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A mandolin string (the part that vibrates) is 32 cm long, and is held in place at both ends; The spced of transverse waves in this string is 282 mls Using the lines...

Question

A mandolin string (the part that vibrates) is 32 cm long, and is held in place at both ends; The spced of transverse waves in this string is 282 mls Using the lines below (which represent the string), draw (any) two possible standing waves that could be produced on this string#ke's intinie # 0f Standing wave A: PorciVae , (weit bit Vu need nodej VoTH ends QVV # (incL: $) O addition nodes in t mlddle Standing wave B:Find the wavelengths of the two standing waves that yOu drew above Dape

A mandolin string (the part that vibrates) is 32 cm long, and is held in place at both ends; The spced of transverse waves in this string is 282 mls Using the lines below (which represent the string), draw (any) two possible standing waves that could be produced on this string #ke's intinie # 0f Standing wave A: PorciVae , (weit bit Vu need nodej VoTH ends QVV # (incL: $) O addition nodes in t mlddle Standing wave B: Find the wavelengths of the two standing waves that yOu drew above Dapendf wat You dreu; Vt whOuld Ve (4Cm Whek e n"i voe ikrr Find the frequencies of the two standing waves that you drew above Dopendy wlut , Vuu drew i pyevidv Q bkt Qner houd he 282 MS F done vit #n freq jhuud Vx Merdert . fb =



Answers

A banjo string $30 \mathrm{~cm}$ long oscillates in a standing-wave pattern. It resonates in its fundamental mode at a frequency of $256 \mathrm{~Hz}$. What is the tension in the string if $80 \mathrm{~cm}$ of the string have a mass of $0.75 \mathrm{~g}$ ? First we'll find $u$ and then the tension. The string vibrates in one segment when $f=256 \mathrm{~Hz}$. Therefore, from (\mathrm{a}) \text { : }}$ and $$ $$ \frac{0.75 \times 10^{-3} \mathrm{~kg}}{0.80 \mathrm{~m}}=9.4 \times 10^{-4} \mathrm{~kg} / \mathrm{m} $$ Then, from, $v=\sqrt{(\text { Tension }) /(\text { Mass per unit length })}$, $$ F_{T}=(154 \mathrm{~m} / \mathrm{s})^{2}\left(9.4 \times 10^{-4} \mathrm{~kg} / \mathrm{m}\right)=22 \mathrm{~N} $$ \begin{array}{l} \frac{\lambda}{2}=L \quad \text { or } \quad \lambda=(0.30 \mathrm{~m})(2)=0.60 \mathrm{~m} \\ v=f \lambda=\left(256 \mathrm{~s}^{-1}\right)(0.60 \mathrm{~m})=154 \mathrm{~m} / \mathrm{s} \end{array} $$ The mass per unit length of the string is

