 Welcome back to our lecture series, Math 1060, Chair Genom, Chief of Students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misildine. In lecture 13, which we're starting in this video, we're going to talk about two very important topics. In this video, we're going to talk about the idea of harmonic motion, the idea of a action repeating over time in an oscillating manner. We'll get to that in just a second. But then the subsequent videos of lecture 13 will actually start chapter 5 about inverse trigonometry, inverse trigonometric functions. Take a look for those videos at the end of this one. As we finish up chapter 4 about graphing trigonometric functions, I want to use the application of simple harmonic motion as a justification for why we are interested in modeling with trigonometric functions. Simple harmonic motion is only going to involve very basic transformations to sine and cosine, but this is a very important application. Like I said earlier, simple harmonic motion, or just harmonic motion for short, is a type of motion where objects move back and forth in an oscillatory manner around some type of equilibrium. Two classic examples of this would be like the pendulum of a grandfather clock. We have our pendulum right here, there's some weight at the end, and it's going back and forth, back and forth. At one moment it might swing up here, and then gravity will then bring it back down to here, and it will continue to swing in the other direction. You have this swinging pendulum back and forth, like so. When it comes to this oscillatory motion, there's this notion of equilibrium. Equilibrium means that if the moving object were stationary at this one point, there would be no motion whatsoever. So as the pendulum just rests there in the position you see right now on the screen, there would be no motion whatsoever. This is opposed to if I were holding up the pendulum, if I were to let go, it would then begin to swing causing the harmonic motion in place. So equilibrium is the place where the moving object wants to come to rest. Another example of simple harmonic motion, imagine we have a spring of some kind, right? And maybe there's some type of weight, some type of bob at the end of this spring right here. And so this spring is gonna be elongated due to gravity, but the reach is a point where if no forces are acting on the spring, it'll stay in a state of equilibrium. What's not moving whatsoever. But then what happens, of course, when I pull down the spring, excuse me, then I'm gonna stretch it out a lot more right, yeah, then you have the bob at the end. When I let go, it's gonna go up and down, up and down, up and down. This is gonna bob up and down, up and down, up and down, keep bouncing beyond that equilibrium spot. Equilibrium happening over here, if it's long, it's gonna want to go back towards equilibrium. There might be a point where it's really squished, but then the tension in the spring is gonna wanna push it back towards equilibrium. And this is gonna happen over and over and over again. These are two examples of simple harmonic motion. The two we're gonna probably play around with the most in our examples in homework. But some other examples that show up as very natural phenomena, oscillation happens in sound, electrical circuits, magnetic waves, just as a few examples from physics and such. If anything ever happens in a periodic manner, we could call this simple harmonic motion. Some natural behavior happens seasonally, and that is based upon, oh, this happens in the summer, this happens in the fall. These things also can be modeled as simple harmonic motion and sine and cosine can be used to model those things. But like I said, we'll mostly be focusing on this example of the spring and the example of a pendulum, just as sort of like the poster child of simple harmonic motion. So if a motion problem can be modeled using simple harmonic motion because there's some type of oscillation around this equilibrium with respect to time, then we can use one of the following two models to correctly match this motion. We can use the position function. So S of T, this is very common notation in physics. S of T is the position function. So it's the location of the object at some given time T. We can set S of T to equal A cosine of omega T or S of T equals A sine of omega T where the number A will take on the role of amplitude as it has done in the past. So the absolute value of A is amplitude, whether it's positive or negative will depend on directions as we've seen before. So we're gonna keep A around for that reason. This number omega right here is what's referred to as the frequency of the oscillation right here. So this is similar to what the angular velocity we saw before we like to use omega in that regard. So it's the same idea here. So from what we've studied previously with modified sine and cosine waves, we see that the period of this function would be two pi over omega, this number omega right here. And so then this number, if you take the reciprocal of it, omega over two pi, this is what we mean by the frequency of the object. For simple harmonic motion, it's typically best to model the motion so that the equilibrium position coincides with the midline of the sinusoidal wave. So as we've drawn these sinusoidal waves, sines and cosines, right? We have this typical midline that shows up. This is typically what you want to