 All around us in the universe, there are phenomena that repeat themselves. They start, they cycle through one epoch of their existence, and then they repeat that epoch over and over and over again. These are oscillatory phenomena. The swinging of a pendulum back and forth, even the light emitted by an atom, a thumbprint of the kind of atom we're looking at, is because of the limitations on the oscillatory behavior of electrons in the atom. Oscillations are everywhere, and we will begin to understand them now. Let us begin to explore oscillatory phenomena. In this section of the course, the key ideas that we're going to encounter are as follows. We will learn to identify a kind of motion known as periodic motion. We've seen it before, but now we're going to really learn to describe it and be able to do so in a way that when we observe it in the natural world, it will be more and more obvious to us what we're seeing. There's a subset of periodic motion, which is known as simple harmonic motion, and we're going to especially learn to identify that. We're going to learn to describe such motion. We'll go through some of the mathematical details of how one does this description. And then finally we will relate oscillatory motion back to the energy and force concepts, and we'll see that we can view these particular phenomena from the perspective of force and energy and gain new insights into the natural world whenever these phenomena are present. The universe is utterly filled with phenomena that repeat over a unit of time. Here are just a few. Some are extremely familiar. The sun rises, the sun sets, and it rises again the next morning. This is an extremely what is called periodic phenomenon, one with a very predictable unit of time over which it occurs. We're used to the cycle of the sun rising and setting and rising again, happening over about 24 hours. But look at a tree. The branches sway in the wind in a breeze. They wave back and forth, back and forth in the breeze. And when they return approximately to their starting position, you can consider that one cycle of the wave. The seasons themselves will cycle from winter to spring to summer to autumn and then to winter again. And really seasons are a oscillatory phenomenon in the average temperature of the region that we are living in. We tend to think of the winter as much colder. We tend to think of the summer as much hotter. Of course, what we mean by cold in an absolute sense may be very specific to where we live. Cold in equatorial regions of the world is very different from cold in polar regions of the world. But nonetheless, we think about a rise in temperature as we go into the summer and a decline in temperature as we head into the winter and the whole phenomenon repeats again. Biological cells, they divide, they grow, they divide again. This is a cyclic process. And even something as fundamental as the structure of the atom in which electrons orbit a central nucleus, the orbits of those electrons are also themselves a kind of periodic phenomenon. And that too can be described using oscillatory mathematical descriptions. Now, some of these phenomena can be represented by a simple mathematical expression. And we will see that today. That expression involves a description of a behavior whose time rate of change can be summarized by either a sine function or a cosine function in time. And such oscillatory phenomena that obey these simple trigonometric functions as regards their behavior in time are a special case of oscillatory phenomena known as simple harmonic motion. And we will revisit and emphasize this point throughout this lecture. Now let's begin by defining the concept of period or periodicity. The period is the unit of time over which one cycle of the oscillatory phenomenon will occur. And this is often denoted by a symbol like capital T. So again, we're looking to identify a phenomenon where it starts in some state. We might define that moment as T0. And then when time equals the period capital T, it returns to that state and the whole phenomenon repeats again. So all we really have to do for an oscillatory phenomenon is understand one period of the phenomenon. And if it just repeats after that, we've understood everything about the phenomenon for all times. We only have to concentrate on understanding the behavior in one unit of time over which the phenomenon cycles once and that is the period. So can you identify the period of the following phenomena? And to give you a hint on this, try finding a physical location in the pictures that I'll show you. Identify what the phenomenon is and identify a location in the picture when the feature of the phenomenon appears to repeat. So for instance, let's take a look at this picture. For those of you who are physically at the SMU campus, if you've ever visited the southern end of campus, this particular sculpture will be a staple of the sites you see there on the southern end of campus by the Meadows Museum of the Arts. Take a look at the picture of this phenomenon and see if you can identify a physical location in the picture where the features appear to repeat. So you might have noted that this sculpture, which is known as the wave, has a cycling behavior in terms of the height to which the sculpture goes. So for instance, here at the back of the sculpture there's some maximum height to which the sculpture rises, then it declines to a minimum, and then it rises again to a maximum. And we see that that repeats. It falls to a minimum, it rises to a maximum, it falls to a minimum, and then it rises back to a maximum. So one period of this phenomenon would be the time required for the sculpture to transition from this maximum to this maximum. If we can understand what happens in between these two points in space, in principle we can describe the entire sculpture. What about this picture? Take a look at this picture and see if you can identify the period of the phenomenon depicted here. This one is much more challenging, and it doesn't matter what it is. We can see that these colored wiggling lines, they seem frantic at first, but then suddenly out of this sort of frantic vibrating