Phase constant

- [Instructor] So say you had a mass connected to a spring and that spring's tied to the ceiling and you give this mass a kick, well, it's gonna start oscillating, will oscillate down and up and down and up, but if I were to try to draw this, all those drawings would overlap, it'd look like garbage. Forget that. Let's just get rid of all this. Let's say we just drew what the mass' position look like every quarter of a second. So, this is where it started at, wait a quarter of a second, now it's here, quarter of a second, now it's here, so this is one mass and we just took pictures of it and we move those pictures right next to each other. If I were to connect the dots, I get a graph which is basically the height of this mass as a function of time. It looks like this. So if I were to plot this legitimately on a y or vertical position versus time graph, you'd get something like this. It starts in the middle and it goes up and then it comes back down and this process repeats and you know what this is. This looks like a sine graph. But here's my question. Let's say we didn't start the mass in the middle and give it a kick upwards. Let's say at t equal zero, we start the mass all the way at the top and we let it drop. In other words, we do this. We start the mass up here. We let it drop. The graph would look like this. This time, if we were to plot this over here, we'd get a graph that looked a little something like this. It would start at the top then would drop down. It reached some lowest point. It would come back up. Get up here. This process would repeat over and over and over again. My question is, are these graphs the same or are they different? Well, they're obviously different but almost everything is the same. Their amplitude is the same. They have the same displacement from the equilibrium, that's the amplitude, that's the same. Their period is the same. The period is the time between oscillations, so the periods are both the same. The only thing that's different is that one graph is shifted compared to the other. In fact, check this out. If I were to take this green graph, the initial one, and just shift it left, we get the exact same graph. They're almost exactly the same. One is just shifted, and so the word physicists use for this idea that two graphs can differ by the amount one is shifted is the idea of the phase. We would say that these two oscillators are out of phase. How out of phase? Well, one whole cycle would be from here all the way to there. They're not that out of phase. They're just shifted by this much and that it turns out, for this case I've drawn here is only a quarter of a cycle. So you could say these are a quarter of a cycle out of phase or if you think of the unit circle, we know one quarter of a cycle would correspond to 90 degrees, either 90 degrees or pi over two. That's what these are. These are out of phase by pi over two radians or 90 degrees. So how do we describe this idea of phase mathematically? Well, if we were to try to write down an equation for this green oscillator, say t equal zero started there, I might write down that, okay, y is a function of time, the height of this oscillator is a function of time would be the amplitude times, since this is starting at zero, I'm gonna use sine because I know sine starts at zero, when t equals zero, of two pi over the period times the actual time. This little t is a variable. This little t represents the actual time at a given moment. So let's make sure that this equation actually works. If I were to start off at t equal zero and plug in t equal zero in here, the sine of zero is just zero, so this whole thing becomes zero and that's what I should have. I should start at t equal zero at a y value of zero, so that's good. And then as t gets a little bit bigger, this inside amount gets a little bit bigger, the sine of a tiny positive amount, be a tiny positive number, so that's why this graph goes up from there. Eventually, it gets to the peak. Where will it reach its peak? It will reach its peak when it gets through a quarter of a cycle. Remember, a whole cycle is this whole amount here. So a quarter of a cycle is just when it gets to the amplitude from zero and that will be at a time, t equals the period over four and the reason is that's a quarter of a cycle. Does this math actually give us that? It does 'cause watch, if I plug in little t, the time variable as the period over four, I'll get the sine of two pi over the period times the period over four, 'cause that was my time. The periods cancel out. I'd get sine of two pi over four is pi over two and pi over two, that's 90 degrees. Sine of 90 degrees or sine of pi over two radians, that's just one. That's the biggest that sine can be. That one times the amplitude is gonna give me a value for the height of the amplitude. This is gonna describe my oscillator perfectly 'cause it's gonna give what the height is at any given moment of time. So that wasn't too bad, but what do we have to change in order to describe the purple oscillator? So now, I wanna describe this purple oscillator. You might say, "Oh, well, easy. "Just make this sine a cosine" and yeah, for this case, it turns out that works, but pretend like you didn't know you can do that or if you wanna make it harder, say this was not shifted a perfect quarter of a cycle, maybe it was only shifted like a ninth of a cycle, then cosine's not gonna do it for you. You need a more general way to adjust how much this wave is shifted and that's gonna be some sort of phase constant in here. Where do we put the constant? You might think, "All right, if I take my green graph, "I wanna shift it to the left." You might think, "Well, should I subtract" "some, like, amount out here?" That's not gonna work. That's gonna take your whole graph and subtract from the value you get for the height, a constant amount every single time. Because of that, that will just take your graph and shift it downward. That's not gonna work. If we added a value of B up here, that's also not gonna work. That would just end up shifting our graph upward. All right, this B value is not gonna do it for us. So, you might want that in certain situations. That's not gonna shift the graph left to right. It turns out, what we have to do to shift to left to right is add a constant within this argument of the sine here. So, if I wanted to describe my purple oscillator, I'd say that y as a function of time is the same amplitude. Let's just, again, use sine. It would be two pi over the period times time and then here's where the phase constant comes in. I have to actually add a phase constant. You might think subtract. You might think subtract because we want it to go left. It turns out, adding a phase constant will shift the graph left. This was counterintuitive to me. That's always freaked me out. I'd always forget this. How come adding a phase constant shifts the graph to the left? Well watch. If we take this equation now and instead of putting phi in there, this is the symbol we use for the phase constant in general, phi, and in this case, we know what it should be. It should be a quarter of a cycle and for a sine graph, a quarter of a cycle is pi over two. So let's just see if this works. At t equal zero, we used to get zero, which is what we wanted, but now for the purple graph, I need to start at a maximum value. So at t equal zero, this whole amount right here becomes zero and I'm just left with sine of pi over two and sine of pi over two is one. That's a maximum value, so times amplitude would give us the amplitude which is what we want. We want a graph that starts at the amplitude. And this is better than just putting cosine or at least it's better to know about because now, even if this phase shift was pi over four or pi over nine or pi over 27, you could shift by any amount you want using sine or cosine. Now you know how to shift these things. Adding a phase constant will shift it to the left. Subtracting will shift it to the right. And the larger the phase constant, the more it's shifted. You don't ever really need to shift it by more than two pi since after you shift by two pi, you just get the same shape back again. So this constant in here, it's pi over two in this case. In general, it would look like this. You'd have some oscillator. It's got some amplitude. You could do sine or cosine plus a phase constant and this phase constant will determine how much this oscillator is shifted left or right. And I should say be careful. Physicists can be sloppy here and use the same word for multiple things. Sometimes the word phase is used just for this little part here, this little added constant part, but sometimes, by phase, people really mean this whole thing inside of here, this whole term that you're taking sine of because this is what's determining where you're at in your actual cycle and these ideas don't just apply to a mass on a spring. You could write down the equation for a wave. You have an extra term in here for space, not just time, and guess what? There'd be a little constant at the end that you could add. That would be the phase constant. So this idea of phase gives you a way to describe how two oscillators or two waves are shifted with respect to one another and it lets you account for all kinds of properties, like we had for this mass on a spring.