion, we understand the bridge's issues. Ultimately, engineers implemented solutions to counteract the oscillations, stabilizing the bridge. This video explains these concepts in detail, using the Millennium Bridge as a real-life example.

Takeaways

• 🌉 The Millennium Bridge in London was closed soon after opening due to swaying caused by the force of pedestrians' footsteps, illustrating the physics of oscillations.
• 🔁 Simple harmonic motion is a type of oscillation that follows a consistent pattern, often described using the example of a ball attached to a spring.
• 🏋️‍♂️ Kinetic and potential energy are key in understanding oscillations; kinetic energy is maximum at the equilibrium point, while potential energy is highest at the amplitude.
• 📈 The maximum velocity of an oscillating object can be calculated using the formula involving amplitude, spring constant, and mass.
• ⏱ The period, frequency, and angular velocity of simple harmonic motion can be understood by drawing parallels with uniform circular motion.
• 📐 The position of an object in simple harmonic motion over time can be described using trigonometry, specifically the cosine function.
• 🔄 The graph of an object's position versus time in simple harmonic motion resembles a wave, which helps explain the wave-like motion of the Millennium Bridge.
• 🤸‍♀️ Resonance, the phenomenon where an oscillation's amplitude is increased by applying force at the right frequency, played a significant role in the bridge's swaying.
• 👷‍♂️ Engineers had to redesign the Millennium Bridge to counteract the oscillations, focusing on the horizontal swaying that was initially overlooked.
• 🔍 The script emphasizes the importance of considering all aspects of oscillation, including both vertical and horizontal movements, in engineering designs.
• 🎓 The episode provides a comprehensive lesson on simple harmonic motion, connecting it with concepts from uniform circular motion and demonstrating its real-world applications.

Q & A

What was the main issue with the Millennium Bridge when it first opened?

The main issue with the Millennium Bridge was that it swayed back and forth dramatically due to the force of pedestrians' footsteps, leading to severe oscillations.

How did pedestrians' actions worsen the swaying of the Millennium Bridge?

Pedestrians leaned into the swaying to keep from falling over, which created resonance and amplified the oscillations.

What is simple harmonic motion?

Simple harmonic motion is a type of oscillation where the motion follows a consistent, repeating pattern, such as a ball attached to a spring moving back and forth.

What happens to the energy of a ball in simple harmonic motion as it moves?

As the ball moves, its kinetic energy increases towards the middle of its motion while its potential energy decreases, keeping the total energy constant.

How is the maximum velocity of a ball in simple harmonic motion determined?

The maximum velocity is determined by the equation: maximum velocity = amplitude * sqrt(spring constant / mass).

What similarities exist between simple harmonic motion and uniform circular motion?

Mathematically, simple harmonic motion is similar to uniform circular motion. For example, if a marble moves in a circular path, its horizontal motion can be seen as analogous to the back-and-forth motion of a ball on a spring.

How do you calculate the period of a ball in simple harmonic motion?

The period is calculated as: period = 2 * pi * sqrt(mass / spring constant).

What is resonance and how did it affect the Millennium Bridge?

Resonance is the increase in amplitude of an oscillation by applying force at the right frequency. On the Millennium Bridge, pedestrians created resonance by leaning into the swaying, worsening the oscillations.

Why did the engineers not foresee the swaying issue of the Millennium Bridge?

The engineers did not foresee the swaying issue because they only considered vertical oscillations, not the horizontal swaying caused by pedestrians walking.

What measures were taken to fix the Millennium Bridge?

Engineers applied a series of changes to the bridge to counteract its oscillations and prevent it from swaying dramatically.

