Example

by Patrick Ford
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Hey, guys, let's take a look at this example. So we're told that a 4 kg mass is on a spring. It's oscillating at two hertz, and it's moving at 10 m per second once it crosses the equilibrium position. So on a mass spring system what? 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 Max is equal to 10 m per second, so I'm gonna start writing everything out. I told that the mass is equal to four. I've got the frequencies equals two and I've got the max is equal to 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 max distance. So, in other words, how long does it take to get from equilibrium all the way out to its max distance? We know that that is one 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 tea over four, 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 F so t and the frequency are in verses 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 over four, that's just gonna be one quarter off one half of a second. And so that equals 1/8 of a second. So that's the answer to part A. So what is part B? Ask us Party asked us to find out what the amplitude is. Someone 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 is well, a is kind of president, 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 anything about the maximum acceleration. We don't know anything about times. We can't use these guys eh? So basically, I'm just gonna have to use thes equation all my max equations. So I've got I don't know what X Max is, because otherwise I would be the amplitude, and I don't know what the acceleration Max is either, but I do know what the V Max is. So let me go ahead and use that equation for V Max, because that's the one that I know most about. So I've got V. Max is equal to a times omega. So if I rearrange for this, I've got that A is equal to v Max, divided by omega. So now I have what V 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 the 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 two pi frequency. Do I know the frequency? Yes, Ideo. So that means to omega equals two pi times f. So omega F is just equal to two frequencies to hurts. So the omega is equal to four pi, so I'm just gonna stick that right back in there. So that means that the amplitude is just 10 m per second divided by right. That's the V max. And then I've got four pi, so I can just go ahead and simplify that and say that it's 5/2 pi, and I'm gonna go ahead and highlight that box it so you guys can see it. So I got five or two pi. Now, this last one is asking me to find a maximum acceleration. So now we're actually gonna go ahead and solve for a max. So which one are we gonna use? We've got this a max over here, but I need to know, okay? And I don't know how the spring constant. So instead, I'm gonna use not this a max. I'm gonna use this a max. So I'm gonna use the a Omega squared. So a max is the amplitude, which is 5/2 pi. And then I've got omega, which is four pi. So if I square this guy, it's just gonna be 16 times pi squared, right? So this is four pi, So omega squared equals 16. Pi squared. So now what happens is I've got a pie on the bottom. The nominator got a pie in the numerator, so they cancel. And then I've got a 16 in the numerator and a two and the denominator. So all of that stuff simplifies. So I've got a max is equal to five times eight times pi. So that means a max is equal to 40 times pi. And that's the answer. So you should definitely become familiar with all these pies popping up all over the place is, I guess, let me know if you have any questions. Let's keep moving on.