pattern we see a rise and a fall, a rise and a fall, a rise and a fall, a rise and a fall, a rise and a fall, a rise and a fall, and then it seems to decay away into this frantic pattern again. In fact, this is a measurement of two stellar corpses known as black holes orbiting each other. They do so ever faster as they grow closer together in space. They merge and become one stellar corpse, one black hole, and then we no longer see them orbiting each other anymore. So this arrival of periodicity in the motion, albeit with a changing time over which the periodicity occurs, is symbolic of an orbit that is decaying rapidly. Two objects spinning around each other, spiraling around each other, suddenly merge into one, and there's no more rotation, there's no more orbiting anymore that we can see in the pattern. So what's the phenomenon? Well, whatever it is, there's a maximum here, and there's a maximum here, and there's a maximum here, and the maxima are growing closer to each other, but one cycle of the phenomenon can still be stated to be between one maximum and the next. Take a look at this picture and see if you can again identify the physical location where the phenomenon repeats. Now this one is much trickier. Clearly there's some kind of maximum here over at a delta T of zero, and then it falls to some kind of minimum somewhere around here, and then it begins to rise again, but it doesn't quite make it back to that original maximum, which is labeled here with the number one. This is a phenomenon that appears to repeat on a time scale from here to somewhere around here, but we're running out of the ability to say anything about the phenomenon once we get over here, probably because we're running out of data. This is a hard measurement to make, and in fact this phenomenon is not orbiting black holes or a waving sculpture. It's actually a subtle effect of quantum mechanics, which is the set of laws that describes extremely small things in the universe, and quantum mechanics makes a very definite prediction about how certain systems behave. These are the boxes shown in yellow and green, and the actual measured data are shown in black, and you see they agree extremely well within the uncertainties of the measurements and the predictions. This is a phenomenon where there is an oscillation, it goes from a maximum to a minimum, and it tries to rise back to its maximum, but it can't quite get there. We'll come back and visit this kind of phenomenon at the end. It's a special case of oscillatory phenomenon known as a damped phenomenon, and we'll look at this in a little bit more detail later in the lecture, but it is an oscillation, it's just an oscillation that is more and more suppressed over time. Now, instead of talking about the time for a single oscillation, one repetitive unit of the motion or of the phenomenon, it is customary to instead talk about the number of repetitions of the phenomenon per unit time. This is also known as the frequency of the phenomenon. Now, frequency is just the inversion of the period. If one cycle of the phenomenon happens over a time capital T, the frequency of that same phenomenon is one divided by capital T. So, for example, let's do a concrete application of this. Let's say there's some phenomenon that we're studying. It has a period of 15 seconds, that is, over 15 seconds the whole phenomenon occurs, and then in the next 15 seconds it just repeats again. Its frequency is 1 over 15 seconds or 0.066 things per second. Another way of looking at this is to say that each second, only about 6.7% or 1.15 of the phenomenon has concluded, and every additional second that passes brings us an additional 6.7% or 1.15 of the phenomenon, until finally after 15 seconds has occurred, one unit of the phenomenon has also occurred. So, the units of things per second or just per second, seconds to the minus one power or 1 over seconds, it actually gets a special name, the Hertz, and it's named after the physicist Heinrich Hertz, who is credited with the discovery that there are electromagnetic waves, a phenomenon you will learn about in the second semester of this course. These are waving electric and magnetic fields that propagate through empty space, they move at the speed of light, and in fact, because he discovered these things, it is now understood that light itself is an electromagnetic wave. It is an oscillatory phenomenon involving electric and magnetic fields. These are things you will study next semester. So, Heinrich Hertz, being so famous, has this unit of frequency 1 over seconds named after him, the Hertz. So, 1 Hertz is 1 over seconds or seconds to the minus one. Now, let's go back and take a look at simple harmonic motion. Simple harmonic motion is a special case of oscillatory motion. General oscillatory phenomena, as we've seen from some of the examples I showed you a moment ago, can be quite complex. They may repeat with a period capital T, but they may not have a behavior that's really easily described by some nice, neat simple mathematical function. Now, in contrast, the subset of general oscillatory phenomena that are known as simple harmonic in nature are exactly describable using simple mathematical functions, specifically the trigonometric functions sine or cosine. So, this is the equation describing the position, x at some time t, for an oscillating phenomenon of some kind. As an example of such a simple harmonic oscillator, here is a video of a mass on the end of a very light spring that I've shown you before. You displace the mass from the equilibrium position of the system and release it, and the mass begins to oscillate back and forth, back and forth over and over and over again. If one uses software to track the y-coordinate of this mass, for example, and then makes a plot as a function of time as the y-coordinate varies, you will immediately observe that this oscillating structure in the y-coordinate looks exactly like a sine or a cosine function. It's incredible how this appears in the natural world and lends itself to such a simple mathematical description as a sine or cosine function that depends only on time and a few other quantities. This