# Tension and frequency of a vibrating violin string

• Dong Min
In summary, the problem statement is investigating how tension affects frequency in a violin string, but the equation used is not entirely accurate. The goal is to use a different equation that is more accurate to better understand the relationship between tension and frequency. The researcher is also interested in exploring the topic more in depth, including investigating any limitations of the mersenne's law.
Dong Min
1. The problem statement, all variables, and given/known data

So I'm doing an IB extended essay on the relationship between frequency and tension of a violin string. As you apply more tension to the string (using weights and pulley), the frequency will be higher, as shown below. There's not too much problem with collecting data, but I'm worried about the simplicity of the topic.

## Homework Equations

v=wavespeed
T= tension force
u= linear density
k= spring constant
x=change length of string
f=frequency
L=lenght of string

v=sqrt (T/u)
f=sqrt (T/u)/2L
T=kx

## The Attempt at a Solution

[/B]
What I want to do to better the depth of the investigation is to use the fact that linear density changes with tension, though minimally, so the equation below may not be entirely accurate.

So linear density would increase:
u=m/(L+x)= m/(L+T/k)

which means the frequency would change:
f= sqrt [T*(L+T/k)/m] /2L
meaning tension will increase slightly more than what we predict with the initial linear density.

Is this correct? and is there a way to further explore my topic in more depth? like any more limitations of the mersenne's law and such.. Thank you!

Ague
T=kx is a a starting point. What is "k" for a violin string? I suggest that you do some background research on Young's modulus which will allow you to calculate "k". Then you can do a theoretical analysis of what percentage of the frequency shift would be due to increasing the tension and what percentage will be due to the elongation on string. Based on that theoretical information, you can draw conclusions whether it is experimentally feasible to separate the two effects and if so, how.

Dong Min
Dong Min said:
1. The problem statement, all variables, and given/known data

So I'm doing an IB extended essay on the relationship between frequency and tension of a violin string. As you apply more tension to the string (using weights and pulley), the frequency will be higher, as shown below. There's not too much problem with collecting data, but I'm worried about the simplicity of the topic.

## Homework Equations

v=wavespeed
T= tension force
u= linear density
k= spring constant
x=change length of string
f=frequency
L=lenght of string

v=sqrt (T/u)
f=sqrt (T/u)/2L
T=kx

## The Attempt at a Solution

[/B]
What I want to do to better the depth of the investigation is to use the fact that linear density changes with tension, though minimally, so the equation below may not be entirely accurate.

So linear density would increase:
u=m/(L+x)= m/(L+T/k)

which means the frequency would change:
f= sqrt [T*(L+T/k)/m] /2L
meaning tension will increase slightly more than what we predict with the initial linear density.

Is this correct? and is there a way to further explore my topic in more depth? like any more limitations of the mersenne's law and such.. Thank you!
Your math looks right. I notice that the pitch of my ukulele strings seems to change with temperature. Maybe temperature affects the spring constant k.

tnich said:
I notice that the pitch of my ukulele strings seems to change with temperature. Maybe temperature affects the spring constant k.
It might. As I understand it, the level of humidity and temperature affect the speed of sound which in turn affects the frequency at constant tension. That's why in a concert hall the orchestra members tune their instruments in the hall itself rather than backstage where temperature and humidity might not be the same.

kuruman said:
It might. As I understand it, the level of humidity and temperature affect the speed of sound which in turn affects the frequency at constant tension. That's why in a concert hall the orchestra members tune their instruments in the hall itself rather than backstage where temperature and humidity might not be the same.
I expect that the speed of sound in air would affect brass and woodwinds since they use air columns as resonators. I am not sure that it would affect a stringed instrument where the primary resonator is a string. Stringed instruments do have secondary resonators filled with air, but they do not determine pitch.

tnich said:
I expect that the speed of sound in air would affect brass and woodwinds since they use air columns as resonators. I am not sure that it would affect a stringed instrument where the primary resonator is a string. Stringed instruments do have secondary resonators filled with air, but they do not determine pitch.
But that is something that could be resolved experimentally.

kuruman said:
T=kx is a a starting point. What is "k" for a violin string? I suggest that you do some background research on Young's modulus which will allow you to calculate "k". Then you can do a theoretical analysis of what percentage of the frequency shift would be due to increasing the tension and what percentage will be due to the elongation on string. Based on that theoretical information, you can draw conclusions whether it is experimentally feasible to separate the two effects and if so, how.

Alright, I decided to use a rubber band instead of a violin string, as it is too hard to prove experimentally the effect of the spring constant on the frequency, the effect is about 1-2Hz.

So I will do two experiments 1. getting the spring constant k of a rubber band and 2. getting the frequency of the vibrating rubber band changing the tension (the weight of the load attached to the rubber band). I haven't done the experiment yet, but I expect to analyze the frequency and tension data with the equation I derived:

[f is frequency, T is tension, L is the length of the string, k is the spring constant, and m is the mass of the rubber band]

f= sqrt [T*(L+T/k)/m] /2L
If we square both sides:
f^2= T(L+T/k)/(4mL^2)
f^2= (1/(4mL^2k))T^2 +(1/(4mL^2))T

In a nutshell, if I graph f^2 against T, I expect to get a quadratic relationship, with a line of best fit a quadratic equation. I want to suggest the validity of the equation by obtaining the percent errors of each of the constants next to T^2 and T ( I will determine the m, L, k values experimentally).

Is this a valid scientific argument? I'm used to doing labs deriving simpler relationships that are easy to modify to be linear, so this is kind of new for me. Thank you!

