Physics 101: Simple Harmonic Motion
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A few weeks back, I visited my friend, Simone, who let me tie a bowling ball to a tree at her workshop in order to demonstrate the conservation of energy, all without smashing my face. And we're going to come back to this most excellent demo because by the end of today's lesson, you and I are going to figure out exactly how high that tree branch is just by watching how long it took the bowling ball to swing. Hello. I'm Dianna Cowern. And welcome to lesson 16 of my class. Today's lesson is simple harmonic motion--
[cow mooing]
--things that go back and forth. I'm going to ask you, and you can think in your head or say out loud to your computer screen, whether you think these things are simple harmonic motion. The bowling ball on a rope swinging back and forth-- Yes. That one was easy. Simple harmonic motion is the whole idea of this lesson. And we started with that example. So yes, how about a spring vibrating, like those hanging spiral decorations at a party moving up and down? Yes. How about the ball in ping-pong? No. Yes, the ball does go back and forth. But it's not simple harmonic motion. So this game maybe isn't that fair yet because I have to tell you more about what simple harmonic motion really means and where it shows up and how it's useful. So to start modeling simple harmonic motion in a useful way, let's use a tool that we already have, conservation of energy.
(Describer) She puts a bowling ball with hooks in it onto her table.
[grunting]
The bowling ball swings in a gravitational field.
(Describer) She moves it.
[grunting]
And it has the most gravitational potential energy here at the end, where it's at its highest point. And then it's all converted to kinetic energy at the bottom, when it's moving the fastest, and then back to gravitational potential energy. Whew. We also know that the ball is going to go to the same height on either side because it doesn't have a rocket attached to it. And of course, we're assuming that there's no air resistance. So what can we say about the time it takes for each swing? Well, we can measure it. Looking at the video at Simone's workshop on my editing software, I measured the time for one full return trip. And it took 4.125 seconds. And it's the same every time. It better be, or everything I'm going to teach you in this lesson is wrong. So it is. This time, the return trip has a name. We call it the period of oscillation, or just period.
[grunting]
(Describer) She puts the ball aside.
Now, I said that the period is always the same. But it's obviously not the same period for every single pendulum. You know that if you've ever seen a Foucault pendulum. Ooh! If you've never heard of these, a Foucault pendulum is this massive pendulum that swings back and forth. And it changes its path. The path turns as the earth rotates over the course of the day. And I got to visit the biggest Foucault pendulum in the world at the Oregon Convention Center a few years back. I was so fascinated that the cleaning staff let me stay after hours to just stare at it and film it. So what do we need to change then about a pendulum to change the period? How about the length of the string? How about the mass of the cow or the mass of Earth
(Describer) A toy cow on a string.
or how about the size of my spherical cow? It turns out it's only two things that affect the period. And those are the length of the string and the acceleration of gravity, g. That's nuts. The mass of your cow has no effect. So if I swung this little cow on the Foucault pendulum versus the big ball they have, it would be the same period. This is crazy. It's something that Galileo first discovered in the 16th century. The story goes that he was really bored in church. And he would watch a giant chandelier on a big chain swinging back and forth and back and forth. And he would think about its period of motion. That's the kind of thing that I do during public presentations too. So he wondered then, and I'm going to ask you now, why does changing the mass and the displacement height of the object not change the period while changing the length of the string does? Hmm. To answer that question is to get simple harmonic motion. So it's going to help us to answer this question to relate the pendulum to another system, the spring.
(Describer) She gets a Slinky.
And I'm going to mathematically show you that the bowling ball pendulum or the cow pendulum when you drop it from a--short distances is mathematically identical to the spring.
[boinging]
(Describer) One end goes up and down.
Cool. So let's do this.
(Describer) She puts aside the Slinky and the cow on the string, pulls a big sheet of paper onto the table and catches a thrown marker.
Say we start with the bowling ball. This is where it's hanging from on the tree.
(Describer) A dot.
And here is the rest position.
(Describer) A line straight down.
Woo. So my bowling ball is here.
(Describer) To the right.
And it is out at some angle theta from its rest position. So when it's up here, it feels a restoring force that pushes it back towards its rest position in this direction.
(Describer) Left.