Is this question? Perhaps this wave that has a peak to peak institute of one centimeter. It has efficacy of 100 per second. It has the in your masters. You have 120 grams per meter in san friendly attention off 90 newton. But a asked us to find the maximum transfer speed of a point. So we already did this equation in chapter 15. And the equation is why n times omega. But I think it will be helpful due to part B to actually write out quickly. Right out. Um said rubbish again. It's because we have This is this we've can be described by one in a sign. Okay, thanks. Minus Omega team. So where would want to hurt you? You is a derivative of white with respect to t. So it's why in I like to fight him. Oh, Mika, I will sign. Okay, Expand Islamic a cheap. So it's maximum value will be. Why am times omega when Coulson equals negative one? Okay, so is this case we have peak to peak is one centimeter. So the amplitude is half of that is five millimeter. We also need to find out what Omegas omega is two pi times frequency, which is 120 hearts. So this gives us a maximum transfer speed of sir 0.7 meter per second. Well, extra pork meat circumstances. Second, we want to play so largest transfers component of tension. What does that mean? So we have tension intentions, always a long history. So but then there is this horizontal component of tension. We want to find some maximum core presentable component of sis tension. Hold on, he finds out. So this can fund if we it's let me draw. Oh, zoomed out across. So for every single component there is a white Sirius X in this tango Seita, um, this in casita can be described by Actually, it's he was a different angle. We can use this bottom ankle Sit up. It's a single set up camp described by tension. Suit up equals gente y over. Don't X person. How is tension related to Cecilia? It's attention is, um say this tension has a transfers component. Two says the transfers component. Sorry, this is the transfers component of the tension, and this is a direction off the tension. So the component of tension in this direction we'll be tension in the transfer. US direction equals total attention Times science Ada. Okay, so we need to use to you I d x. Like we will take the derivative off this equation since time was respect to X to find what tension Seita is. And then we will use that tenderness, ada, and we will take the maximum value off that, um it takes a maximum value of Seita to find what, um, tendon transfer asses. So the top of transfer asses Okay, so, um, let's first take sister relative If we take this with respect to X week, it okay, times Why him? Times sign Co sign. Can you explain this Omega T? So it's a maximum value you could possibly obtain is okay time. So I am so tensions Satyam Maximum, which is also say the Maximus ease. Tension is more autonomously increasing. So you don't equals injuries. Tension off okay, times while I am so we need to find kay first. Then we can find satar. What is kay? K? Is to pi over Linda. But we also do not know what NMDA iss. Oh, so we need to because say okay. He calls to pi over Linda. What's in Linda equals, Um he over? Yes. So because of that, OK equals omega over be. And that's probably these uses expression which he used for now. Okay, so omega is troop I f he is square root, tall, over mute and then tall from you. Is this whole sing again? Times Why? So this impressed tension up to pi Times 120 hearts over square root of 90 Newton over zero point Want you kilogram per meter times Why am is quite a communicator. Our calculator tells us this is Ron 7.8 Seder. So that is a maximum say that we have. And so Laura is a say dies, The larger the transfers component of the tension is so the largest transfers component of tension happens when say that you cause 7.83 degree inside is 90 Newton on site. Uh, seven point. It's three degree and it is 12 points. Three Newton. So we see that, um, here it's actually fascinator and say that itself was actually pretty similar because, um, cedar is very small. Also like sincere intention Sita So don't differ that much. Okay? Now we can move to part see what has part C. C. What's he says. We have to show that the truth. Maximum values, calculate above. Um, they were occurred. It's the same. Free its value for the way. And then we ask, What is the transfers? Displacement off? Why of the stream that these faces? So that is asking us to you. Fine, since the place Well, firstly, it's easy to say. Why's it occurred at the same face? Because boats of some. If we look at this, both of them are do provocative Off this wave equation one is with prospective tea and wise res respect your ex and both of them get their maximum one co sign K X minus omega T get to a maximum. Um, so that is the explanation for part C. Um, it is if we add a five over here and we were out of five over here, the conclusion is still the same. There will be It's a face. Whatever faces that maximize this co sign K X minus omega T plus five. Because that's the first part off. See, the second part of C is that we won't you find out a displacement of why at set point, That is also not so hard because we have co sign. Okay, X minus omega T pass by equals Clouseau minus one. It's these points. And if co sign equals puzzle minutes one who knows that sign K X minus omega T plus fi will be equal to zero. So why e calls? Why equals against this equation? Why? Because when I am times, Times something well, I will equal to zero as well. So, um, part c transfers, displacement. Who? I will be zero. Okay, not party. What is the maximum rate of energy transfer along the street? Um, with maximum rate of energy transfer along the street, well, we can find what, um, he is. So it is. He equals work. So that's like the rate of energy transfer. Um, possessions. Amount of work is similar, so we can use work equals 1/2 of envy square. But it's easier since we already finds up the transfers. Tension is equal to, um, punk transfers. So this is it Transfers tension time still away because force times distance is the amount of work you do isn't over Delta T. So what does it become gentle. What over Egypt? T is you. So this is Tom transfers times you and this obtained a maximum when most how you gets a maximum, which we know were occurred at the same time. So this is equal to 0.3 newton times frequent a meter per second equals 46. What market? So since I'm running out of space, I'll get you a new page. Um, let me just make sure I have all the answer over here. 40 part see, and party. That's baby CD. And not I will start a new page to do the rest of the question. Okay? No, it's time to look at heart E for Part E. You said, What is the transfers? Displaced woman? Why wins this maximum transfer a curse? Well, that is just there, O meter, because we know that's a maximum Occurs when boasts, um, bimbos, um Z transfer. Still honesty and the transfers. Attention gets to the maximum, and we know that wins out happens. The displacement is zero. So here, with the maximum power, it's about time. Displacement is also sickle part s part f. We ask what is a minimal Rita vintage transfer a long history. Okay, again, we can go back to use ah past equation. Well, actually, we original. What it is is probably zero because, um, he is again. It's you times tension you times tension and both. So you and they're transfers Tensions that have a co sign cakes. Puzzle madness will make a t possibly have a Cole sat inside. So previously when we were looking for the maximum body was set off, these coast has to be one. Well, well, give us a minute. Workout. Well, purchase system to be zero. It sends a total power will be zero. And you can also just visualize it When the wave is here, there is no transfers component door. Everything's like horizontal, so we don't have any transfers. Work going on. Oh, f g. And another way of thinking about it is that it's these points, you know, like it's a maximum place. Um, so instantaneous speed zero. So we don't have any kinetic energy, so because we have no energy, um, he's this case like there was no energy and there is no power. So again, somehow be zero off these peak points. Finally, um, is part G and if a party would ask. Well, what is the transfers? Displacement. Why was this minimus about new happened while when co sign is equal Thio zero sign K X minus omega T plus I will be equal to Maximus. So if we go back to this equation, why will be at it's a maximum value possible when these covers and he could actually be most positive on negative. So because you have no tips, a maximum value which has posted a connective, um, 0.5 centimeter. Okay, so is used to Arza solution for this questions. This is partly Judy is a part e to G.