be the equilibrium, the equilibrium point of the experiment. If you can do that, it makes life so much easier for you. It appears to me I didn't finish spelling the word, all squished in there, whatever. You wanna set the midline to be equilibrium. Thus, if the moving object begins at equilibrium, you would use the sine function to model it. More likely though, like if you think of like our pendulum, if it starts at equilibrium, you need something to push it, right? The easiest way to push it is actually just to lift it up and then let go. And so most likely, if the object starts its motion, what would be necessarily the maximum distance from the equilibrium, which would necessarily also give us the amplitude, we can model this with cosines. So more often than not, we actually use cosine to model simple harmonic motion. But if per chance it did start at equilibrium, which is very less likely, you could use sine in that situation. So let's look at two examples of modeling motion using this simple harmonic model. Suppose that a ball is attached to a coiled spring, such that it's suspended from a vertical spring, excuse me. So this is like the picture we're drawing earlier. We have our spring and then there's this ball that's at the bottom of it. If pulled down five inches, so if we pull this thing down by five inches, this is extending the spring from its natural length. The natural length due to gravity of the spring is its equilibrium position. That's where it would stay at rest. Then we're gonna let go. And so then the time for one complete oscillation is four seconds. This is gonna be a measurement of the frequency. And so let's take this information we've learned here. So the maximum distance beyond equilibrium, we stretched it five inches, that's significant, that's gonna affect the amplitude. And then the statement about one complete oscillation in four seconds, that's a statement about frequency and period. So how do we unravel this? So the first thing to do is we wanna come up with an equation that models the position of the ball at the time T. And so there's some decisions we have to be made here. So as explained, the ball has an initial position, S sub zero is equal to five. So at the beginning of the experiment, we extended the spring by five inches. So that's the location of the ball at the very beginning of it. Let's say that if the spring is longer, that's gonna be considered positive distance. And if the spring is actually compressed, we'll call that negative distance. So if you go above where the ball's rest position is, that's negative. If you go below, we'll consider that positive. So as the coil begins in this stretched position, since we're starting in the stretched position, let's use cosine to model this behavior. So S of T is equal to A cosine of omega T. We don't start in equilibrium, we start at the extension. So as we stretch the coil out, the five inches we stretched it is the farthest we can ever get from equilibrium. So that five inches also represents the amplitude here. We get five inches as well, which again justifies what we're using cosine to model this. Now, if we decide to make down positive and up negative, then we get A is gonna equal five. If you switch things up, which you are allowed to do, maybe you think going down should be negative and thus going up would be positive. That would actually say that A is equal to negative five, but it also means your initial position is negative five as well. It really doesn't matter because whether you're doing positive or negative, it's just a preference on direction. As long as it's clear what that means, you can do either one and we'll stick with the model decision that we've made so far, all right? So then we've learned that one complete oscillation occurs in four seconds. This is information about the period, right? So it takes four seconds for one complete cycle to occur. Therefore, omega, which is gonna equal two pi over P right here, we would then get two pi over four and we end up with pi over two as our coefficient omega right here. Therefore, the model we can use to map out the location of our ball at any moment would be S of T is equal to five cosine of pi over two times T. So this is the model we then can use to answer questions about this harmonic motion. So for example, where will the ball be located at the moment 1.5, or wrote on it a little bit, sorry about that, but where is it at 1.5? So basically we're asking ourselves, what is S of 1.5? So we're gonna compute five times cosine of pi halves times 1.5 and feel free to use a calculator for this type of calculation. A scientific calculator would definitely be sufficient. We're gonna take pi halves times 1.5. Now if you want to 1.5, you could write that as a fraction, three over two. So this is the same thing as five times cosine of three pi over four. The reason I say that is that this one is actually a convenient one. We actually don't need a calculator, but most likely you would need a calculator to do this type of calculation here. But cosine of three pi over four, just think about that one, three pi over four is in the second quadrant. So you're gonna get negative five times cosine of pi fours and cosine of pi fours is root two over two. So you get negative five root two over two. But again, you're probably gonna get a calculator at this moment anyways, because it's like how big is this distance? This