So this force is my restoring force, mg sine theta. And I got the mg sine theta-- if you remember our free-body diagrams, because my gravitational force is downward with mg, I get the component restoring back to its original position. Now I get to use one of my favorite little physicist mathematician tricks. So grab your calculator. And make sure that your calculator is in radians and not degrees
(Describer) The top dot is the vertex of theta.
because it does not work in degrees. Now, plug in a small angle, like 0.1 radians, which is approximately 6 degrees. And then when I hit the sine button, I get 0.0998, which is approximately my original angle value, 0.1. So I got my angle value back, almost. And this is true for all small angles. For all angles up to about 15 degrees, in fact, the small angle approximation is accurate to within 1%. And this makes math really easy for physicists-- small angle approximation.
(Describer) She puts the calculator aside.
And that means that the restoring force on the ball is directly proportional to mg theta if my sine theta is approximately theta. And that is for small angles. And then I can do another cool trick. Using geometry, if my string has a length L and I pulled up a distance x-- so we're going to say that this distance here is x-- then theta equals x over L. And you might remember that from geometry on a circle, where you take the arc of a circle. That arc x equals theta times the radius of the circle. And in this case, the radius is L. I could either consider x as this arc here or as the line straight across; it doesn't really matter because of the small angle approximation. Those distances are about the same. So theta equals x over L. So then plugging that in here, I get force equals mg times x over L. Now, let's hang out with this equation for a second. It's pretty neat because it says that the restoring force is proportional to the displacement, x. F equals mg x over L. The restoring force also depends on the ball's mass, m, g, the gravitational acceleration, and L, the length of the string. Now, since m, g, and L are numbers we know-- they're constants-- that means that the restoring force is linear, meaning that we graph it as a line, just like a spring. If you remember from the spring lesson, our graph looked something like this. This is my displacement. And this is my force. It was linear. So both the pendulum and the spring involve a linear restoring force proportional to this displacement x. The mathematical models for the pendulum and the spring are starting to look very similar, particularly when the pendulum is at small angles. Now, back to our original question, we want to find out what the period depends on. So let's bring in Newton's second law. So my force equals ma. So I'm going to say this mg over L x equals ma. So I've got m on both sides. Those cancel. So for the pendulum, the masses cancel. And the acceleration of the ball at any point in its motion, a, is going to be g over L times x. This is a key step because we've shown that the bowling ball's acceleration does not depend on mass. The m is gone. It just depends on gravity and the length of the pendulum and, of course, the displacement, which is a stone's throw away from showing that for the bowling ball's period. Now, a side note before we relate this to the period-- notice that for the spring acceleration equation, the mass doesn't cancel. So check it out. My spring acceleration equation over here is F equals minus kx. Then if I set that equal to ma, I don't have m's on both sides. So I can't cancel the mass. So if I set that equal to ma, my acceleration is minus k over m times x, which means the spring's acceleration is actually proportional to k over m times the displacement. So I've got this model now for the swinging bowling ball's acceleration, which is linearly proportional to its displacement. Time for the big mama-- let's relate the bowling ball's acceleration to its period. To do that, we have to use a little calculus. It's not scary calculus. But we do need to talk about derivatives for a hot second. And if you know calculus, you know how hard it's going to be for me to try to explain what a derivative is in a couple of sentences. But here it is.
[tinging]
The derivative is a function that tells you how fast another function is changing. So it tells you the slope of another function. That's a derivative in two sentences. And that's basically what all of calculus is. There's the integral, too. But derivative is a big part of it. So how does a derivative apply here? Well, the acceleration, it turns out, is the second derivative of position. Acceleration is the derivative of velocity, meaning it describes how fast an object's velocity is changing. We write that like this, the derivative of velocity with respect to time, how velocity changes,
(Describer) dv over dt.
change in velocity over change in time, meaning it describes how fast an object's velocity is changing, which you know. And velocity is actually the derivative of position, dx dt, which means it describes how fast the position is changing, as you know, or as I like to say, acceleration describes the rate of change of the rate of change of position. We could also write that as "acceleration equals d squared x dt squared."
(Describer) ...over...
[trilling lips]
And interestingly, there are two very common functions that are their own second derivative like this-- so where we would take the derivative of something twice and get back the same function, almost-- the cosine function and the sine function. It turns out cosine is the function that tells you how fast sine is changing. Derivative of sine is cosine, and vice versa. The negative sine is the function that tells you how fast cosine is changing. Derivative of cosine is negative sine. The negative is the reason that I said almost. So then the rate of change of the rate of change of sine is negative sine. And the rate of change of the rate of change of cosine is negative cosine. Say that three times fast. I want to write all that out. So d squared, dx squared of cosine of x
(Describer) ...over...
is minus cosine x. And d squared, dx squared of sine of x
(Describer) ...over...
is minus sine of x. So my pendulum oscillating can be described using one of these sinusoidal functions.
