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Let'$ explore why the spring model in part 3 was invalid, starting mathematically- Comparing the commonly used value for the speed of sound in air (c 343 m/s) ...

Question

Let'$ explore why the spring model in part 3 was invalid, starting mathematically- Comparing the commonly used value for the speed of sound in air (c 343 m/s) and the theoretically determined value of that you calculated using the not- quite-valid resonance model (you should have Cth 286 m/s), we can write:C =Cth =Vv4.1: We can define Y to be some arbitrary constant value that satisfies the above equation. Solve for Y From a purely mathematical standpoint, then_ it looks like we need to inc

Let'$ explore why the spring model in part 3 was invalid, starting mathematically- Comparing the commonly used value for the speed of sound in air (c 343 m/s) and the theoretically determined value of that you calculated using the not- quite-valid resonance model (you should have Cth 286 m/s), we can write: C = Cth =Vv 4.1: We can define Y to be some arbitrary constant value that satisfies the above equation. Solve for Y From a purely mathematical standpoint, then_ it looks like we need to include some multiplicative factor (Y) inside our resonance calculation to make it valid. But what does this Y term mean physically? Recall that within any closed system_ the amount ofenergy is a sum of the system'$ internal energy, work done on the system by its surroundings, and work done by the system on its surroundings



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Consider the mass attached to four identical springs, as shown in Figure 5.7 (b). Each spring has force constant $k$ and unstretched length $l_{\mathrm{o}},$ and the length of each spring when the mass is at its equilibrium at the origin is $a$ (not necessarily the same as $l_{\mathrm{o}}$ ). When the mass is displaced a small distance to the point $(x, y),$ show that its potential energy has the form $\frac{1}{2} k^{\prime} r^{2}$ appropriate to an isotropic harmonic oscillator. What is the constant $k^{\prime}$ in terms of $k ?$ Give an expression for the corresponding force.

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.

But okay, So far apart, they're here. We have that. Our equation of motion because there's no damping is given by M times wide over prime plus k times y equals zero. Sorry Equation of motion with the values of M and K is wide, double prime plus 25 y equals zero and the general solution he's going to be given by why equals C. One Times Co sign of five p plus C. Two off sign times five t. The particular solution will be why P is equal to 0.3 Times Co. Signed five t plus point 02 sign of five t They should actually be a minus 0.2 times Sign of party. Okay, now for Part B. How we will attain the minimum time required for the spring. To reach the equilibrium position is by selling y equals zero or Y p in this case equal to zero. Then we can solve for the minimum time, and it will be approximately point 301 seconds over part. See, the frequency is given by the inverse of the period of the motion. So that's one over to pie over five, because five is the value here inside the argument. And so this is equal to five over to pie, and that's the frequency in seconds.

In this question of your use TV uncertainty Preschool To finally see minimum energy off the harmony oscillator. So GST energy expression for you. I want else later. Well, we gotta find what, his pian lunch you many moment very or P and X And we can't do that. You think the uncertainty principle That's a Peter start. Our X must be greater or it goes to over to nobody. Really? These two together. Well, we know that the average momentum and average ah, position must be so right. It's symmetrical about zero. So the p x sorry dp square value can be approximated by the uncertainty. The answer Did he scream Same way 50 x squared value. Now what we want is to substitue in what is x squared in terms off peak. So we know that from this expression that ah dot are x This must be Greedo. It goes too far over two times still top p. So for a minimum X, this is basically, uh, don't tar square the minimum it's going there seeing spots square over four times Delta P square. Now we substitute this in into our expression. 40 energy goes to peace, girl for two m look Kill the two times X squared, which is ish bars gray over four times piece way. But I'll be briquettes. So this is our minimum energy because this is our minimum X square. So this is energy. Must be great. Oh, equals two peace crap over to end, plus key ish bar square over it. Pete. Scrap No, to actually minimize this function if respect to pee key Ah, you're different. Shipped with respect Toe piece square. It says if you medium ice p square were obviously minimizing p meaning my step we differentiate and every expression by P square Get one of the two m plus key ishbo her eat times. Uh, negative one one over p to put full. So to get the minimum point, you must create this to Ciro. So we get ah, people for goes to tow em over eight times K h bar square, she becomes P square. Must be equals two bar time. Skirt off. Aim time skated by two. So this is the minimum value off P square. Obviously very or P square, which will give us the minimum energy. So substituting this very api square into oh initiate expression. What's your minimum? Energy goes to, uh, two skirts off key. There's two in peace. Go to him, plus K Shebaa Square. But if I eat over P square, so this is to offer each bus screwed em key. So after opening up the market and simplifying a bit catfish over four times care and for both terms, so both you can see any potential energy all the same, actually, finally we get to times square it off cabling, and if you remember, kill them is equals to make us great. So they sexually Hisham Omega over to So the medium of energy is half issue omega.

Okay. Hi friends. For free oscillations of the system to be damped. Oscillation so Omega one sculptor root off America not square my desk. Well am I squared by four and mega not into wrote up on minus one upon for two square three christian. But the verdict is given. Mhm. Mhm. Yeah. two days to Omega It called to make governed by five. So omega is called omega not plus minus May govern by 10. So the frequency the average power. So at that frequency. Yeah. Yeah. Rich power. His half the maximum power. Oh okay. Mhm. So we can write Mhm. Have not squared when we were not Upon two K. Q. Into burn upon omega not by omega minus omega but omega not only square Plus one upon QE Square. It's cool to have have not a square cube upon to every. Maybe not Simplify it and we can write one upon omega not by omega minus omega. Omega not wholly square plus. Burn upon Q square equal to half Q square solving every question we can right one upon chewy square is called omega not by omega minus omega by omega not Police credit. This is your 3rd question from, we will find omega, one is square upon omega, not square is called tobon minus upon upon for QE square, that is four Q square minus burn upon or the square. So you you will get okay one upon 4 into 1 -1 Omega one upon omega not it's quick. Yeah, So from three and 4 we can write uh huh, fall into one minus mega, one is square upon a mega notice where it's got a mega, not by omega simplifying for this equation for omega vernon, omega you will get or maybe or not, one is called mhm the one point judo judo fight or maybe I want the world, the factor of the system, Jewish cult, 01 upon to route of one minus mega burn upon omega not only square. So you will get to into root talk on minus Born upon 1.005 is square. So it is to be 5.0 and see part damping factors is called to them into cover that is m into omega not by cube that s M into root of came by em. Yeah. Into one. Very cute. That is point to root of game. Yeah, we have to plot up graph. Yeah. Yeah. No. Okay he fought or it's called me or not a maximum. It's slightly off by Omega one as Are approaching 20. So from one weekend right he upon have not by K. D Escoto. Omega not by omega upon the rooftop America not America. My America by America not square blocks gone upon to square. So the draft will be like this amplitude versus time graph. Yeah. Mm hmm. Yeah. And they're so thanks for watching it


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