This lecture is going to be about simple harmonic motion, which has a very precise definition that's going to be important to understand. The definition is: periodic motion around a central point of equilibrium where there is no net force. There is a restoring force that acts in the opposite direction proportional to the displacement from the equilibrium position, and acts in the direction opposite to that displacement. So that definition is pretty advanced. I'm going to take you through it one step at a time. Two of the most common objects that occur in simple harmonic motion are the spring and pendulum, and both of these have what's called an "equilibrium position." And at the equilibrium position there is no net force, that's one of the definitions of simple harmonic motion. There has to be a certain point where there's no net force on the object from the system. So you can see there are different reasons for why this is the case. With the spring that's just the spring's natural length, so it doesn't put a push or a pull on the orange box. And on the pendulum at that exact point the force of gravity is being perfectly balanced up by the force of tension, so there's no net force at that point either, so this is the first big idea of simple harmonic motion: any object in simple harmonic motion goes back and forth around an equilibrium position where there's no net force, so these objects are going to get farther away from and then closer to that equilibrium position as they move back and forth. So as this spring is displaced farther away from its equilibrium position, you can see that there is a bigger and bigger spring force pulling the object back toward the equilibrium position. So the farther away you go from the equilibrium position, the more the spring tries to push or pull you back to that equilibrium position. In a previous video on elastic potential energy I talked about Hooke's Law. I'm going to bring that up again here, and I'll link that other video in the description of this video. So when we're describing a spring, delta X is the displacement of the spring from its equilibrium position. Fs is the force applied by the spring on the object, and in that Hooke's law equation that you can see at the bottom. The spring force is equal to the spring constant times x, and that's negative just because the force always points in the opposite direction as the displacement from the equilibrium position. In the picture in the slide you can see that the displacement is to the right, but the force is pulling it back left. That makes sense. However you displace the spring it's going to pull opposite to that displacement, so looking back up at my definition of simple harmonic motion, the definition says that there is a restoring force that acts on the object that is linearly proportional to the displacement from the equilibrium position and acts in a direction opposite to that displacement. So here I can see that Fs is playing that role. It gets bigger as the displacement from the equilibrium position gets bigger, and it pushes back toward the equilibrium position in the opposite direction of the displacement. And Hooke's law is linear because it obeys the form y = mx + b you can see that it fits that form. So if you were to graph that on a linear graph, -k would be your slope and 0 would be your y-intercept. So a spring fits the definition of simple harmonic motion. There's this restoring force that gets bigger in a linear way compared to the displacement. There's a linear relationship between the force that pulls the object back and the displacement away from the equilibrium position. So we're now going to look at the pendulum. So the pendulum's displacement occurs at a certain angle from the equilibrium position like this, and the farther away from the equilibrium position it gets the more unbalanced the force of gravity on the pendulum will be. I'm going to show you why using this animation so you can see that there's a force of gravity going straight down on this pendulum, and if I split this up into an x and y component along the angle of the pendulum I can see that this component is going to be the force of gravity times sine of the angle of the pendulum, and the other force, the other component, is pointing in a parallel direction to the tension of the rope of the pendulum. And because the pendulum isn't falling up or down in that particular green direction, that means that the forces are being balanced out, so both of these must be Fg times cosine of the angle of the pendulum. So because those are being balanced out, the only net force on this pendulum is Fg sine theta. That's the force that's pulling the object back toward the equilibrium position, and just to prove that this is linear I'm gonna write down a quick proof of this linear relationship to the displacement from the equilibrium position. I just need to prove to you that this definition also works for pendulums where there is a linear relationship between the restoring force and the displacement from equilibrium, and there's that relationship and you can see that it fits y = mx + b. One important vocab word for simple harmonic motion will be amplitude. The amplitude of simple harmonic motion is the maximum displacement from the point of equilibrium that the object achieves. This may sound a little simple but it's actually very precise. So as the spring oscillates back and forth this green line is showing me the maximum displacement from the equilibrium position, and as it goes in the other direction it's displaced in that same amount, but again in the opposite direction, so the amplitude is the maximum