[tinging]
Oh, this is, like, a really big moment. I'm going to tell you why it's big, because sine and cosine are these wavy functions, like this. It looks like a sine or a cosine. And that is the motion we're describing in this whole lesson. It's a big deal. Back to the math--so I'm going to pick cosine to describe my position with some angular frequency omega because-- oh, by the way, we've only really used angular frequency described going in a circle. But it works the same for a cycle. This is a cycle.
(Describer) Back and forth.
It just describes cycles per time in this case. And I'm picking cosine because for the cosine function, when time equals 0, cosine is at the maximum. And that's true of my bowling ball too. At time equals 0, the displacement is also at its maximum. Then it swings and goes on. So I'm going to say x equals cosine of omega t. So then the first derivative of this is going to give me its velocity at any time. My velocity equals dx, the change in my position over change in time. And that's going to be negative omega sine omega t. And then the second derivative of x, the acceleration, equals negative omega squared cosine omega t, which gives me my position function back. I get my cosine omega t back with this negative omega squared in front of it. So then I can plug this in and rewrite this as negative omega squared x. And now let's go back to algebra. So acceleration equals negative omega squared x. Let's go back and equate that to our other acceleration formula, which you may have forgotten by now. But it was a equals g over L times x. So we've got negative omega squared. And I'm going to give it a negative because the restoring force is in the negative x direction. So it's minus g over L times x. And my x's cancel. Displacement doesn't matter. And these negatives cancel. So I get omega squared equals g over L. And now I know the relationship between angular frequency and period, which is what we're looking for. Period equals 2 pi over my angular frequency, omega. So now I can find the period of my pendulum.
(Describer) ...plugging in the equation for omega.
And I get that it equals-- T equals 2 pi square root of L over g. And there--there it is. And by getting to this equation, my friends, we have shown what we set out to do today. We've shown that the only way to change the period of our pendulum is to change the length of the string. Period equals 2 pi square root of g over L, where L is length of the string, because I can't change 2, I can't change pi, and, unfortunately, I can't change g unless I go to the moon or something. And actually, now we have all the tools now to do the thing that I promised you at the very beginning of this lesson. We can figure out how high that tree branch is on Simone's tree just by using the period. So if we assume that, at rest, the bowling ball is just above the ground, which it's not--it's about a meter up. But we can use that too. So I'm going to draw the tree.
Here's my branch. [chuckling]
It's a weird tree. Here's my bowling ball. If I assume that it's just above the ground, then L is going to be the height of my branch. But it's--it wasn't. It was about a meter off the ground. So then the period of my swing is this T equals 2 pi square root of L over g. And I know the period because we measured it to be about 4 seconds. And I know, obviously, g is about 10 meters per second squared. So 4 seconds equals 2 pi square root of the length I'm looking for over 10 meters per second squared. Do all this math. And you get L, the length of my string, equals 4 meters. Well, that's eerie. We started with 4 seconds for our period. And we got 4 meters for the length of our string. Oh, yeah, the height of the tree branch is that plus 1 meter. So all in all, the tree branch is 5 meters high. Done--amazing--just by knowing the period-- But back to this eerie business--so 4 seconds and 4 meters. That's pretty intriguing. And in fact, if you plugged 1 meter into this equation, 2 pi square root of g over L, you would get really close to 2 seconds using 9.81 meters per second squared for g. You actually get 2.006 seconds, which means it takes almost precisely one second for a pendulum to swing from one side to the other. And you can try this at home. Cut yourself a meter-long string. And attach something pretty dense to it, like some coins or some washers.
(Describer) She gets the cow.
And then use a clock to time it. And you'll get that one swing takes approximately one second
(Describer) She swings it.
to go across and then another second to swing back.
[cow mooing]
What is going on? Is it the stars aligning in the seventh house of the age of Aquarius? No. This is actually not a coincidence. In the late 18th century, during the French Revolution, when the French Academy of Sciences were trying to define the standard meter for the entire country, they strongly considered using a pendulum set by the second. If it swings to one side in a second, it's a meter long. And people did that informally. But for various reasons, when it came to the formal definition, they decided instead to define the meter as one 10-millionth of the distance from the North Pole to the equator measured along the meridian that passes through Paris. I wonder why they did that. Now, of course, the meter is defined in terms of the speed of light. So none of those standards are used anymore, anyways. It's still a cool thing to try. And now we've also just described how a grandfather clock works. The pendulum of Big Ben is actually made of concentric tubes of steel and zinc that shrink and expand to oppose each other when the temperatures change so that the pendulum length stays constant, which, as you know now, is very important. And great. Now you have all the tools to understand simple harmonic motion. And you can answer these questions. Is a vibrating string on a guitar plucked simple harmonic motion? Yes. How about a bouncing ball? No. The shape of a bouncing ball's motion doesn't look like this, which is our simple harmonic motion
(Describer) Waves.
sine and cosine. It looks like this.