equation will describe those positions. We need only determine what the quantities are in the equation. Well, the position as a function of time is equal to the product of a number, xm, and a cosine function. xm has a special meaning. It's the biggest displacement in x the phenomenon will ever have. And then the cosine function, and we'll visit the pieces of this in a moment, this describes the sort of time structure of the change in position. So let's look at the pieces of this equation. As I said, x as a function of t is the displacement as a function of time of the phenomenon from its original position. So at time zero, it's at some original position x, and then as time moves on, the position evolves, but it oscillates. xm is the maximum displacement of the phenomenon that it can ever achieve, for instance, from its original position or really from any position. Omega is a new quantity that will go into more depth in a second known as the angular frequency, and it's defined as two pi times the frequency, which is one over the period of the phenomenon. So omega, and again we'll go into more depth on this in a second, is two pi over the period or two pi times the frequency, and you'll see why in a moment. Now phi is something that is known as the phase angle, and it's a number, it's an angle, and it allows us to describe such a phenomenon, an oscillatory phenomenon, regardless of when we started timing the phenomenon. So for instance, it's possible that a given phenomenon is not in its maximum displacement location when time is zero. So we start our clock, it's at some position in space, but not its maximum position. The phase angle lets us describe that, and we'll go back and look at that in more detail in a few moments. Let's begin with angular frequency though, and because this concept is a very new concept for us, we need to dig in and take a slightly closer look at it. Now I want you to keep in mind that the goal of describing simple harmonic oscillations is to describe a system that begins in some state of displacement at time t equals zero, and it returns to that very same state whatever it was at time equals one period of the motion, capital T. With that in mind, let's look at angular frequency. Let's recall the features of the cosine function. When the argument of the cosine function is zero, the value of the cosine function is one, and that is the biggest value it will ever take. The cosine of zero being one is the largest value cosine can ever return. If we then change the argument to 90 degrees or pi over two radians, the cosine function becomes zero. Now this is not its smallest value. The smallest value occurs when we then additionally increment the angle from pi over two to pi or 180 degrees. The cosine of pi, or the cosine of 180 degrees, is negative one. This is the smallest value that the cosine function will ever take. Finally, we keep swinging around the circle here. We've gone now from 180 to 270 degrees or three pi over two radians. Cosine of three pi over two is zero. We're back to zero again. We've returned to zero. Finally, we get full circle, 360 degrees, two pi radians. The cosine of two pi is one. We've returned to our maximal starting position. The cosine function and its related function, the sine, they have this nice oscillatory feature. They start at a value. They then move to different values, but they never go more negative than they went positive, and they return back to their starting point at the end of one full cycle. For instance, if the phenomenon begins in its maximal displacement at t equals zero, clearly then at that time, we need the value of the cosine to be one because we want x to be equal to its maximal displacement, and at time zero, that means the cosine of zero needs to be one. Clearly, we need the cosine to depend on time in some linear way so that when t equals zero, in this case, the cosine of zero can be equal to one, its largest value. Now, we then need it to return to that value when time gets to one period of the motion, capital T. So we need to get cosine of two pi when t equals capital T. When time gets to one period of the motion, we need the argument of the cosine to be two pi. So you might have thought, ah, the argument of the cosine function is probably going to be something like frequency times time so that when time is one period, capital T, we get back to one, the maximum value of the cosine function. Except the problem is that when you take f and you multiply it by the period, you just get the number one, and the cosine of one is not one. The cosine of one is something else. One radian is not the same as two pi radians. And that's where angular frequency comes in. What we do is we redefine the argument of the cosine so that we achieve two pi when time hits one period. So we define omega to be two pi f such that the cosine of omega times the period gives us what we want. So two pi f times capital T gives us just two pi times one, and the cosine of two pi is one. So angular frequency conceptually is just the amount of one complete cycle, literally a circle in radians, represented by the frequency in hertz. So if the frequency of a particular phenomenon is five hertz, it means that there are five cycles of that phenomenon per second. Okay, so we need to make five complete circles conceptually in one second. We need to go two pi, two pi, two pi, two pi, two pi, and we need to do that in one second. So the angular frequency of that will be two pi times one over five, or two-fifths pi radians per second. Now what about the phase angle? If the angular frequency is the amount of a complete circle represented by the frequency f of the phenomenon, then the phase angle is the amount of one circle that you've got to shift the cosine function by in order to describe the actual state of displacement of the system when the clock starts ticking, a t equals zero seconds. So consider the graph I've constructed over here on the left. I'm showing you displacement on the vertical axis, and I'm showing you the number of radians, so the number of degrees through one cycle, that a phenomenon has concluded by a particular part of the cycle. What