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A guitar; the lowest toned string is usually strung t0 the E note which produces sound at 82.4 Hz. The dlameter of E lengthguitar = strings is Svpicsl 0500 inches a...

Question

A guitar; the lowest toned string is usually strung t0 the E note which produces sound at 82.4 Hz. The dlameter of E lengthguitar = strings is Svpicsl 0500 inches and the scale length between the bridge and nut (the effective of the string) 5 inches Various musical acts tune their E strings down to_produce a "heavier" sound or to better fit the vocal range of the singe er; Asa quetarrng you want t0 detune the E on your guitar to D (73.4 Hz). If you were t0 maintain the same tension in

a guitar; the lowest toned string is usually strung t0 the E note which produces sound at 82.4 Hz. The dlameter of E lengthguitar = strings is Svpicsl 0500 inches and the scale length between the bridge and nut (the effective of the string) 5 inches Various musical acts tune their E strings down to_produce a "heavier" sound or to better fit the vocal range of the singe er; Asa quetarrng you want t0 detune the E on your guitar to D (73.4 Hz). If you were t0 maintain the same tension in string as wilh the E string, what diameler of string would you need t0 purchase t0 produce the desired note? Assume all strings available to you are made of the same material: Number inches Unfortunately: none of the strings in your collection have such a large diameter. In fact; the largest diameter you possess is 0.05307 inches If the tension on your existing string denoted Tbefore; by what fraction will you need t0 detune (that is lower the tension) of this string to achieve the desired D note? Number Mter before Tools 10?



Answers

Scale length is the length of the part of a guitar string that is free to vibrate. A standard value of scale length for an acoustic guitar is 25.5 in. The frequency of the fundamental standing wave on a string is determined by the string's scale length, tension, and linear mass density. The standard frequencies $f$ to which the strings of a sixstring guitar are tuned are given in the table: $$ \begin{array}{l|llllll} \text { String } & \text { E2 } & \text { A2 } & \text { D3 } & \text { G3 } & \text { B3 } & \text { E4 } \\ \hline f(\mathbf{H z}) & 82.4 & 110.0 & 146.8 & 196.0 & 246.9 & 329.6 \end{array} $$ Assume that a typical value of the tension of a guitar string is $78.0 \mathrm{~N}$ (although tension varies somewhat for different strings). (a) Calculate the linear mass density $\mu$ (in $\mathrm{g} / \mathrm{cm}$ ) for the $\mathrm{E} 2, \mathrm{G} 3$, and $\mathrm{E} 4$ strings. (b) Just before your band is going to perform, your G3 string breaks. The only replacement string you have is an E2. If your strings have the linear mass densities calculated in part (a), what must be the tension in the replacement string to bring its fundamental frequency to the G3 value of $196.0 \mathrm{~Hz}$ ?

We have a guitar string with a fundamental frequency of 330 hertz with length of 63 centimeters for part a of the problem. We need to find a new fundamental frequency whenever the string is flocked at the lower 2/3 of the street. So we're going to write the equation, which has the new fundamental frequency which will call FM. And this equation is going to be ever been times the length Evel Times 2/3. To get the newly and this is equal, she's given fundamental frequency times the length, so we'll have everything. Time. 63 centimeters times 2/3 is equal. 2 330 hertz time. 63 7 years. So we take this equation sulphur effort then and we'll get a new fundamental frequency of 495 fruits. Enough report. Be the strings plucked at about 16 of the way along its length from the rich. And we need to find the new frequency when it is plugged at this location. So you get this frequency. We can do that by taking given fundamental frequency F and most client by three. So this new this new frequency. He's gonna three times two given fundamental frequency, which was 330 hertz. And so this will give us 900 and 90 hurts.

