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the figure above shows a string with one end attached to an oscillator and the other end attached to a block there is our block the string passes over a massless pulley that turns with negligible friction there's our massless pulley with it that turns with negligible friction for such strings ABC and D are set up side by side as shown in the figure in the diagram below so this is a top view you can see oscillator is the top view of the oscillator string pulley mask system and we have four of them each oscillator is adjusted to vibrate the string at its fundamental frequency F so let's think about what it I'll keep reading but then we'll think about what fundamental frequency means the distance between each oscillator and the pulley L is the same so the length between the oscillator and the pulley is the same and the mass of each block is the same so the mass is what's providing the tension in the string however the fundamental frequency of each string is different so let's just first of all think about what the fundamental frequency is and then let's let's think about what makes them different so the fundamental frequency one way to you could think about it is it's the lowest frequency that is going to produce a standing wave in your string so it's the low it's the frequency that produces a standing wave that looks like this it's the standing wave where the string is half a wave length this is I guess there's two ways to think about it's the lowest frequency where you could produce a standing wave or it's the frequency at which you're producing the wave the standing wave with the longest wavelength so the string at the fundamental frequency is just going to go is going to vibrate between those two positions and you see here that that the that the wavelength here is twice the length of the string and if you wanted to see that a little bit clearer if I were to continue with this wave I would have to go another length of the string in order to complete in order to complete one wavelength or another way to think about it you're going up down and then you're going down back it's it's reflecting back off of this end here and as we mentioned essentially what the the mass is doing is providing the tension the force of gravity on this mass is provide the tension in this string so the oscillator is vibrating at the right frequency to produce this the lowest frequency where you can produce this standing wave so let's answer the questions now and we have four of these setups and they all have different fundamental frequencies the equation for the velocity V of a wave on a string is V is equal to the square root of the tension of the string divided by the mass of the string divided by the length of the string where F sub T is the tension of the string and M divided by L is the mass per unit length linear mass density of the string and hopefully this makes sense it makes sense that it's going to be that if if the tension increases that your velocity will increase the tension you could think of the atoms of the string of how much they're pulling on each other and so if the if there's higher tension well they're going to be able to move each other better I guess you could say accelerate each other better as the as the wave goes through the string it also makes sense that the larger the mass if you hold everything else equal then you're gonna have a slower velocity mass is under its it's a its you could view it as a measure of inertia it's how hard is it to accelerate something so if those if the string itself especially the mass per unit length if there's a lot of mass per unit length actually let me circle that because that's actually the more interesting thing if there's a lot of mass per unit length it makes sense that for a given amount of the string it's going to be harder to accelerate it back and forth as you vibrate it and so you just this part right over here would be inversely related to the velocity it's not proportional though you have this square root here but there definitely if the if the linear mass density increases then you're going to have a slower velocity and if your tension increases you're going to have a higher velocity so hopefully this makes some intuitive sense and they asked what is the difference about the fourth string shown above that would result in having different fundamental frequencies explain how you arrived at your answer and then Part B a student grass frequency as a function of the inverse of the linear mass density will the graph be linear explain how you arrived at your answer let's answer each of these so a part a what is different about the four strings because they all have different fundamental frequencies so the fundamental frequency let's just go back to what we know about waves that the velocity of the wave is equal to the frequency times the wavelength of the wave or you could say you could divide both sides by lambda you could say that the frequency of a wave is equal to the velocity over the velocity over the wavelength so if we're talking about the fundamental frequency if we're talking about the fundamental frequency fund let me just let me write frequency and then let me write fundamental realistic draw small here fundamental the fundamental frequency is going to be the velocity of our waves divided by the wavelength is going to be twice the length of our string divided by 2l and if you look at the expression that they gave us for the velocity of the wave on the string well this is going to be equal to the square root of the tension in the string divided by divided by the linear mass density divided by the linear mass density and then all of that is going to be over 2l now all of them have different fundamental frequencies but let's think about what's different over here they all have the same tension they all have the same tension how do I know that well what's causing the tension is the mass is hanging over the the pulleys so the weight of those masses so that's all going to be the same for all of them and they all have the same length they all have the same length so these are all the same so the only way that you're going to have different fundamental frequencies is if you have different masses so different masses so that has to be different different so what we can to answer their question these strings must have different mass well they would have different linear mass density x' but since they're all the same length they would also have different masses so let me write this down strings strings must have different different masses and mass densities and mass densities densities since all other variables driving driving fundamental frequency are the same fundamental frequency are the same are the same all right let's tackle Part B now a student graphs frequency as a function of the inverse of linear mass density will the graph be linear explain how you arrived at your answer so student grasp frequency let me write let me underline this they're graphing frequency as a function of linear mass density so we actually can write this down so if we want to write frequency as a function of linear mass density so we could write as a function of M over L well this is going to be equal to it's going to be equal to well we could rewrite this expression actually we could just leave it like this this is the same thing as 1 over 2 L times the square root of the tension divided by the linear mass density and or you could view this as being equal to you could say this is a square root of the tension over 2l and I'm putting all of this separate because we're doing it as a function of our linear mass density so we can hope we can assume that all of this is going to be a constant and then times the square root of 1 over the linear mass density so if you're if you're plotting if you're plotting frequency as a function of this it not going to be the graph is definitely not going to be linear you see here one I have the reciprocal of it and then I take the square root of it so so so let me write this down this F of our graph of f of 2l definitely not linear let me write it that way F of M over L graph definitely definitely not linear and we could you could point out that it's as a reciprocal and so square root involves involves square root and and inverse or I could say and reciprocal of the variable and reciprocal reciprocal of the linear mass density of M over L so definitely not linear alright let's tackle Part C the frequency of the oscillator connected to string D is change so that the string vibrates in its second harmonic on the side view of string D below mark and label the points on the string that have the greatest average vertical speed so one way to think about the fundamental frequency that's our first harmonic so if they're talking about the first harmonic of the strip and the string is what I what I showed before that's the that's the that's the lowest frequency that produces a standing wave or so frequency that produces the largest standing wave I guess you'd say the one with the largest wave length and if it was the first harmonic the parts the the part of the string that moves the most is going to be that Center but our second harmonic is the next highest frequency that produces a standing wave and that's going to be a situation it's in which the wave instead of having a wave like twice the length of the string it's gonna have a wavelength equal to the length of the string so now it's going to look like this so it's going to vibrate between this so let me draw it a little bit neater than that it's going to vibrate between between this and so it's going to vibrate between that and this and this right over here so when you see this this version of it where now that we have half the wavelength the wavelength is exactly the wavelength of of it's actually L this time the part that moves the most R is here so that's going to move the most and here I didn't draw it perfectly but one way to think about it's exactly 1/4 and 3/4 of the way exactly half way is not going to move as much it's not going to move because it's a standing wave now or it's going to be moved very imperceptibly these two places where you're going to move the most at the first I'm sorry at the second harmonic the second harmonic being the next highest frequency that produces a standing wave again