(Describer) The tops are curved, but the bottoms come to a point where they bounce.
So it is harmonic motion. But it's not simple. This is a cool place to stop and describe what simple harmonic motion means. So if you play music, you've heard of harmonics. And you know that you've got the main frequency of your note. And then you've got these higher frequencies above it. Those are harmonics. And they add to your note and make it sound different. With a simple harmonic motion, what we're talking about is just one frequency, one simple sine wave. That's the motion of our object. So how about then a marble rolling back and forth inside of a semi-spherical bowl? Yes. How about a doorstop if you hit it and it vibrates.
[vibrating]
Yes. That's just like the spring. What about the Newton's cradle? Nope. The motion of one of the individual balls is not simply harmonic. It is motion that looks something like this. It has part of this swing, like this part of the swing. But then it stops. So it goes like--
[moaning]
--and then stops for a while. And then it does the other part of the sine and stops, and so forth, just like that.
(Describer) The end balls swing but stop when they hit the others.
[click clacking]
But other examples of simple harmonic motion are the motion of a bobbing duck as ripples in the pond go by, the motion of air as you get a bottle to whistle as you blow across it, a swing, a tuning fork vibrating or anything that you twang, like a piece of spaghetti, or your motion falling back and forth through a tube that goes directly through the center of the earth, or car suspension, and, of course, the pendulum of a grandfather clock. You can also build a simple clock using other systems that oscillate steadily, like a mass on a spring. So the period of a spring is proportional to, and this is another useful period to remember-- so the period of a spring is proportional to-- I'm just going to give it to you-- T of a spring is proportional to the square root of the mass divided by the spring constant--so square root of m over k times 2 pi. Now, let's work a quick problem with that to get it. And by the way, I always get confused about which goes on the top or the bottom, m or k. But I think to myself, what should it do? If I have a higher coefficient or a higher spring constant, it'll be a stiffer spring. So it'll oscillate back and forth faster. So the period is going to be shorter. So that means k should be on the bottom. And if I put a big mass on it, then it's harder for it to accelerate. So it's going to swing back and forth more slowly, making the period longer. So m should be on the top. So this equation makes sense. Now, for the final problem, you can get a rough idea of the spring constant of the springs in your car by thinking about the frequency of oscillation when you're on a bouncy road and thinking about how often you go up and down. So it turns out 1 hertz is a pretty typical oscillation frequency for mass-produced cars. So if my frequency of my car oscillating is 1 hertz and I'm assuming a mass of about 1 ton--
[horn honking]
--or 1,000 kilograms, we can work this out pretty quickly because 1 hertz is one oscillation per second, which conveniently also then means one second per oscillation. So that means that my period is also then one second. So then plug it all into this equation. Oh, I should square this.
(Describer) She draws around the equation.
It's a big one. So T of my car is going to equal 2 pi times the square root of my mass of 1,000 kilograms over k. My T of my car is the one second. I do all this math. And I get that k equals 40,000 newtons per meter.
[horn honking]
That's huge. But remember that there are four wheels in your car. And the springs are in parallel. So they add. So really, we get about 10 kilonewtons per meter, 10,000 newtons per meter, for each spring. And that's also huge. But it seems pretty reasonable. Now, you can use this model of a spring to build a clock in space. A clock based on a pendulum won't work in space because you don't have gravity to restore it. You don't have that restoring force. So yeah, you could use a spring. And one last thing I want to show you-- this is a really cool simulation of the pendulum's position as it swings back and forth. So the position of, say, the bowling ball on the string follows the cosine function. And I'm going to overlay on its velocity. The velocity, as we know now, follows a sine function. So it's slightly offset compared to the position. The equation shows us that. But it also makes perfect sense if you think about it because at time equal to 0, you're at the top of your swing, where you have no velocity. But that's also your maximum displacement. At the bottom of the swing, when you have your maximum kinetic energy, you're at position 0. But your velocity is maximum. And then you swing to the other side here. Velocity and position are changing both sinusoidally, but just out of phase. And then here is the acceleration overlaid. And it is even more beautiful because the acceleration is in phase with the position, but exactly opposite. That's where the negative comes in. When is your maximum acceleration? Well, when you're displaced the most. When is your minimum acceleration? It's at the bottom. Think about that for a second. There's no acceleration at the bottom because there's no more gravitational potential energy at the bottom to be converted into kinetic energy, and also because all of your gravitational force is pointing down tangential to your path. So there can be no restoring force. So I want you to think about that, about all of this, next time you're in a playground and get on a swing, as I still do. As you swing through the lowest point, think about the fact that there's actually no acceleration on you at that point. And then as you're moving up, you can feel the gravitational potential energy accelerating you backwards until you get to the max height. And you have no velocity for a split second. And then you're accelerating backwards. Think about them apples swinging.