I think this is a problem based on vibration. Off spreads strict on thermal expansion, off material all hitting. Here it is, given we strength off the guitar made off his team having got entry 7800 k G per meter cube. Let 63.5 centimeter, that is white 635 m and your diameter while 406 millimeter. That is four point Gino six in tow, 10 to the power 4 m. It is vibrating with fundamental frequency off 2 47 hurts. In the first part, we have to find the tension in this strict and second part. We have to show that change in frequency upon original frequencies. Cotto Delta F upon f your data f is changing force attention industry and see part. We have to fight the change in frequency off vibrating straight, which is tuned at frequency off 2. 47 hurts at initial temperature 18.5 degree. Such as If temperature Toby increased to 29.5 degrees searches Here it is given Mm smallness of the wire is two into 10 to the power 11 paschal and coefficient off. Linear expansion is 112 into 10 to the power minus pi for degree. Senses first back. Fundamental frequency off expressed. A string is given by burn upon to it root off upon you here, um, you his leader Mars density of the while, which is also defined as density into area course section of the district. An area perception is by the square white food so fundamental frequency we can also right. It is to be full f upon Bye rope into the square. So from here, attention industry you will get Bye. Route de square l E squared, if not squared substitute. The value density is 7800 diameter is going 4.6 10 to the power minus four. Land this 0.635 square and frequency is 2 47. So on solving it the force you will get what So force? We will get 99.4 Newton. This is the answer off part A No. In part, B frequency is given by born upon to well, route off have upon mute so data f can be return. It's one upon to well, brew talk mute one by two. Yeah. Data African Look up divided by the frequency, but oh, we will get. Did not have a point to area see part due to increase in temperature there between Thurman expansion. So data l will be l in tow. Alfa in today 30 an angle Nurses to find us Have a point, you know, eight Delta l upon and so have upon he I'll find to delta t so force people get by a hand for the static. This is actually Delta f Delta F is gatto by Alfa Delta t substitute. The value by is given two into 10 to the power 11. Alfa is given 1.2 in tow. Tend to the bar minus pipe. Increasing temperature 29.5 minus 18.5 and to Alia piety square by foot. Pipe diameter is given 2.3 Tend to the par minus four square. Yeah, on solving it, it is to be negative 3.4 because there is a thermal expansion Newton Delta F one f you would get 3.4 upon 99.4. That is 0.34 So changing frequency data Have a call. Aphis Curto. Did you ever want to F That is, while zero three full upon to hence data, if you will get minus 0.17 and 2 to 47. So it is to be minus 4.2 herds, that's all. Thanks for watching it.

