Hey guys, let's check out this example here. So we've got a 0.25 kilogram mass. It's oscillating on a spring. We're told the period's 3.2 seconds. Now we're told that at a specific position, the speed is 5 meters per second. So it's a lot of numbers. I'm just gonna start writing stuff down. So I've got the mass is equal to 0.25, the period's 3.2, at x equals 0.4, I've got the velocity at that specific position is 5. And what 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 A's in them. So let's take it from the top. You only use these equations when you're told something about the force or the acceleration. We're not. So we're not gonna use these, which means we can't use their max values either. Now, in these second row equations, we can only use them if we have a time to plug in. Right? Because remember these are 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 v max or a max or anything like that, so I can't use those.
So whenever all this fails, we need to use our energy conservation. So let's take a look at which equation I'm 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'd have to figure out what k is, which I might be able to do, but I don't have the mechanical energy. So that means I can't use this first part. Right? Okay. But what about the second equation? In the second equation, I don't have what v max is. So and I if I don't have that and I don't have the mechanical energy, I can't solve it. And in this last equation over here, 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 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 as a function of x is, and I also do know what that x position is. So all I really have to do in this case is solve for omega. So let's see if I can do that all over here. So omega is equal to what? Well, I've got this big omega equation I'm gonna use over here. I've got 2 pi frequency, and then I've got 2 pi over period, and that's equal to square root of k over m. So let's take a look. I don't have anything about the frequency, the linear 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, omega is just gonna be 2 pi over the period. I have the period as 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 radian or you're gonna get 1.96 radians 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 vx is equal to 5 and the omega is 1.96. And 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 to 0.4. Okay. So, now I'm just going to go ahead and divide over the 1.96 and it'll become 2.55, and then I've got the square roots of A squared minus 0.16. And let me go ahead and write that, 0.16. Okay. So now I've got this nasty square root in here. So to get rid of it, I've just got to square both sides. So we've got 2.55 squared equals the square so the square root 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 gonna be A squared. So go ahead and plug this into your calculator and then just remember to take the square root and we'll get an amplitude that's 2.58 meters.
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 going to 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 1 half k A squared. So let's start out with that one. So in part b, the mechanical energy I'm gonna use is 1 half times kA 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, 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 2 pi f and 2 pi over t, but I don't want that. But I've got that that's equal to the square root of k over m. Now in this case, I know what omega is and I know what m is, so I can 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 got to square both sides, so 1.96 squared is equal to k over 0.25. So that means 0.25 times 1.96 squared is equal to k. And so if you plug that in really carefully, you're gonna get, 0.96 as a k constant. Right? Newtons per meter. So that is what k is equal to. Now I can just go ahead and plug it right back into my mechanical energy formula. So I've got that this total mechanical energy of the system is 1 half, 0.96, and then the amplitude's 2.58, and you got to square that. So if you plug that in, you should get 3.20 joules.
Alright. So that's pretty much it. You could have also figured it out by using this equation, but it's just a little bit of extra steps. So let me know if you guys have any questions. Let's keep going on for now.