would you say is the amount by which the maximum, so for instance here, or here, is shifted from being at time equals zero? Notice that the curve does not make it back to its maximum point at t equals zero, so when the angle is zero. This is called the phase shift. So what is the angle by which this phenomenon has been shifted in order to avoid having the maximum be at zero? Well if you really stare at this, you'll notice that we should have expected maximum at zero, which we don't have, and at two pi, which we also don't have. Here's the height of this blue curve at two pi. But the maximum is actually over here, about one eighth of a complete cycle away from where it's supposed to be. So notice the following. If I start counting from this maximum to this maximum, I have to go one, two, three, four, five, six, seven, eight units on this horizontal axis in order to go from maximum to maximum. If I had intended my maximum to be here at two pi, but I see it's over here one unit to the left, then I'm one eighth of a circle shifted away from the maximum. And so the answer here is that I'm a shift of 45 degrees, one eighth of a complete circle, or pi over four radians away from having the phenomenon actually hit its maximum point at, for instance, time equals zero. It's much easier to see this if I overlay the unshifted curve in red. So see where the blue maxima are. See where the red maxima are. The red represents the non-phase shifted curve, describing the oscillatory phenomenon. The blue represents the phase shifted curve. And here I mean shifted by a phase angle of pi over four radians or 45 degrees. Okay, so we have a phenomenon that can displace by an amount x at some time t. That displacement can be related to its maximum ever achievable displacement, times a cosine function involving angular frequency, time, and a possible phase shift. Can we say how fast the thing is displacing as a function of time? Sure. Of course, we can do all these things. We can use calculus and take a look at how the displacement is changing with time, and we can start defining velocity and acceleration for this oscillatory phenomenon. We just have to use the standard, old definitions of velocity and acceleration to do this. So, for instance, velocity is the first derivative with respect to time of displacement, acceleration is the first derivative with respect to time of velocity, which in turn means it's the second derivative with respect to time of displacement. So let's start with velocity. What's the velocity of the displacement of a simple harmonic phenomenon? All we have to do is take the first derivative with respect to time of the cosine function that represents the time behavior of our oscillatory phenomenon. Before we do that, let's remember some features of the time derivative of the cosine function. If I have cosine of some number a times time plus some number b where a and b are constant in time, so the derivative of the cosine of at plus b will simply return negative a times the sign of that same argument, at plus b. And again, remember, a and b are constant in time, so t is the only thing here that depends on time, and this is just an application of the derivative. So if this is not fresh in mind, it's wise to revisit Calc I, the first semester calculus, to brush up on what happens when you take the time derivative or the derivative of any cosine or sine function of some argument. So let's go ahead and apply this to our specific case. We want to know the speed of the oscillatory phenomenon, so we want to take a look at the time rate of change of its displacement. So this is the time derivative of the function I showed. The maximum displacement that it can ever have, which is just a constant, times the cosine of omega t, the angular frequency times the time plus phi, the phase angle. So again, I have a situation where I have a constant times cosine of a constant times t plus a constant. So if I do the derivative dutifully at this point, I will find that this velocity is equal to negative omega, the angular frequency, times the maximum displacement, times the sine of omega t plus phi. Now the quantity omega xm is known as the velocity amplitude. It's the maximum velocity that the oscillatory phenomenon will ever physically have, in this case, in the x direction. This tells us at what rate displacement is changing at an exact time, this equation, v of t. Omega xm is the maximum velocity that this phenomenon will ever have. Now let's in turn look at acceleration. Acceleration is the time rate of change of velocity. So all I have to do is take that equation for velocity I just obtained and do the time derivative of that one more time. And if you do this, and I encourage you to try this on your own, you will find that you get negative omega squared xm times the cosine of omega t plus phi. Now look what's happened. We have returned back to a number times our original displacement function. So we can just substitute xm cosine of omega t plus phi with x of t. Acceleration is equal to the negative of the angular frequency squared times the displacement at any time. This is cool. I don't have to keep taking the time derivative of simple harmonic phenomena to find the acceleration. All I have to do is know the angular frequency of the phenomenon, take the displacement at any time t, multiply by negative omega squared, boom, I have the acceleration. Now this simple relationship between the displacement of an oscillatory phenomenon and its acceleration is a key characteristic of simple harmonic motion. If simple harmonic phenomena can be easily described by sine or cosine functions, their accelerations are also as a result of this, as a direct consequence of this given by negative omega squared times the displacement. It's a feature of this way of being able to describe a periodic phenomenon. So if you observe a phenomenon where this above relationship holds, without knowing anything else about that phenomenon, you can already say it's simple harmonic in nature. And that gives you a tremendous amount of power and simplicity in describing what may at first seem like a very unfamiliar or complex