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.