Hey, guys. Let's take a look at this example. So we're told that a 4-kilogram mass is on a spring. It's oscillating at 2 hertz and it's moving at 10 meters per second once it crosses the equilibrium position. So on a mass spring system, when it's oscillating back and forth, as it crosses the equilibrium position right here, we know that the speed is maximum. So that's what they're telling us. They're telling us that v_{\text{max}} = 10 \text{ m/s}. So I'm going to start writing everything out. I know that the mass is equal to 4. I've got the frequency is equal to 2, and I've got v_{\text{max}} = 10. And what I'm supposed to find is I'm supposed to find the time it takes to get from equilibrium all the way out to its maximum distance. So in other words, how long does it take to get from equilibrium all the way out to its maximum distance? We know that that is 1 quarter of the period. One full period is the whole entire cycle, so we're just looking for that quarter. So if we're looking for t/4, we might as well just find what t is. So let's just use our equations to find out what time is. Well, I've got the big omega equation down here, but I also know that t and the frequency are inverses of each other. And I have what the frequency is. So that means that the period is just one half of a second. But that's not what I'm looking for, I'm looking for a quarter. So if I got t/4, that's just going to be 1 quarter of 1 half of a second, and so that equals 1 eighth of a second. So that's the answer to part a. So what does part b ask us? Part b asks us to find out what the amplitude is. Let me write that down here. So we're supposed to figure out what A is. Let's look at all of our equations and figure out where A is. Well, A is kind of present in all of them, so let's rule out the ones that we can't. We don't know anything about the mass. We don't know anything about the maximum acceleration. We don't know anything about time, so we can't use these guys. So basically, I'm just going to have to use this equation, all my max equations. So I've got I don't know what x_{\text{max}} is because otherwise that would be the amplitude, and I don't know what the acceleration max is either. But I do know what the v_{\text{max}} is. So let me go ahead and use that equation for v_{\text{max}} because that's the one that I know most about. So I've got v_{\text{max}} = A \times \omega. So if I rearrange for this, I've got that A = v_{\text{max}} / \omega. So now I have what v_{\text{max}} is. I don't know what omega is, so let me go ahead and find that. So let me check that. I've got v_{\text{max}}. Now I just have to go over here and find out what \omega is. So let's use my big omega equation. Omega is equal to 2\pi frequency. Do I know the frequency? Yes, I do. So that means 2\omega = 2\pi \times f. So \omega f is just equal to 2. The frequency is 2 hertz. So the omega is equal to 4\pi. So I'm just going to stick that right back in there. So that means that the amplitude is just 10 meters per second divided by, right, that's the V_{\text{max}}, and then I've got 4\pi. So I can just go ahead and simplify that and say that it's \frac{5}{2\pi}, and I'm going to go ahead and highlight that and box it so you guys see it. So we've got \frac{5}{2\pi}. Now this last one is asking me to find the maximum acceleration. So now we're actually going to go ahead and solve for a_{\text{max}}. So which one are we going to use? We've got this a_{\text{max}} over here, but I'm going to need to know k, and I don't have the spring constant. So instead, I'm going to use not this a_{\text{max}}. I'm going to use this a_{\text{max}}. So I'm going to use the a \omega^2. So a_{\text{max}} is the amplitude which is \frac{5}{2\pi}, and then I've got omega which is 4\pi. So if I square this guy, it's just going to be 16 \times \pi^2. Right? So this is 4\pi. So \omega^2 = 16\pi^2. So now what happens is I've got a \pi on the bottom in the denominator. Got a \pi in the numerator so they cancel. And then I've got a 16 in the numerator and a 2 in the denominator. So all of that stuff simplifies. So I've got a_{\text{max}} = 5 \times 8 \times \pi. So that means a_{\text{max}} = 40 \times \pi, and that's the answer. So you should definitely become familiar with all these pies popping up all over the places. Let me know if you have any questions. Let's keep moving on.
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17. Periodic Motion
Intro to Simple Harmonic Motion (Horizontal Springs)
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