phenomenon. In the context of all of this, let's revisit springs and go back to Hooke's law and now contemplate simple harmonic motion in the context of an extended spring. So in the earlier lecture on work in kinetic energy, we explored the spring force and specifically Hooke's law which for a spring within its elastic limit will have a constant relationship between force and the displacement at which a force is exerted. An ideal spring is any device where the force it exerts is proportional to its displacement from its resting position. And we observed that a spring so displaced and then released will oscillate back and forth around that original equilibrium position where force is all balanced. The force opposes the displacement at all points in the displacement and that is the statement of Hooke's law. Force is the negative of the spring constant times the displacement. So we have a situation where the acceleration of the displacement opposes the displacement which looks identical to the above. In fact, employing Newton's second law of motion for the body of mass M in simple harmonic motion, we would find that force at any time T is equal to the mass of the object in simple harmonic motion times its acceleration at any time T. Well, for a body that executes simple harmonic motion, we know what the acceleration is in terms of displacement. It's negative omega squared times the displacement at any time T. So now grouping terms, what do we observe? The force at any time T is equal to some number, negative M omega squared times the displacement at any time T. Look at Hooke's law. Force equals the negative of a number times displacement. Look at simple harmonic motion. Force equals the negative of a number times displacement. Springs, ideal springs can execute simple harmonic motion. In fact, we can then make the identity that for a spring, the spring constant is defined as the mass times the angular frequency for a spring. This would then lead us to conclude that the angular frequency is the square root of the spring constant divided by the mass. Well, this opens up all kinds of relationships. So without knowing anything about the actual motion of a spring with spring constant K and mass M, we can already define its angular frequency without having ever set it into motion. It will be given by the square root of the spring constant divided by the mass. That also means that the periodicity of its motion is exactly determined by its spring constant and its mass. It doesn't matter how much you displace it at all. The period is a fixed property of the spring and its mass. And the period of its oscillatory motion will be given by 2 pi times the square root of M over K. Now, let's consider the situation when mechanical energy is conserved, as is the case for an isolated and closed system exposed only to conservative forces. We can understand mechanical energy using all of this as well in relation to displacement and other quantities such as angular frequency for some mass M that's undergoing simple harmonic motion. All right, so for instance, let's have a mass affixed to a nearly massless spring of spring constant K and it's undergoing simple harmonic motion. We're ignoring friction and other dissipative forces. No drag, no friction, no way for energy to leak out of this system. It's closed and isolated. There's no force that can act on this system. In this case, mechanical energy will be defined at the beginning and it will never change. It may change forms, it may bounce between potential and kinetic, but the sum of potential and kinetic energies, which is defined as the mechanical energy, will itself never change. So if we do that, mechanical energy is K plus U. Well, the kinetic energy at any time T is one-half times the mass times its velocity squared at that time T and the potential energy at any time T is one-half times K times its displacement from equilibrium squared, again, at time T. Well, remember, we have equations now for the velocity of a phenomenon undergoing oscillatory motion and its displacement in terms of cosines and sines. So let's go ahead and substitute our velocity equation and our displacement equation that we have from earlier in the lecture. So the mechanical energy will be equal to K plus U, which is equal to one-half Mv squared plus one-half Kx squared. Plugging in for v and x, we get these relatively unpleasant-looking equations. Here is the velocity equation. We have the negative omega xm times the sine function. That whole thing will be squared. And over here we have the maximum displacement, xm times the cosine function, and that whole displacement will be squared. This isn't looking too promising, but let's square some things and group some terms together and see what we can learn. So let's do that. If we square that, I find that I now have, in the first term, M omega squared xm squared. I can substitute in for the spring constant with omega and M. And if I do that, I find out that K is equal to M omega squared. And then I have xm squared and cosine squared. Notice I have m omega squared xm squared and m omega squared xm squared. These constants multiply the sine and cosine functions, and they're common to both of them. So I can pull them out in front, just like I pulled out the one-half that's common to the kinetic energy and the spring potential energy. Okay, so let me do that. I have one-half m omega squared xm squared all out in front, and then I have the sum of sine squared and cosine squared of the same argument. Well, remember your trigonometry. Sine of x squared plus cosine of x squared is 1 always. I have sine of omega t plus phi all squared. Cosine of omega t plus phi all squared. Same argument, sine squared plus cosine squared. The sum of these is 1. And I find that the mechanical energy of the system is entirely defined by one-half m omega squared xm squared, the maximum displacement that the system can ever have. And that's just one-half times the spring constant times that maximum displacement squared. So if I want to know for a mechanical system what the total stored energy and mechanical energy will be in the system so that I can apply energy conservation