Simple Harmonic Motion: Crash Course Physics #16
Summary
TLDRIn 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
- đ 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.
- đ 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.
- đïžââïž 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.
- đ The maximum velocity of an oscillating object can be calculated using the formula involving amplitude, spring constant, and mass.
- â± The period, frequency, and angular velocity of simple harmonic motion can be understood by drawing parallels with uniform circular motion.
- đ The position of an object in simple harmonic motion over time can be described using trigonometry, specifically the cosine function.
- đ 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.
- đ€žââïž 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.
- đ·ââïž Engineers had to redesign the Millennium Bridge to counteract the oscillations, focusing on the horizontal swaying that was initially overlooked.
- đ The script emphasizes the importance of considering all aspects of oscillation, including both vertical and horizontal movements, in engineering designs.
- đ 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
đ 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.
đ Energy and Motion in Simple Harmonic Systems
To understand the ballâs 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, itâs all kinetic. The ballâs total energy equals half the mass times the maximum velocity squared. Combining these energy equations helps answer the ballâs maximum velocity, which is the amplitude times the square root of the spring constant divided by mass. This analysis shows the ballâs energy dynamics and provides an equation for maximum velocity, revealing a deeper understanding of the ball's oscillation properties.
âïž 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 ballâs 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 2Ï 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 2Ï. This comparison elucidates the shared characteristics of these motions.
đ Trigonometry in Simple Harmonic Motion
To find the ballâs position over time, analyze the marbleâs path along the ring. The horizontal distance from the center, seen edge-on, matches the ballâs distance from equilibrium. The cosine of the angle equals this distance over amplitude. With angular velocity, the ballâs position equation becomes x = A cos Ït, 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.
đ§Ș 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.
Mindmap
Keywords
đĄMillennium Bridge
đĄOscillations
đĄSimple Harmonic Motion
đĄKinetic Energy
đĄPotential Energy
đĄAmplitude
đĄResonance
đĄPeriod
đĄFrequency
đĄAngular Velocity
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 2Ï times the square root of mass over the spring constant.
The frequency of simple harmonic motion is 1 over 2Ï 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 didnât 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 itâs lying there, at rest, itâs 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, whatâs its maximum velocity?
To better understand whatâs happening to the ball, letâs start with its energy.
As the ball compresses and stretches the spring, both âkinetic energyâ and âpotential energyâ come into play.
Kinetic energy is the energy of motion, and as the ball moves, there are two points --
the turning points -- where itâs NOT moving:
One point is where the spring is compressed all the way, and the other is where itâs stretched all the way.
And the distance between either of these two points, and the equilibrium point, is called the âamplitudeâ.
At those two turning points, the ball wonât have any kinetic energy, since it isn't moving.
Instead, all of the ballâs 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 itâs 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 ballâs motion -- at the equilibrium point -- its potential energy goes down to 0.
The ball is back where it started, so the spring isnât 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 ballâs maximum velocity, and this equation tells us, that itâs equal
to the (amplitude), times the (square root of the spring constant) (divided by its mass).
So weâve 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 thereâs 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 weâve already talked about them in our episode on uniform circular motion.
And thatâs 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 ballâs 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 itâs 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, letâs take this comparison a step further.
Letâs assume that the radius of the ring is the same as the amplitude of the ballâs motion on the spring.
And the marbleâs 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, youâd find that the equation for the marbleâs 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, letâs 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 marbleâs speed.
And! The radius of the circle is the same as the ballâs 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 marbleâs speed -- which, again, is the same as the ballâs 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 marbleâs motion around the ring is equal to (two pi) times (the root of m) over (k).
Now, weâve also talked about the frequency of uniform circular motion:
Itâs the number of revolutions the marble makes around the ring every second, and itâs 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 thatâll apply to the ball on the spring, too.
Because the rules are the same!
Finally, thereâs angular velocity to consider.
In uniform circular motion, weâve 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, itâs 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 ballâs simple harmonic motion as it oscillates on the spring.
But thereâs one more question: How does the ballâs position change over time?
To find out, weâll have to analyze the marbleâs motion along the ring again.
And the answer will involve some trigonometry.
But itâs not particularly complicated trig so, it'll be fine.
At any given point along the marbleâs path, itâll 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 ringâs radius.
We already know that the radius of the ring is the same as the amplitude of the ballâs motion along the spring.
And if you turn the ring so that it looks like a line again, you can see that the marbleâs
horizontal distance from the center of the ring is the same as the ballâs distance from the equilibrium point.
So, the cosine of theta is equal to the (ballâs position) divided by its (amplitude).
In other words, the ballâs 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!
Weâll be talking a lot more about waves in our next three episodes.
But for now, itâs 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 bridgeâs 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 didnât realize that theyâd 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 thereâs one thing you donât want your bridge to be doing, itâs 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.
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