So we know Dad's the combination off traveling ways in this case gives gives us standing way that is forming sound on the given guitar. So why off Accident E, which is with a standing way they function, is equal to buy one for the traveling wave on the reflected wave wave function which is why to accent e and we know that one want people's see Sinus only that Wandy and then Sinus que Ron eggs. And for the seconds we have T c Sinus, I'll get to the Sinus. OK, do backs. So it seems that the forms for these two ways are for himself, Stan. Anyways, and this is the combination of the standing with So in this case, we know also that only the one is related to the way number. Okay, one, and also that this stands true for the on the to. So the string is fixed on both ends. So all for all her Monets, no matter committed them. There is the ends Our noses of do knows that their shirt for every harmonic and they just made it The first harmonic will help to notes me that the first carbonic is going to be only half for the full installation on one half of you over the full wave. Therefore, the second Chronicles that'll fit the whole full wave. And for the second car, morning, the Wayland. Big for the length of the strength. That means that but for the party of the problem, as we know the notes first globally equals zero. That that's the place where the position where this zero for anything. And since we know that that we have to ends and strings for comment, no matter, no matter what kind of way for is here, this will be no. And this will be X equals zero, and this would be axed equals l Therefore the answer from two parties. It's for the mental frequencies at random, first fundamental frequency because notes only at the end, what other has other knows, Plus these on the end. So for the first one, there are only two notes, and these are acts equals zero, and eggs equals l for two part b of the problem. Like I said previously, since the first harmonic will have two notes, then only one half of the bay for will fit on the length of the string. Therefore, because to know it's basically means this. This is the first notice, the second and the first harmonic Locally. This and this isn't helpful for pull for me something like this, where we have tree notes. So then and this means that the second harmonic will be the one that's the hold cost. Actually, the whole wavelength off this May on the string and it'll help tree notes, and the snows will be zero they'll have on dealt. So the final answer is X equals zero l do l over do and or one car length and the whole length This is these air positions for the notes off the second harmonic. And this do the answer for the part B enough for part C willing to graph certain situations so and equals zero. And also 18 on the fundamental first frequency, Then one fort over the fundamental forest frequency then created from the mental first frequency and one half of from the mental first pregnancy. So for the first, which is to AIDS, this will be the sketch of the graph. So this is X. This is why displacement, uh, their units with investment So who on Go and Mike Swan minus two. And you're here. Units off, blinked to tree and so on. So for just 1st 18 or the first to the men from since it is divided into AIDS from therefore for the first for the first harmonic, the baby form would be something like this, but no. For the first time. Morning. But for for 18 for the first demonic, this would probably probably fit the eight harmonic, but her mind. But the frequency low over eight times. Then the first. Good morning. Not great. Very thanks to the first carbonic. Then for the one border frequency off the fundamental first demonic, we'll draw the same time. Oh, the graph. These units of link the recording. So this will be just one actually. This for this tree times at one pregnancy. DeGraff A little something like this. Minus dude. Andi in itself land here, one to treat. And here we will have exactly on between wanted to for him down bad and then the rising here and then from here, whether something social for dropping exactly Austin Treat. And for the final, she's one car fundamental frequency sketch will be like it is here, in a very form, will go down and tree will be here. So is it the sketches for these four cases that the lesser than the fundamental harmonics? We can see the debate forms? They're not clear, like when we actually care. First, Carmona's second harmonic preacher full. They're basically costing full or semi full constellation. So the displacement off each particle of the string, bad user quarters and eights and house off this first for the mental pregnancy. Also, for for the frequency for these last given the frequency the ground can be, it can be sketched like this, and it's in the from the 1st 32 total on the bay. Then now and then it will go up here. It's a nautical, gleeful full bay for and for the part deal. The problem. What's the Samels air standing to extending weeks? Well, the answer is no. Some of the two standing ways on different frequencies will not be a standing way. We see from these graphs, sketches, photographs that the way form certain, not symmetrical. We're Cheryl locations. All the notes and the anti notes will very in time through different in time. And this is actually the answer 44 for this part of the problem. Because when when the varying time, then the baby section moving. You can convict. Imagine, just simple traveling Wait or is entirely away boats. This is why this is X on the tee at some tea. Do you one the buses it is. And it has nodes, Cree nodes to auntie notes and in die empty too. So the X This will be why this way for will be, for example, here, convicted for position. So and this is a traveling wave. So the same captains we ought to together to standing waves that have different frequencies. We have noted nothing what's changing their places. And basically we can consider that as, ah, traveling way. Even though there is reflection and there is superposition, different outcome is traveling ways.