and know the energy at any other time total in the system, I need only know the maximum displacement that in this case this spring is ever brought to. And based on that, the fact that it will begin oscillating when I let go of it after stretching it out or compressing it and then letting go, it will begin oscillating back and forth in a repetitive way. And because it executes that oscillatory behavior, the most energy it will ever have is one-half times k times its maximum displacement squared. Now this isn't a big surprise. From the earlier discussion of conservation of energy, we know that if we put a whole bunch of potential energy into a spring and then we let it go into other forms like kinetic energy in a isolated and closed system with conservative forces like the spring force acting, we never can get more energy than we started with. And so big surprise, we will never have any more energy than we put into expanding or compressing the spring at the beginning, which also coincidentally will happen to be the most displacement, the biggest stretching or compressing it will ever have from equilibrium. Now what happens when mechanical energy is not conserved? Now it's impossible in a class like this to go into an exact treatment of these situations because they are quite nasty, but I will give you a taste of the complexity that arises from this simple question. What happens if I have a system where energy can be lost to non-conservative forces? So let's consider such a situation. Energy is depleted from a system by a non-conservative force, and this process of draining energy out of a mechanical system that can undergo oscillations but is subject to a force that saps mechanical energy from the system is known as damping. An oscillatory phenomenon in this situation will execute oscillations, but they will be damped, and you'll see what this means in a moment. The force that does this is known as the damping force, and for instance, it can be described as having the following form. A damping force can be proportional to linear velocity. So the slower the phenomenon, the less the damping force. The faster the phenomenon, the greater the damping force. Damping is proportional to speed. And it's a force that depletes. It acts against the motion, so it's negative something times velocity. B is known as the damping constant. It's independent of velocity, and depending on the situation where an it can be bigger or smaller. I'll give you an example of this in a moment. Now note that drag force, which depended on velocity squared, is not a good example specifically of this kind of damping force, but in and of itself it would obviously damp oscillations, but it would do so at a rate of velocity squared. It's quadratic, not linear on speed. But this is a new kind of force that we can explore, one that's proportional to speed, not speed squared. So consider a case where simple harmonic oscillation is occurring but in a system subject to such a damping force. We can consider that system from the perspective of Newton's second law. The sum of the forces will be equal to the mass times the acceleration of the system. Well, what are the forces? Well, we have two forces. We have a damping force, negative bv. We have a spring force, negative kx. So we have two forces, the damping force and the spring force. Well, we can now substitute into this equation with the appropriate derivatives of position, x, and we can obtain a new equation. So let's do that. We have mass times acceleration. That's mass times the second derivative of position with respect to time. We have velocity over here. Well, that's the first derivative of position with respect to time. And then we just have position. So we can just write position with respect to time here. Let's group terms all on one side of the equation and get zero on the other. So we have mass times the second derivative of position plus the damping constant times the first derivative with respect to time of position plus the spring constant times position itself equals zero. Well, this looks horrible. What absolute fresh hell is this thing? This is a very special kind of equation known as a differential equation. We are not looking for a number that solves this. A quadratic equation would be like x squared plus x plus a constant equals zero. That's a quadratic equation. A number solves that. This is an equation where we have time derivatives of a function x and we want to know what function solves this equation. This is a differential equation. Its solution is not a number. It's a function of x with respect to time. What function satisfies this equation? Whole courses are taught in solving differential equations. This is not a course in learning to solve these equations. I will simply state the form of such a solution. It's not very pleasant looking, but if you plug it in here and grind through the derivatives, you'll see that it works just fine. So here we go. A function that solves this equation is the following. The displacement at any time t is given by the maximum displacement times an exponential function that depends on the damping constant and time and mass and the cosine of some angular frequency, which I'll talk about in a moment, times time plus a phase shift. This function plugged in dutifully up here to this equation will solve this equation. You'll see that in fact when you do all these derivatives and add all this stuff together, you will get zero, but it's not very pretty looking and it's not easy to set up and solve these equations, which is why I simply state the solution. What do you notice before I talk about omega prime? Well, we have our old oscillation equation, right? We've got xm, we've got cosine of some angular frequency times time plus phase shift. That all looks familiar, but then we've got this extra function of time stuck in here. What does it do? Well, when time is zero, we have e to the zero and anything raised to the power of zero is one. So at time zero, this function's value is one, and this equation looks exactly like the oscillation equation we saw earlier. When time goes away from zero, gets bigger than zero, one second, two