Outlines

00:00

🌉 The Swaying Millennium Bridge: An Engineering Challenge

In June 2001, London officials unveiled the Millennium Bridge, a pedestrian bridge over the River Thames. Initially celebrated for its utility and design, the bridge had to be closed almost immediately due to severe swaying caused by pedestrian footsteps. As people walked, they leaned into the sway, exacerbating the movement until the bridge took on an S-shape. Engineers spent nearly two years fixing the problem. The issue was rooted in oscillations, specifically simple harmonic motion, which follows a consistent back-and-forth pattern. Physicists often describe this motion using the analogy of a ball on a horizontal spring. When the ball is moved and released, it oscillates, showing both kinetic and potential energy at different points in its cycle.

05:05

🔄 Energy and Motion in Simple Harmonic Systems

To understand the ball’s motion, consider its energy: kinetic energy at the middle of its path and potential energy at the turning points. The distance from these points to the equilibrium is the amplitude. At the turning points, the energy is all potential; at the middle, it’s all kinetic. The ball’s total energy equals half the mass times the maximum velocity squared. Combining these energy equations helps answer the ball’s maximum velocity, which is the amplitude times the square root of the spring constant divided by mass. This analysis shows the ball’s energy dynamics and provides an equation for maximum velocity, revealing a deeper understanding of the ball's oscillation properties.

⚙️ Simple Harmonic Motion and Uniform Circular Motion: A Comparison

Simple harmonic motion shares properties with uniform circular motion, such as period, frequency, and angular velocity. By comparing a ball on a spring with a marble on a circular path, you can see that from the side, the marble's motion appears as a back-and-forth line, similar to the ball’s oscillation. Assuming equal amplitudes and maximum speeds, the equations for velocity are identical. The period (time for one full cycle) is the circumference divided by speed, simplified to 2π times the square root of mass over the spring constant. Frequency (revolutions per second) is 1 over the period, and angular velocity (radians per second) is frequency times 2π. This comparison elucidates the shared characteristics of these motions.

📐 Trigonometry in Simple Harmonic Motion

To find the ball’s position over time, analyze the marble’s path along the ring. The horizontal distance from the center, seen edge-on, matches the ball’s distance from equilibrium. The cosine of the angle equals this distance over amplitude. With angular velocity, the ball’s position equation becomes x = A cos ωt, graphing as a wave. This wave-like motion mirrors the bridge's sway, caused by resonance, which increases oscillation amplitude when force is applied at the right frequency. Pedestrians leaning into the sway created resonance, amplifying the oscillation. Engineers initially accounted for vertical but not horizontal oscillations, leading to severe sway. They later added counteracting forces to stabilize the bridge.

🧪 Conclusion and Further Learning

This video explained simple harmonic motion, its energy dynamics, and how uniform circular motion concepts help find period, frequency, and angular velocity for a mass on a spring. Additionally, it showed the position-time relationship as a wave, connected to the Millennium Bridge's oscillation issue. Engineers overlooked horizontal swaying in their design, leading to resonance problems when pedestrians walked. Future episodes will delve deeper into waves, building on this foundational understanding. The episode credits PBS Digital Studios and mentions other educational shows and the production team.

Keywords

💡Millennium Bridge

The Millennium Bridge is a pedestrian bridge spanning the River Thames in London. It is central to the video's narrative as it serves as a real-life example of oscillations and resonance, which caused it to sway dramatically due to the force of pedestrians' footsteps.

💡Oscillations

Oscillations refer to the repeated back-and-forth movement around an equilibrium point. The Millennium Bridge swayed due to oscillations caused by pedestrians, highlighting the importance of understanding these movements in engineering and physics.

💡Simple Harmonic Motion

Simple harmonic motion is a type of oscillation where the movement follows a consistent pattern, like a ball on a spring. The video uses this concept to explain the predictable nature of the oscillations that affected the Millennium Bridge.

💡Kinetic Energy

Kinetic energy is the energy of motion. In the context of the ball on a spring, kinetic energy is highest at the equilibrium point. This concept is used to explain the energy changes in simple harmonic motion.

