17. Periodic Motion

Energy in Simple Harmonic Motion

# Example

Patrick Ford

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Hey, guys, let's check out this example here. So we've got a 0.25 kg mass. It's oscillating on a spring. We're told the periods 3.2 seconds. Now we're told that at a specific position the speed is 5 m per second. So it's a lot of numbers. I'm just gonna write starts, start writing stuff down. So I've got the mass is equal to 0.2 25 periods, 3.2 at X equals 0.4. I've got the velocity at that specific position is five. And now we're supposed to do is figure out what the amplitude of the system is. So I'm looking for capital A. So let's go to my equations. Right. But unfortunately, almost all of them have Aisin them. So let's take it from the top. So you only use these equations when you're told something about the force or the acceleration. What we're not so we're not gonna use these, which means we can't use their max is either. Now, in these second row equations, we can only use them if we have a time to plug in right, because remember these air functions of time now we don't have a time to plug in. Which means we also can't use the max values of these equations. Either we're not told anything about the max or a max or anything like that, so I can't use those. So whenever all this fails, we're going to use our energy conservation. So let's take a look at which equation and specifically gonna use. So if I'm looking for the amplitude, that's gonna be capital A and that appears in both of these equations. So in this equation, I have the mechanical energy that I'm gonna need to know. And let's take a look at all of these components here because I've got a bunch of different equations so I could figure this out. The problem is, I have to figure out what K is which I might be able to dio. But I don't have the mechanical energy, so that means I can't use this first part. Right. Okay. What about the second equation in the second equation? I don't have what v. Max is so and I don't have that. I don't have the mechanical energy. I can't solve it. And in this last equation over here, um I don't have the K, and I don't have the mechanical energy either. So let's just take a look at what I know, right? I can't use that. But I do have this velocity as a function of X. So let's take a look at my equation at the very bottom. So I've got this velocity as a function of X. So maybe I can use this one. It's also the simplest one, so let's just try it. So let me write it out. I've got that. The velocity is a function of X is equal to omega times, the square root of the amplitude minus the X at a particular position squared. So let's take a look. I've got my amplitude that I'm trying to solve for And I do know what this velocity is. A function of X is and I also do know what that exposition is. So all I really have to do in this case is solved for Omega. So let's see if I could do that all over here. So Omega is equal to what I've got this big omega equation I'm gonna use over here. I've got two pi frequency and then I've got two pi over, period, and that's equal to square root of K over em. So let's take a look. I don't have anything about the frequency, the linear, your frequency. But I do have something about time. I do have one of those variables, and I also don't You know, I don't really know anything about K, so let's just not even worry about that. If I'm trying to figure out what Omega is va Omega is just gonna be two pi over the period. I have period of 3.2 seconds. So I've got that right over here. Let's just go ahead and plug that in. If you do that, you're gonna get a radiance, or you're gonna get 1 96 radiance per second. So I'm just gonna plug it back into that formula. So now I'm just gonna start plugging numbers in because I know what all of these things are. The V X is equal to five, and the Omega is 1.96. Then I've got square root of a squared minus this particular position squared, right? That's 0.4. I'm told that X is equal 2.4. Okay, so now I'm just gonna go ahead, divide over the 1.96 it'll become 2.55. And then I've got this square roots of a squared minus 0.16 and let me go and write that 0.16. Okay, so now I've got this nasty square root in here, so get rid of it. I just got a square, both sides. So we've got 2.55 squared, equals the square. So the square, we will just go away, and I've got a squared minus 0.16. So that means that 2 55 squared right? Plus, that 0.16 is going to be a square. So I couldn't plug this into your calculator and then just remember to take the square root and we'll get an amplitude. That's 2.58 m. So that is the answer to part a. Let's take a look at part B now. So Part B is asking us to figure out what the total mechanical energy is. So let's go to our equations Now, Fortunately, this is pretty straightforward. We know we're gonna have to use this mechanical energy equation now. It's just now The question is, which part do we use? So let's take a look. I've just figured out what this amplitude is. I've just figured out what a is. So if I wanted to figure out the mechanical energy, all I have to do is figure out what the K constant is. Okay, so that might be a place to start now again, I don't know what the V Max is, so I can't use that guy. And what about the third one? Well, in the third one, let's take a look. I have what the position is. I have the mass and the velocity. But in order to solve this, I'm gonna need to figure out what K is. So if both of them I need to figure out what K is, I'm just gonna use the simplest one the one half k squared. So let's start out with that one. So, in part B, the mechanical energy I'm gonna use is one half times k a squared. I just figured out what a is. All I have to do is just go over here and figure out what K is. So how do I figure out K? Well, I've got that K. I can find out if I have the, um, the forces or the acceleration, but I don't have any of those, so I can't use that. So what I could use is I could use this omega equation, right? So I've got that Omega is equal to two pi f and two pi over tea. But I want that, but I've got that's equal to the square root of K over em. Now, in this case, I know what Omega is, and I know what em is. So I could go ahead and figure that out. So Omega I know is 1 96. We got square roots of K over 0.25. That's the mass. And then I've just gotta square both sides. So 1 96 squared is equal to, uh, k over 0.25. So that means 0.25 times 1.96 is squared is equal to K. And so, if you if you plug that in really carefully, you're gonna get 0.96 as a k constant right, Newtons per eater. So that is what K is equal to now. I could just go ahead and plug it right back into my mechanical energy formula. So I got this. Total mechanical energy of the system is one half 0.96 then the amplitude is 2.58 You gotta square that. So if you plug that in, you should get three points, 20 jewels. Alright, so that's pretty much it. So you could have also figured it out by using this equation. But it's just a little bit extra steps. So let me know if you guys have any questions, let's keep going on for now.

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