seconds, ten seconds, twenty seconds, whatever the number is, the exponential function with its minus sign in front of time gets smaller and smaller and smaller. e to the minus one is a bigger number than e to the minus two and that's a bigger number than e to the minus ten. And there's a consequence of that, even though the oscillatory motion continues over time, its amplitude, its degree of maximum displacement declines over time. This is damped oscillation, damped in the sense that over time, while oscillation may continue, the maximum degree of displacement from say equilibrium will decline as a function of time. If you look back at that third example I gave at the beginning of the video with the oscillatory phenomenon that begins at a maximum of one, falls to a minimum, then rises again, but never makes it quite back to the maximum, that is a damped oscillation. Now, what is this angular frequency omega prime? Well, because of the nature of the solution of this problem, omega prime is not just a simple function of k and m anymore. It's k and m and the damping coefficient b, it's a mess. This is complicated. This is why there are whole classes on solving differential equations and I'm merely showing you the solution here. Now, we may use this to solve problems, but you can see that this is a complex situation already. Nonetheless, it's an extremely important one physically. Where are damped oscillations important in the world around us? Think about the systems on an automobile. You're driving along on a car, you hit a bump. There are springs, the so-called suspension system, in your car, so that if you drive over a bump, the whole car doesn't jump up and down. Rather, the wheels can move up, which compresses a spring, and then when you go over the bump, the wheels go back down as the spring pushes the wheels back down into the ground. The idea here is to keep the body of the car as level as possible to make the ride comfortable while still allowing you to drive around. The problem is though that if you hit a bump, that's an impulse and it would impart energy into the springs that's stored and then the springs would bounce up and down and up and down oscillating. That would be extremely uncomfortable. You'd probably get motion sick in the car if that happened. What the engineers that design these suspension systems do is they build a damping fluid into the spring so that you get the oscillation but not more than one cycle of the oscillation that you feel. Damped oscillations are extremely important for smooth and comfortable rides in automobiles, and there are other applications like that, of course, in industry and other places in the world around us. Now, a force like this is going to sap mechanical energy from a system because we have a force that is resisting the motion no matter what the direction of the motion is, just like drag, just like friction. So what is the mechanical energy doing as a function of time? Well, you can go ahead and try to answer that question exactly, but it's a bit of a mess. Going back and working through that sum of potential and kinetic energy like we did before, it's a lot messier than the simple spring with a conservative force in it. I will give you an approximate equation that describes this change in energy, and this is approximately what the mechanical energy does as a function of time. It's still 1 half k times the maximum displacement squared, and at time zero, that's the most energy you can ever have, but you'll notice that at later times, it's damped away by this exponential function, e to the minus bt over m. As time goes on, the energy is drained from the system, and in the limit that t goes to infinity, the mechanical energy goes to zero. And that's exactly the limit of this function. It declines in size over time. It tends to zero as time goes to infinity, and so the mechanical energy is driven to zero, which in a system that's truly having energy sapped out of it by a non-conservative force would actually happen. Now, what happens when an external force, rather than taking energy away from a system that can execute oscillatory motion, instead supplies energy at some, perhaps, steady rate to that system? We can reverse the process that we've just thought about, and instead have a situation not where the amplitude, the maximum displacement of the phenomenon decreases in time, but perhaps increases in time to some new maximum. The force that does this is known as a driving force, and oscillations in the face of such a driving force are known as driven oscillations. Now, if you thought damped oscillations were complicated, forced oscillations are complicated in their own way, and so much so that I'm really going to give them a very cursory treatment here. I'll give you the highlights. They're quite complicated, but we can provide some examples to help you think mentally about driving forces in situations you may already be familiar with. So some of you may have experienced earthquakes before. Earthquakes occur when there are shifts in the position of the ground, literally the surface of the Earth, due, for instance, to the slipping of a fault. You have tectonic plates on the Earth's surface. They're sliding over each other. Sometimes they stick together. Sometimes they slip. When they stick for a long time and store up energy in the compression, and then the energy is released and the plates slip, you get an earthquake. That's the very simplistic view of how this happens. Buildings are rooted in the ground, and so when the ground shifts, buildings move. But buildings are complicated things. Buildings have mass. They're tall. Not all of them are rooted directly to the ground. The roof of a building may be 80 stories up from where it's actually rooted to the ground. And it's not a perfectly rigid structure. The whole building doesn't all move at once, especially if you have a really high impulse over a short time, so you get a big force due to the shifting of the bottom of the building in a short time. That may not be transmitted right away to the top of the building, and so the structure will begin to stretch horizontally. And this creates