💡Potential Energy

Potential energy is the stored energy due to position. In simple harmonic motion, potential energy is highest at the turning points of the oscillation. The video uses this to explain the energy transformations in the ball-spring system.

💡Amplitude

Amplitude is the maximum displacement from the equilibrium point in an oscillating system. It is used to describe the extent of the ball's movement on the spring and is analogous to the bridge's swaying distance.

💡Resonance

Resonance is the phenomenon where an oscillating system experiences increased amplitude due to matching external forces. The Millennium Bridge experienced resonance when pedestrians' footsteps synchronized with its natural frequency, exacerbating the swaying.

💡Period

The period is the time it takes for one complete cycle of oscillation. The video explains that both the ball on the spring and the pedestrians on the bridge have a period that can be calculated using uniform circular motion principles.

💡Frequency

Frequency is the number of oscillations per unit time. The video describes how the frequency of the ball on the spring and the pedestrians on the bridge can be determined, showing its importance in predicting and understanding oscillatory behavior.

💡Angular Velocity

Angular velocity is the rate of change of the angle in uniform circular motion. The video relates this to simple harmonic motion, explaining that the angular velocity of the ball on the spring helps determine its oscillatory characteristics.

Highlights

In June 2001, officials in London unveiled a striking new feat of engineering: the Millennium Bridge, a pedestrian bridge spanning the River Thames.

The Millennium Bridge had to be closed almost immediately because it swayed back and forth dramatically due to the force of pedestrians' footsteps.

As people walked on the bridge, they leaned into the swaying to keep themselves from falling over, which only made the swaying worse.

The swaying motion of the bridge became so severe that it took on the shape of a giant 'S,' essentially a horizontal wave.

The bridge had to be closed, and engineers took nearly two years to fix the problem.

The physics that caused the swaying of the Millennium Bridge involves oscillations, specifically simple harmonic motion.

Simple harmonic motion is described as oscillations following a consistent pattern, often explained using a ball attached to a horizontal spring.

In simple harmonic motion, the ball on the spring has kinetic energy when moving and potential energy when at the turning points.

The maximum velocity of the oscillating ball is equal to the amplitude times the square root of the spring constant divided by its mass.

Simple harmonic motion shares mathematical similarities with uniform circular motion.

The period of simple harmonic motion is equal to 2π times the square root of mass over the spring constant.

The frequency of simple harmonic motion is 1 over 2π times the square root of the spring constant over mass.

Angular velocity in simple harmonic motion is equal to the square root of the spring constant over mass.

The position of an object in simple harmonic motion changes over time and can be described using trigonometric functions.

The swaying of the Millennium Bridge was exacerbated by resonance, where pedestrians leaning into the swaying created resonance, amplifying the oscillation.

Engineers had to implement changes to counteract the oscillations, ensuring the bridge no longer exhibited such dramatic swaying.

Transcripts

In June 2001, officials in London unveiled a striking new feat of engineering: the Millennium Bridge

-- a pedestrian bridge spanning the River Thames.

It promised to be very useful, and it was cool to look at, but it had to be close almost immediately.

Because when people used the bridge, it swayed back and forth dramatically, due to the force of their footsteps.

Undeterred, people kept using the bridge, but as they walked they began leaning into the swaying to keep themselves from falling over.

And that only made things worse.

Eventually, the motion of the bridge became so severe, that the bridge took on the shape of a giant S.

Essentially, a horizontal wave.

The bridge had to be closed and the engineers took nearly two years to fix it the problem.

So, what was wrong with the Millennium Bridge?

And why didn’t the engineers foresee the problem?

The answer lies in oscillations.

[Theme Music]

The physics that caused the swaying of the Millennium Bridge has to do with oscillations, or back-and-forth motion.

More specifically, it has to do with simple harmonic motion:

where oscillations follow a particular, consistent pattern.

But before we had the Millennium Bridge as a real-life example,

physicists often described simple harmonic motion in terms of a ball attached to a horizontal spring, lying on a table.