All right. So, uh, in this problem, we have a string attached on the force. So this is a P. This is a cue. And according to the problem, we know that the density of the rope is a 0.4 kilograms meter. So there's a name. You is Ah, 0.4 kilograms a meter. And ah, we also know that the Speedo transfers way for the rope is Ah, 12 meters per second. So these 12 meters per second and then ah, impart a We want to find out attention to rope. So we know their relation that Ah, the velocity of transfers Waving the rope equals guru t over me. All right. So teased attention and muse the density so we can't end attention based on this relation. And ah so t kuo Ah, according to this problem is 12 squid times 0.4. Right, So let me should never see the result. So 12 times 12 10.4, which is a 57.6 Attention forces of 57.6 notice. All right, so there's a party, eh? And part B, we want to find out Ah ah With what frequency must a rope vibrate to create the transverse waves Waves that way. Blind self, two meters. So ah, because we already have to waive the speed of the wave which is 12 meters sec it. And now that we also have the wavelengths which is two meters. So we have turned that to the frequency Kuo, the velocity over the wavelength Wait right So this equal seekers six herds and then suppose we have to Ah, we have to supporting studying wave on wave less full meters at 3.2 meters and who's harmonica numbers are consecutive integers. So I want to find out the lens of the rope. Excuse me. So ah, use the other page. So we know that it will have a standing wave in a rope. It'll be something like this rank. So this is the lowest, lowest vibration mode and this is the ah So as you can see, it's supposed to less off the robots out. Right, So this Ah, the wavelengths of this one will be equal to ah to our over one day. And this one will be, ah to well over two. And ah, if we have one more higher, so it'll be like this. So this will be to over three. All right, So if you look at the denominator 123 These are the vibration mode number and ah, since ah, for the ah witless Landa equal four meters and the 3.2 meters. Ah, this number is the consecutive. It means that we can just assume that this number is and 44 meters and then plus one for 3.2 meters. Right? Then we could just equation to our over end equal four meters in a two hour over n plus one. They quote 3.2 meters. All right, so we can't because we have this two questions and we we we have two women verbals Aylin. And so we can resolve this to all known variables and pretend that l equal eight meters and equal fool. Excuse me. So, uh, let's see. So the lens of the rope as we found here, unless the rope is eight meters. Okay, So see part II ail April 8 meters and ah ah, the second part, the mass of the rope. So now that we have the lens of the rope, the mess of the rope is easy. We just utilize m equal a lot, has me. All right. So Mary is the density of the rope. It's a muse 0.4. So this gives you a 3.2, uh, kilograms, right? And ah, for party. We didn't find a homeowner number off the four meters standing wave. So this is the one that we found over here and before. All right. And the party on the diagram above draw a sketch of a four meter standing wave labeling the nose and enter notes. So because ankle four for the four meters standing wave. So come back to the diagram. We need to draw the ankle. Four ankle. Four case, we will have three notes inside. All right, so 12 and three. So the wave would be like this. Go up and go down like this, right? And because they did a standing wave. So just the vibrates. Ah, without moving. Hurry