displacement from the point of equilibrium. So one of these green arrows, one of those lengths would be the amplitude, because it's the line drawn from the equilibrium point to the far edge. Some students get confused by this and assume that the entire length of the path, the full forward and back motion, is the amplitude, but this is not the amplitude, that's actually twice as large as the amplitude because the amplitude is only from the equilibrium position to one end, not the entire length. All objects in simple harmonic motion have a period and a frequency. Just as a reminder period is the amount of time an object takes to complete one full cycle and it's measured in seconds, whereas the frequency is the amount of repetitions a cycle completes in one second and it's measured in Hertz. And Hertz or Hz really just means events per second. And because of their definition period is equal to one over frequency. So we can calculate the period and frequency of objects in simple harmonic motion, and in the simple harmonic motion the period is specifically the amount of time it takes the oscillation to go to both extremes and then go back to its starting point, not just to one extreme and back. So I'll show you what I mean by that. I'm going to start to time the oscillation of this spring. So this is the starting point of the spring. This is one extreme. it goes back to the starting point, gets to the other extreme of its motion, and then comes back. So that was one full period of motion and I can see that that took 12 seconds of oscillation, so the period would be 12 seconds. One common mistake is for students to start the object at the starting point, allow it to oscillate to one extreme and then go back, and stop the time as soon as it gets back to its starting point. But this would not be correct. This says that the period is 6 seconds, which is only half of the full period, so this is what I mean by the amount of time it takes the oscillation to go to both extremes and then back, rather than just to one extreme and then back to its starting point. So the period of this spring would definitely be 12 seconds, not 6 seconds. we use many different types of graphs to show many specific properties of objects in simple harmonic motion at different points along their path. We'll start with the position-time graph of an object in simple harmonic motion. So what I'm going to be doing here is measuring the displacement of the spring from its equilibrium position as time passes. So my y-axis shows the vertical height of the spring and the x-axis shows the amount of time that has passed as the spring's oscillating. And as the spring oscillates you're going to notice that the graph of its displacement from its starting point over the time that has passed forms a sine wave. There we go. So all graphs of simple harmonic motion will work this way. Pendulums also produce sine wave graphs, and any other object in simple harmonic motion also produces a sine wave or cosine wave graph. We can get some useful properties of the motion from this graph. For example the height from the center is the amplitude, because that's the total distance from where the center point of the object is, where its equilibrium position is, to one extreme. So that total height of the wave from the center to the top would be the amplitude. Just notice that I'm not measuring from the very highest point on the wave to the very lowest point on the wave. That would not be the amplitude here. Just like in the example of amplitude I showed earlier. We can also find the period of the simple harmonic motion graph. If time is on the x axis the period is going to be 1 full completion of that sine wave going all the way up, all the way down, and back to where it started. So again a common mistake here would be for students to look at the sine wave and say "oh the period is 6 because that's how long it takes the wave to get back down to its starting position" but the period is 12 seconds specifically, because it has to go up to one extreme, back to the other extreme and then back to its starting point. That full wave pattern is one period, not just half of the wave pattern. Okay, this is going to be kind of a throwback, but we're going to use the information that the displacement time graph of an object in the simple harmonic motion forms a sine wave to find the velocity time and acceleration time graphs of an object in simple harmonic motion, and you'll remember that the slope of a displacement versus time graph is equal to the velocity, and the slope of a velocity time graph is equal to the acceleration, so looking at these slopes I can see that the slopes of the displacement time graph start positive and go to 0, so that means that the velocity is also going to start positive and go to zero. Next the slopes become more and more negative, and then become less negative until they hit zero again. So the same thing will happen to the velocity. It becomes more and more negative and then less negative until it hits zero again. Finally the slopes are getting bigger and bigger and bigger, so that's what the velocity will do until it hits some maximum. So that is what the velocity time graph of an object in the simple harmonic motion would look like. So in this particular case that's actually a cosine wave. We can now translate from a velocity time to an acceleration time graph using the fact that the slope of a velocity graph is equal to the acceleration. So looking at the slopes here I can see that the slope starts at zero, becomes very negative, goes back to zero, becomes very positive, and hits zero again. So this is what the acceleration