While it’s lying there, at rest, it’s in equilibrium.

And when you move the ball so that it stretches the spring, then let go,

the ball keeps moving back and forth forever... in a frictionless world.

That back-and-forth motion caused by the force of the spring, is simple harmonic motion.

Now, we want to know two things about this oscillating ball:

What kinds of energy does it have?

And, what’s its maximum velocity?

To better understand what’s happening to the ball, let’s start with its energy.

As the ball compresses and stretches the spring, both ‘kinetic energy’ and ‘potential energy’ come into play.

Kinetic energy is the energy of motion, and as the ball moves, there are two points --

the turning points -- where it’s NOT moving:

One point is where the spring is compressed all the way, and the other is where it’s stretched all the way.

And the distance between either of these two points, and the equilibrium point, is called the ‘amplitude’.

At those two turning points, the ball won’t have any kinetic energy, since it isn't moving.

Instead, all of the ball’s energy will be potential energy from the spring:

(half of the spring constant), times the (amplitude squared).

Now, as the ball moves toward the middle, its kinetic energy starts to increase,

because it’s moving faster and faster.

And at the same time, its potential energy decreases, keeping its total energy the same.

And exactly in the middle of the ball’s motion -- at the equilibrium point -- its potential energy goes down to 0.

The ball is back where it started, so the spring isn’t pulling on it anymore.

Its kinetic energy, on the other hand, has reached its maximum.

Which means that at that point, the total energy of the ball will be equal to

(half of its mass), times its (maximum velocity squared).

Now we have two equations for the total energy in this oscillating spring, which we can combine into one equation.

And if we use algebra to move around its variables, we can start to answer the second question we had about the ball.

We wanted to know the ball’s maximum velocity, and this equation tells us, that it’s equal

to the (amplitude), times the (square root of the spring constant) (divided by its mass).

So we’ve answered our two questions about the ball on the spring!

We know about its energy, and we have an equation for its maximum velocity.

But there’s a lot more going on with this ball than just its energy and velocity.

It also has properties like a period, a frequency, and an angular velocity.

Plus, its position changes with time.

You might recognize those terms, because we’ve already talked about them in our episode on uniform circular motion.

And that’s no coincidence!

Simple harmonic motion is actually a lot like uniform circular motion, mathematically speaking.

You can see this for yourself, if you compare the ball’s motion on the spring to an object in uniform circular motion --

say, a marble moving along a ring at a constant speed.

OK, I admit: It might seem like kind of a weird comparison at first.

For one thing, the ball on the spring is moving in one dimension,

while a marble moving along a circular path is in two dimensions.

But what if you take that ring, and look at it from the side?

The marble keeps moving along its circular path.

But to you, it looks like it’s just moving back and forth along a straight line.

Not only that, but it looks like this marble is stopping momentarily as it changes direction,

and moving faster as it gets closer to the middle.

Which is exactly the same way the ball was moving on the spring.

Now, let’s take this comparison a step further.

Let’s assume that the radius of the ring is the same as the amplitude of the ball’s motion on the spring.

And the marble’s constant speed along the ring is equal to the maximum speed of the ball on the spring.

In that case, if you did the math, you’d find that the equation for the marble’s velocity

-- when you look at it edge-on -- is exactly the same as the equation that described

the velocity of the ball on the spring.

So, let’s recall what we know about uniform circular motion, to see what it can tell us about simple harmonic motion.

We know that the time it takes for the marble to move around the ring once is called the period.

We also know that the period will be equal to the circumference of the ring, divided by the marble’s speed.

And! The radius of the circle is the same as the ball’s amplitude on the spring.

So its circumference will be equal to two times pi times the amplitude.

This means that the period will be equal to 2 times pi times the amplitude,

divided by the marble’s speed -- which, again, is the same as the ball’s maximum speed as it moves on the spring.