All right. [exhaling]
Now is the part of the video where I tell you about how what we just learned is a simplified model because the pendulum is only approximately a simple harmonic oscillator when it's swinging at a small angle. We had to use that small angle approximation. But that doesn't mean that this was all some dumbed-down, useless concept. In fact, it's so incredibly useful because every type of oscillatory motion to first approximation is simple harmonic motion, like the bouncing of your car on a bumpy road. There's maybe some other interesting oscillations added in. But there's one main frequency, one main period of oscillation, like the vibration of a plucked guitar string-- there's one main frequency--or a swing set, which is basically a pendulum. That is today's lesson on simple harmonic motion. When they ask you today what you learned on YouTube, here are your two key takeaways. One, simple harmonic motion is all about a linear restoring force. And two, the period of a pendulum is 2 pi square root of L over g. And the period of an oscillating spring is 2 pi square root of m over k. And now here are all the problems that we did in today's lesson because, as you know, the best way to make sure you understand these concepts is to practice applying them yourself.
(describer) Problem one: A. Find the length
(Describer) Problem 1: A: Find the length of a pendulum that swings with a period of 4 seconds on Earth. B: What would be the period on the Moon? Problem 2: A: Your 1000-kilogram car is bouncing up and down on its four identical suspension springs with a period of 1 second. What is the spring constant of those springs? B: What would be the period on the Moon?
of a pendulum that swings with a period of 4.0 seconds on Earth. B. What would be the period on the moon? Problem two: A. Your 1,000 kilogram car is bouncing up and down on its four identical suspension springs with a period of 1 second. What is the spring constant of those springs? B. What would be the period on the moon?
(Dianna) And so go do all these practice problems again on your own. And now, finally, a message from a special guest.
(Describer) Kyle Norbert:
Hey, everyone, Kyle here. And I have a bachelor of science and physics with an emphasis in astrophysics. I'm also a member of the United States Space Force. And I'm currently at the Air Force Institute of Technology, studying operations research. Now, I just want to let you guys know that in the United States Space Force, we use physics all the time in a lot of really cool ways. We use Keplerian motion, which is the behavior of two orbiting objects about each other. We use concepts of time dilation when we send information from Earth to our satellites and back down. We use Newton's laws all the time in launching and recovering our rockets. And we even use orbital mechanics because things in the vacuum of space behave a little bit differently than they would down here on Earth. So physics is really awesome-- just want to let you know that if you apply yourself, you work really hard, have some fun along the way, physics is going to take you far. So it's great that you're watching these videos. Keep at it.
(Describer) Titles: Executive Producer/Host/Writer: Dianna Cowern Lead Writer/Course Designer: Jeff Brock Producer: Laura Chernikoff Video Editor: Rachel Watson Researcher/Editor: Sofia Chen Researcher/Editor: Erika K. Carlson Videographer: Levi Butner Curriculum Consultants: Lucy Brock and Samantha Ward Accessibility provided by the US Department of Education.
Accessibility provided by the U.S. Department of Education.
Now Playing As: English with English captions (change)
Simple harmonic motion is defined as a periodic motion of a point along a straight line, such that its acceleration is always towards a fixed point on that line and is proportional to its distance from that point. In this episode, host Dianna Cowern tackles the concept of simple harmonic motion and works through three related word problems. Part of the "Physics 101" series.
Media Details
Runtime: 27 minutes 16 seconds
- Topic: Mathematics, Science
- Subtopic: Mathematics, Physics, Science Methods
- Grade/Interest Level: 10 - 12
- Standards:
- Release Year: 2021
- Producer/Distributor: Physics Girl
- Series: Physics 101
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