graph will look like. So now we can use these graphs to find more information about objects in simple harmonic motion. One thing I notice is that the velocity is highest at the equilibrium point, and zero at the extrema, so the equilibrium point is whenever the displacement from the equilibrium point is equal to zero. So there are three places on these graphs where the object hits the equilibrium point. Here, here, and here, and you can see that when this happens, when the displacement is zero, the velocity is actually at a maximum size. It's either very positive or very negative, but it's never zero, and here at the extrema where the displacement is at its largest. I can see that the velocity hits zero, so this fits our experience of simple harmonic motion because when this object is moving in simple harmonic motion it always moves very fast through the equilibrium position. So that's where the v-max will occur, but then as it gets closer to that maximum or minimum its velocity goes to zero, and then it goes very fast and then it hits zero at the other maximum again, and it keeps oscillating up and down like that. So the fastest velocity always occurs at the equilibrium position and the velocities of zero occur at the extrema of the motion. We can also see that the acceleration is highest at the extrema where the object is not moving and zero at the equilibrium point where the object is moving the fastest. I found that this one is a little more difficult for students to understand, and it's understandably difficult because it's moving very fast through that equilibrium point ,but what this is saying is that the acceleration is zero where the velocity is at its highest value. So whenever the velocity hits a maximum at the equilibrium position the acceleration actually hits zero, so I'll show you why that is in just a moment. What's extra confusing is the places where the velocity is zero are actually the places where the acceleration is the highest. Again I'll show you why that is in just a moment. So looking at this spring I know that here the acceleration is zero, and up here the acceleration is at its max pulling it back down, and over here at the other extrema the acceleration is at its max pushing it back up. So if we think about where the forces are occurring on this object in simple harmonic motion it makes sense that at the equilibrium position there's no force. So as a result there's no acceleration either, and at the maximum points the force is going to be at a maximum pushing the object back toward the center, so that means the acceleration will also be a maximum pushing the object back toward the center. Another way to understand this is that acceleration being zero absolutely does not mean that the object is not moving. Acceleration is a change in velocity so if the acceleration is zero that just means that the velocity is not changing as it coasts very quickly toward the equilibrium position it's just going through unrestricted, so it just moves at a high constant velocity until it gets to the other side of the equilibrium position and begins to slow down due to the acceleration pointing in the opposite direction, and the point where the velocity is changing the most is where it hits zero, changing from positive to negative, so that's where the acceleration is at a maximum. So that's where the acceleration is at a maximum, at the two maximums of the movement of the displacement, and the acceleration points in the opposite direction of the displacement. So those are all really important central facts to know about simple harmonic motion going forward. Just have a few more basic ideas to cover. First of all, if we made a graph of the force on this object in simple harmonic motion it would look the same as the acceleration graph, just with an M multiplied by all the values because force is directly proportional to acceleration. Energy graphs are also interesting and important in simple harmonic motion. Energy is a scalar and cannot be negative, kinetic energy is proportional to velocity, and potential energy is proportional to displacement, so that means that if we were to graph the kinetic energy and potential energy of this spring, that same spring we were graphing before it would look something like this and like this. So the farther away you are in equilibrium the more potential energy you have, and the faster you're moving the more kinetic energy you have, and these graphs can't go below zero. They can just be either very positive or zero. One interesting pattern here is that because of the conservation of energy, if you add up the kinetic energy and the potential energy together you'll notice that their total energy always stays the same. If you add the height of the red line and the green line at any point you'll notice that it always makes the exact height of that purple line which is just a flat constant line. The reason why this is happening is because energy cannot be created or destroyed. There's no outside work being done on this object in simple harmonic motion, so that means that all of the energy is staying inside of the system. So whenever it gains kinetic energy, whenever it goes faster, it loses potential energy. It becomes closer to the equilibrium position. For springs the potential energy we're talking about is elastic potential energ,y and for pendulums the potential energy is gravitational potential energy. So which type of potential energy we use will be different depending on the object, but the kinetic energy is always just kinetic energy. And that's everything that you have to know about basic simple harmonic motion for this course.