For the birthday of this problem, we will derive detention for us from the general formula. Is the end mode on the frequency order for the mental frequency that were given in this case so ever Neagle's and over if you ill and then a squared off detention for us over the density linear distance. So since and because one for the given case we know that have it was one over to L over the dance T When me arise, vote both sides to the power of to and your range the equation of brackley you obtained that f equals four times Franklin C squared time slain square times leader density Also believe your density is mess over L, which is then said the more evidence determines the area times the length over length For this to cancels out than your pain road times A. Which is this case for the given string radius. There's the times five times, right, He's squared so we can say that af equals. We know the frequency four different demand full and first harmonic. So it is his four times 247 herd's times. No, we need me to square of course. Now we need to apply the body of the linked he meters and square it and then we should have this. So this is equal. Do 7800 kg permitted skew times by times on this will be the radiance is very small. So radius for this problem. He's basically how the older given diabetes which is zero point for 06 millimeters over two and then squared. And when we evaluate this your pain that this this linear audacity is one 01 kilograms meters. But I'm just that this will be times 10 to the negative. Terry Power kilograms for meters killed just meters. Sorry. So this way you can that be substituted here they can say this 0.1 constant negative third power squared and course the end dimension will be Newtons. And when we place this foot, it's into calculator. You obtain that people's 99 point or Newton's now for depart, be within two bruv relationship between the small change. Delta uh f frequency somehow relates to changing and exactly how it relates. And to prove that given four million the problem. So we'll start from the general for for the frequency and then we will express the changes. The frequency will be the change of the whole function. No dessert. This is the cost of term. So this is just want to l No, no. The square of the bath off the leaner density. This is also cost the term. So it is one over to l square off the density Andi have here Delta off square off. Scared off. Now, since we need to go to the other half, we should note that if we have some function off, some worry able, resemble ffx Delta affects the small change in his function will be the first derivative love that functional Rex times Delta X So for the force have here we can say that this is the derivative of the square. The fast over f times are tough and now it is. It is the real issue here. He's actually derivative of F off on one half over the and these derivative from the elementary rules, self motivation will be one cause f minus wrong. Half the F which is this minus is plastic pieces one over F. So this is a good one one over square. The fast. I don't have. This should be substantive back on. Then we finally turned. The delta frequency is one over to l square into the density. And then one powerful changing force square with force. The ratio from here we go that the ratio delta frequency or frequency will be one over to l squared density among Karpov does ask whole Graf and then all of this over one. There were do now and then this square with half over the nasty. What? We cancel out everything as we can cancel up this term this term this term busy going to everything except one half. And this racially intended. This is ableto one from Delta F over half. And by this the formula is proven before they're given in the problem for the bar to be for the barn. See, we have this change in the temperatures on and this change in temperatures are contributing. Teoh the linear expansion that we hear letting their expansion coefficient given as well. So we can first relates the changing tension force with the linear expansion. You too be the change off the temperature. So we need to. Why? Coefficient of expansion, Cross sectional area and the change in temperature. We can also substitutes what is since that, given the problem. Some incident to be two times despite insurrection early This is the young young's Mothers off steal this scary. We shouldn't maybe better say this week, expressing with big Why, so that it doesn't get may stop weed the function of a function or vertical displacement. So here we have two times stand for 11 power basketballs. Since next models, he has seen dimension so pressure times. And here we have this professional in every expansion which is 1.2 times and the negative fifth power one ourselves years and then we can the area cross sectional area, which is by times zero point tree 20 to 3 times 10 to the negative third power meters and then squared from then times. The changing deputy reaches from the daughter of the late changes basic 11 degrees. This that is obtained then be subtracted given bodies from temperature so 11 degrees tells use. This isn't one over Celsius. We can use cells is here. And when we put this forest into calculates, we obtained that this change forced entry means negative three points for Newton's. Now we know the relation that I have over after is one off off Delta Force over force frequency and forced change relationships so we can use this relationship and say the frequency or frequencies able to well, one half of negative one point for mutants own word 99 point for mutants. Of course, one half of this is in the product, so this will be equal to this. Will people just somewhere around 0.34 over two. Negatives on Finally, when it could, despite the car crashed through a fence. Zero Negative 0.0 70 No, from this body you you obtained that the frequency is pregnancy times negative 0.0 self team, which is equal June 247. It's times negative, 0.0 17. And when we place this one isn't a calculator and evaluate, we obtained that the change frequency is negative. Four points to hurts. This is the end of the problem. I hope you find it's helpful and they hope to see you in another lesson.

Higher. Prevent. This is the problem based on formation of standing Bates over the suppressed via here, it is given density of stained by A to B 7800 kg kilometer to the Tension Express movie seven into 10 to the power eight Newtons per meter square. The mass of the spring is foreign to 10 to the power minus three kg and maximum tension to be 900 putrid. Mm hmm. Area minimum area projection. We have to fight the forces Cartoon express into area maximum forces Niland it stresses 72 10 to the power eight. So area you will get 1.29 10 to the power minus 6 m square and eight years for two Pioneer square. So radius people get 6.4 into 10 to the power minus four metre. That is 1640 millimeter. Now in the B part Just a moment. Bottom up to viability mass. Upon Dean Street Moss is foreign to 10 to give our minus three kg dense trees 7800. So volume of the wire is 5.13 went to the bar minus seven literature health length of the pipe will be a u N Your land. You will get 5.13 10 to the power minus seven upon area That is 1.29 10 to the power minus six. So length you will get 60.39 99 m mhm be part frequency or fundamental mode of vibration. First, we have to find mosque per unit length. Marcel Pavel is given four and 2 10 to the Power Majesty length we have calculated 0.399 So 0.0 one Europe Fiji part written and fundamental frequency is put upon to a root of have upon you substitute The value of land is quite yeah. 399 Force is 900. New is 9000.1 So fundamental frequency, you will get 3. 75 hertz. There's all thanks for watching it


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