an oscillation in the building, and it will move. The bottom displaces first, the top displaces later, and then it catches up, but then the bottom might have slipped back in the earthquake and then the top catches up again. This is terrifying. It's an oscillation that's driven by the earth moving, and if the frequency of the earth's shaking drives the oscillation at just the right frequency, the whole building can come apart. Have you ever been on a playground swing? These are the seats that hang on chains. They're attached to a bar at the top, and you can sit on them, and if you swing your legs, pump your legs back and forth, back and forth, you can get yourself to swing with a greater and greater amplitude from your original position, which was zero. I mean, this is just like a pendulum, right? This is just like the bowling ball connected to the cable, except that you're driving the bowling ball higher and higher and higher as you pump your legs forward and back. What you're doing, of course, is you're taking your stored chemical energy and you're converting it into the kinetic energy of your legs, and this, in turn, transfers energy to the swing itself. If you pump your legs at just the right frequency, you can actually drive the maximum height of the swing with each swing that you make to a higher and higher point over time. If you pump your legs, however, at the wrong rate, and you'll feel it when you do this, you won't be transferring energy from your kinetic energy into the swing as efficiently, and so the swing doesn't respond as well. In fact, you can swing in such a way that you completely drive the motion to zero. So this is another example of a driven oscillation, and one that maybe as a child, if you ever played on a toy like this, you would be somewhat familiar with. So what's going on? In these previous examples, the playground swing, for instance, being efficiently driven to some maximum height, some maximum amplitude, or a building that winds up getting ripped apart because the transfer of energy through the oscillatory motion, the driven oscillation from the ground shaking, is at some sweet spot that rips the materials of the building apart. These are both demonstrations of resonant phenomena. Resonant phenomena occurs when the frequency of the driven oscillation is at a frequency determined by the structure that's being moved that is just right to achieve the maximum possible amplitude of motion. And more amplitude, more acceleration, more force, more destruction, or at least that's the potential of it all. So we have a driving force that achieves some special value, omega sub d, for the frequency of the driving force, such that the amplitude of the resulting oscillation of the system hits a maximum. And in fact, in a perfect driven situation where there's nothing that can damp out the motion, you can drive the system off to a huge amplitude, which can, again, rip it to pieces. This point in amplitude versus driving frequency, which is kind of illustrated over here, so the transmissibility of energy into the oscillation as a function of the frequency at which you drive the system, and this is the, at one, you're at the sweet spot. You're at that perfect natural frequency of the system where the maximum energy transfer can occur. This is where the most damage can be done. And so engineers that design buildings, they take into account these kinds of resonant frequencies, and they try to design systems to damp out these oscillations before they can get so big that they rip a building apart. Resonances also occur in the subatomic realm, in atoms, and even in fundamental systems of particles. These would be explored in a later course, for instance, in our department in particle physics. So let's review the key ideas that we have explored in this part of the course. We've learned to identify periodic motion. This is a phenomenon that starts in a state at time zero, returns to that state at some later time capital T, and then repeats the whole phenomenon all over again. A subset of such motion is simple harmonic motion, where this change in, for instance, displacement in space or some other quantity can be described by sine and cosine functions that are themselves functions of time. We've used this idea to try to describe such simple harmonic motion, and we've looked at some of the aspects of this motion, not just displacement as a function of time, but velocity and acceleration. And we've seen that the acceleration in simple harmonic motion is related very neatly to the displacement through the angular frequency of the phenomenon. And we've revisited energy-enforced concepts, Newton's second law, conservation of mechanical energy, or non-conservation of mechanical energy, in light of this kind of motion, this oscillatory motion. And we've even looked at some special phenomena where we drain energy out of a mechanical system that can execute oscillations, damping, or we put energy into a system that can have oscillations, driven oscillations. And we've seen that these are complex phenomena. They have real-world consequences, but the most gritty details of these phenomena lie just outside the boundaries of a course like this. So I've given you a taste of these phenomena without getting too much into the weeds of the mathematical descriptions and so forth. The ideas that we have explored here are crucial in understanding phenomena like waves, be those mechanical waves that you might find in water or in elastic devices that can stretch and restore their sizes. Sound is a wave, it's a compression of air, so you have regions of high density and low density in air. This creates an oscillatory structure that can be described with sines and cosines. Light is a wave. It's a wave that travels always at the same speed, 2.998 times 10 to the eighth meters per second. And waves, in fact, fundamentally describe all of matter. The electron itself can be described as a wave. So oscillatory phenomena are everywhere, and we have seen the most basic tools that you could use to lay a foundation for understanding those phenomena wherever you encounter them in the great breadth of the cosmos.