is gonna give you roughly negative 3.54 inches. So this is a negative 3.54. So really what that means is we really should be saying it's 3.54 inches above equilibrium here. That's where the ball would be located using this model. Could we determine the frequency of this? Could we find the frequency of this thing? Absolutely. Remember frequency as we saw before, F right here, this was supposed to be omega over two pi, like so. For which case you're just taking the reciprocal of the period, just so you know. I mean, frequency is just the reciprocal of the period. So this turns out to be one fourth. And it's important to think about the units here. We get one fourth oscillations per second. So we're saying that every second that occurs, one fourth of a cycle would have occurred, all right? So that's the frequency. Let's look at another example. This time the harmonic motion is already given to us. The model I should say S of T is equal to eight sine of three T. So suppose that an object oscillates according to this model where T is measured in seconds and S of T is measured in feet. Can we analyze this motion? What do I mean by that? What do I mean to analyze? Let's dissect it. We don't necessarily need to know whether it's a pendulum or a spring or anything like that, but let's break it apart. So because the motion is modeled with a sine, notice how we're using sine this time instead of cosine, the object begins at equilibrium. So we have S of zero equals zero. So we're gonna start at equilibrium here, okay? What else can we say? Well, the amplitude is given as eight. So what does that say? If the amplitude is eight, that implies that the object will initially move in the positive direction by eight feet. So first, we're gonna move in the positive direction. If it was a negative eight, we would have moved in the negative direction. So we move in the positive direction whatever that is. I need some more reference before I could do that, but we're gonna move in the positive direction eight feet. Then of course, you're gonna move back to equilibrium. Then you're gonna move in the negative direction another eight feet. And then after that, you move back to equilibrium again. And then at that point, you repeat the process because that's the description of a single cycle right there. All right, so it's gonna continue to repeat that self over and over and over again. What can we say about the period? Well, the period is gonna equal two pi over three. So two pi thirds, which that's approximately 2.09 seconds. Again, I used a calculator to help me out on that one. So in other words, if we think of the frequency, the frequency would be three over two pi, which is the reciprocal of this number. This would be 0.47. And so this is gonna be 4.7 oscillations per second. So when you work with harmonic motion, you often talk about these frequencies, these oscillations per second, that's a mouthful. So this is often abbreviated as a Hertz, which is denoted as capital H lowercase z. Hertz here is exactly just a measurement, a unit to describe oscillations per second. So the frequency of this phenomenon would be 0.477 Hertz. So about a half a Hertz is what's going on right here. In the previous example, the frequency turned out to be 0.25 Hertz. So as we end our very brief coverage of simple harmonic motion, I wanted to compare it to a more advanced form of harmonic motion. Because when you think of the pendulum swinging or the spring bobbing, if we were to watch these phenomenon long enough, eventually it stops, it slows down over time. And that's because as these things are moving around, there's friction and air drag that gradually is retarding the motion. So eventually it disappears over time. We'd say that the harmonic motion is actually damped. It dampers over time. And so how could you model a damped harmonic motion? It turns out you use the same basic idea. You're gonna have A cosine of omega t or A times sine of omega t. But if you want it to damp over time, you're gonna insert the expression E to the negative Ct where T is still our variable of time. C as a parameter depends on how quickly does it damper. If it dampers very rapidly, then C would be a very large coefficient. If it's a very slow dampening over time, then C would be a very small number in that regard. So we're using this E to the negative Ct because it has the property as a function, it eventually goes off towards zero. And so the amplitude shrinks over time because of this exponential expression. So if you're to look at the graph of a damped harmonic motion, you would see something like the following. At the very beginning, the amplitude's very big, but the amplitude is shrinking as a variable of time. So the amplitude with this damped harmonic motion, A here is this little A times E to the negative Ct. So again, as T gets big, E to the negative Ct will get small, which then shrinks this number A by the appropriate factor. And so you can see that it eventually damps out, dampers away towards zero. And so this could be a very good model when drag or friction or something else kind of slows down the oscillation further. Now for this lecture series, our examples will just focus on simple harmonic motion, I didn't want to introduce the idea of damped harmonic motion as how you can combine the things we learned in trigonometry with other algebraic tools to get much more accurate real-life models.