What? Ah, good day, ladies and gentlemen. Uh, again, we're looking at problem number 33 here. And in this problem again, we're not really emphasizing the solving method so much, um, or more, uh, setting up a problem and then looking at what we get, um, we get it. So a lot of ah, this is actually using things from 32. I didn't technically need to do I'm here, so I'm just sort of grabbing, um, information from 32. So if any of this stuff you're not aware of, it's because you haven't looked at problem number 32 or you don't know how to solve, um, previous problem. So I'm sort of just summarizing pretty much what you didn't regard. Um, so in the first case, in the first part of it, when you set up the ordinary differential equation, um, in the way it is away, how can I put this? So the 10 here, which is as ah, mass of the object. Um, this is tthe e m. Why? Double prime term? In other words, this is the force term. Um, this here. Ah. See, remember, correctly, um is is this the k or you. I'm getting now. Which terminus is, um So once the K and one's the beach? No, I This script, I can't remember what it is. Um, was variable that I should have it here. Okay, so, uh, this is really the I think that. See you. Yeah. So the 60 here. Sorry. This is the B y prime. And it's instant, um, I guess there's accent. It's, uh it refers to the the damping term, um, which which comes from friction and then the stiffness term. McKay, uh, that comes from this spring. So this is sort of the spring constant. So you just I mean, the center of the problem is really to just input those terms into the equation. And the previous one, um, has no, um is un down. So this is a case for you have damping. The other one is undammed, and you're sort of twisted pair between the two of them. What happens, eh? So the first case. So now if we look at what the difference equation is, um yeah. Ah, Uh, uh. Okay. So what I want to do is just mention here, um, that when you go through the boy solving and everything you're going to see. Um, this is your equation here. Um, and then we'll me, we want to talk about setting it to zero. Okay, um, we just sat y equals zero solve. And what you get is this tangent 42 b negative three halves and then that tells you that t is equal. Thio this. But you're, um the, uh This is not four divided by pi, by the way. It's for pie. It was supposed to be four pi. This number is negative. Um, which doesn't mean anything, but you can add four pi to it. You can ask basically, by adding numbers a pie multiples of pie to this to this initial value of tea I was looking for the first time is that, um, this would be positive, so I can't remember what the number wasit was. Something like negative 14 or something. Um, which was, like, negative 14 seconds or something. And so, by adding four pi to it, I get back. Um, the 1.6 tree, which is the first time it adds to its positive. So this tells me basically that in 1.63 seconds, it gets back to zero. Okay. And, um so that's the first time it it passes. Zero. Um, and then the frequency here, as defined lips in the previous problem, the frequency comes from the, um hola. Say this the B term. So are the beta term. So remember, when you solve this ordinary difference equation, you're going to get a e l tha t sometimes, you know, co sign bait. Ah, t a plus sign beta t. So, um, the frequency, as defined in the previous problem, is dis beta term. And in that case, it is actually beta divided by two pi, which, in this case, just to over pi. Um and it makes sense in the other problem, because it's just a It's just a non oscillation population. Eri um how would you say that? Like a looping spring. So spring all the springs really doing is going, you know, something like lumps, uh, Springs. Just going back and forth. Boy is terrible. Uh, yeah. Okay. It's not gonna do what I wanted it to do. Essentially, what I want to do is this. So spring starting year going over here, you know, sort of going back and forth. So the oscillation here or the frequency basically is looking at How long is it, Uh, how long is it going going to take to get back to here? In that case, it makes perfect sense. Is has pie. Um, in this case, um, in the damp term, because what's happening is the spring will start out sort of going wide, and then it will sort of damp and so sort of decrease in some sort of go eggs. This basically sort of like that s o the the, uh, what you call it. It's not measuring how long it's taking to get back at this point, but how long it's taking to go from its furthest 0.2 between its two furthest points of oscillation. In this case, it's, um, you'll notice that in the undead apt frequency or in the undammed, um, case that has a higher frequency habits. Because this guy's losing energy with time. Um, then in the damp cave. So So this case is losing energy, going to the damping. So, um, and that's that's that's really it is that the damping term basically causes the system to lose energy because it's official. Um and that causes it to lose energy, which causes the motion to just sort of fall Go to zero. Okay, so that's really it for here. Um, you know, I I had to read, um, about this and section, um, 4.1 and then a little bit more. 4.3. So, um, you know, I it's probably useful to actually read about how what these varying terms mean and so on and so forth. It's probably helpful pouring a little bit about it, but, I mean, it's basically what it tell it gives you is a little idea of what to expect your solutions toe look like, even if you know, you don't necessarily care. I got but, uh okay, thank you very much.