And we can simplify that equation, since we know that the maximum speed of the ball is

equal to the (amplitude) times (the square root of the spring constant) divided by the (mass).

So: the period of the marble’s motion around the ring is equal to (two pi) times (the root of m) over (k).

Now, we’ve also talked about the frequency of uniform circular motion:

It’s the number of revolutions the marble makes around the ring every second, and it’s equal to 1, divided by the period.

In this case, the frequency will also be equal to 1 over (2 pi) times (the square root of k) over (m).

And that’ll apply to the ball on the spring, too.

Because the rules are the same!

Finally, there’s angular velocity to consider.

In uniform circular motion, we’ve described it as the number of radians per second that

the marble covers as it moves around the ring.

And angular velocity is just equal to the frequency times 2 pi.

Which means that in the case of the ball on the spring, it’s equal to the square root of k over m.

So now, with the help of our knowledge about circular motion, we can understand the period,

frequency, and angular velocity of the ball’s simple harmonic motion as it oscillates on the spring.

But there’s one more question: How does the ball’s position change over time?

To find out, we’ll have to analyze the marble’s motion along the ring again.

And the answer will involve some trigonometry.

But it’s not particularly complicated trig so, it'll be fine.

At any given point along the marble’s path, it’ll be at a certain angle to the right-hand side of the ring.

And the cosine of that angle will be equal to its horizontal distance from the center of the ring, divided by the ring’s radius.

We already know that the radius of the ring is the same as the amplitude of the ball’s motion along the spring.

And if you turn the ring so that it looks like a line again, you can see that the marble’s

horizontal distance from the center of the ring is the same as the ball’s distance from the equilibrium point.

So, the cosine of theta is equal to the (ball’s position) divided by its (amplitude).

In other words, the ball’s position is equal to (the amplitude), times (the cosine of the angle).

And we can simplify this equation, too.

In the same way that distance is equal to velocity multiplied by time,

the angle is equal to the angular velocity multiplied by time.

So, we can write the equation for the position of the ball as x = A cos w t.

And when you graph this equation, something interesting happens: It looks like a wave!

We’ll be talking a lot more about waves in our next three episodes.

But for now, it’s helpful just to see the connection here:

For an object in simple harmonic motion, the graph of its position versus time is a wave.

Which is why the swaying of the Millennium Bridge looked like a wave.

Speaking of the bridge: now we can better understand what happened to it.

The bridge’s shimmy was the result of oscillation, but it was made worse by another culprit: resonance.

Resonance can increase the amplitude of an oscillation by applying force at just the right frequency --

kind of like how you can get a kid to swing higher by pushing at just the right moment.

The engineers of the Millennium Bridge were reminded of that, the hard way.

When pedestrians on the bridge started to lean into its swaying, they created resonance.

They amplified the amplitude of the oscillation.

And the engineers of the bridge did account for oscillations caused by resonance when they designed it.

But they only considered vertical oscillations -- the kind that would have made the bridge bounce up and down.

They didn’t realize that they’d also have to factor in the horizontal swaying caused by people walking.

So, it was only a tiny bit of swaying at first, but it got a lot worse because people were leaning into their steps, causing resonance.

In the end, engineers had to apply a series of changes to the bridge that applied force to counteract its oscillations.

Because if there’s one thing you don’t want your bridge to be doing, it’s The Wave.

Today, you learned about simple harmonic motion -- the energy of that motion, and how we can use math

of uniform circular motion to find the period, frequency, and angular velocity of a mass on a spring.

We also described how the position of an object in simple harmonic motion changes over time.

Crash Course Physics is produced in association with PBS Digital Studios. You can head over

to their channel to check out amazing a playlist of the latest episodes from shows like First Person, PBS Game Show, and The Good Stuff.

This episode of Crash Course was filmed in the Doctor Cheryl C. Kinney Crash Course Studio

with the help of these amazing people and our equally amazing graphics team, is Thought Cafe.