Good day in this question we are given the waves industry. So first we are asked to solve for the speed and transfers waves. So speed in transfers wave is equal to the square root of then shown in the spring, divided by the mass per unit land. So our tension given is equal to need to eat. Point to your done. Our mass is equal 2.5 grabs, Converting two kg. We have 1000 g for one kg is equal to five times 10 to the negative for a kg. Our land given is 50 cm converting to meter. We have one m is 170 m. Therefore 0.50 m. Now substituting the parameters to the transference wave speed we have speed equals 88.2. New done divided by five Time stand to the -4 kg divided by 0.5 m. Therefore the speed of the transfer S wave is equal to 200 97 m/s. No, we are asked to solve already frequencies for each fundamental. So using the figure 22 dash to we have the fundamental for for the fundamental. The first overtone and the second overtime we need to find the frequencies. So for the fundamental Length is equal to 1/2 lambda. Since our land is .5 m Lambda or the wave land is equal to one m. There were substituting the Waveland to the frequency equation of frequency. It was speed divided by the Waveland where our speed is the speed that we got from the previews You have to 97 m/s divided by one m frequency. For the fundamental is equal to 297 Hz. Next for the first overtone racist overtone. Our land is equal to two times one half lambda. Therefore our Waveland or lambda is equal to why five you through this. So solving where the frequency we have speed divided by Waveland. It was 297 The 30 per second invited by .5 eaters. Frequency is equal to 594, Urge and lastly for the second over it all we have second overtone where the length is equal to three times 1 half of them now or the one have loved the is our the stands for the and the dudes therefore our we've learned is equal to 0.33 meat fish. Now solving for the frequency of with us. I was speed divided by B. In England we have 297 m/s Divided by .33 meters frequency for the second, overtone is equal to 891 hurts. Thank you


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