You are assuming that the propagation velocity of transverse waves in a rubber band obeys the same equation ##v=\sqrt{TL/m}## as in a steel wire. The tension in a rubber band is ill-defined because rubber bands exhibit "hysteresis" effects. This basically means that a tension in a rubber band depends on its past history, i.e. what was done to it. Do a simple experiment: Load a rubber with equal weight increments and measure how much it stretches after each addition. Then go backwards and take away the weights in decrements remeasuring the stretch distance. You will find that not only you don't get the same stretch distance for the same load, but also that what distance you get for a given load depends on how long you wait before you measure the length. The longer you wait the more the rubber band will stretch under the load. Rubber bands are notoriously difficult to quantify with any precision. I would stay away from them.
Dong Min said:
Alright, I decided to use a rubber band instead of a violin string, as it is too hard to prove experimentally the effect of the spring constant on the frequency, the effect is about 1-2Hz.
If you have frequency shifts of about 1-2 Hz from the reference frequency ##f_0##, you might be able to measure the shift by using a pure frequency ##f_0## from a loudspeaker hooked to a function generator (loudspeaker of your laptop?) and record beats against the vibrating string. The beat period would be of order 1 second and should be easily detectable. Or you can do a "null" measurement and use the frequency generator to match the frequency of the string at which point the beats should disappear. Then you can compare the actual frequency with the predicted frequency. A lot depends on what kind of equipment you have available for your use.

kuruman said:
You are assuming that the propagation velocity of transverse waves in a rubber band obeys the same equation ##v=\sqrt{TL/m}## as in a steel wire. The tension in a rubber band is ill-defined because rubber bands exhibit "hysteresis" effects. This basically means that a tension in a rubber band depends on its past history, i.e. what was done to it. Do a simple experiment: Load a rubber with equal weight increments and measure how much it stretches after each addition. Then go backwards and take away the weights in decrements remeasuring the stretch distance. You will find that not only you don't get the same stretch distance for the same load, but also that what distance you get for a given load depends on how long you wait before you measure the length. The longer you wait the more the rubber band will stretch under the load. Rubber bands are notoriously difficult to quantify with any precision. I would stay away from them.

If you have frequency shifts of about 1-2 Hz from the reference frequency ##f_0##, you might be able to measure the shift by using a pure frequency ##f_0## from a loudspeaker hooked to a function generator (loudspeaker of your laptop?) and record beats against the vibrating string. The beat period would be of order 1 second and should be easily detectable. Or you can do a "null" measurement and use the frequency generator to match the frequency of the string at which point the beats should disappear. Then you can compare the actual frequency with the predicted frequency. A lot depends on what kind of equipment you have available for your use.
Do you know any string-like material that obeys the Hooke's law but also stretches noticeably with tension?

Dong Min said:
Do you know any string-like material that obeys the Hooke's law but also stretches noticeably with tension?
I do not. You might try monofilament fishing line. You will need to do some experimentation to see if it exhibits the kind of hysteresis effects I already mentioned.

Dong Min
kuruman said:
I do not. You might try monofilament fishing line. You will need to do some experimentation to see if it exhibits the kind of hysteresis effects I already mentioned.

Thank you so much for your help! But I don't have much time and resources, it's a high school essay after all, and want to continue with a rubber band, while acknowledging the limitations.

I did some research and found out a method to make the rubber band behave closely to a spring. (https://www.wired.com/2012/08/do-rubber-bands-act-like-springs/) Since the hooke's law is not a main part of my essay (obtaining the frequency is) I wish to mention this as a limitation, but still use the equation above assuming the band followed the Hooke's law. What do you think of this?

Sure, go for it. If you get a chance, post you graph here. I am curious to see what it looks like. Note my comments below.
Dong Min said:
Since the hooke's law is not a main part of my essay (obtaining the frequency is) ...
But it is important if you use Hooke's law to estimate the extra length as T/k and use this to see how the frequency changes.
Dong Min said:
So linear density would increase:
u=m/(L+x)= m/(L+T/k)
You mean "decrease". The mass is the same but the string gets longer and the same grams are distributed over more meters.
I have one more question, what is your setup? Specifically, is the wavelength fixed or does it change with the length of the string? Your equations imply that the wavelength is fixed. If your setup looks like something like this. the wavelength will be fixed but the total length of the string will increase as the hanging weight increases.

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Dong Min said:
1. The problem statement, all variables, and given/known data

So I'm doing an IB extended essay on the relationship between frequency and tension of a violin string. As you apply more tension to the string (using weights and pulley), the frequency will be higher, as shown below. There's not too much problem with collecting data, but I'm worried about the simplicity of the topic.

Hi! I'm an IB student too, and I was thinking of doing something similar to this for my Physics Internal Assessment I thought of doing it with a guitar string or some nylon string, but I'll see. I think I will see how the difference in tension and in length (or only one of them) affects the pitch. Maybe I'll work on how temperature affects it too.

What do you think? I would love to see your results, and how did everything went!
Were you able to get it with the rubber band?
Did hook's law affected that much?
Were you able to ingnore the fact that the linear density changed with the tension?

I hope I can get some answers. Thanks!

Ague said:
Hi! I'm an IB student too, and I was thinking of doing something similar to this for my Physics Internal Assessment I thought of doing it with a guitar string or some nylon string, but I'll see. I think I will see how the difference in tension and in length (or only one of them) affects the pitch. Maybe I'll work on how temperature affects it too.

What do you think? I would love to see your results, and how did everything went!
Were you able to get it with the rubber band?
Did hook's law affected that much?
Were you able to ingnore the fact that the linear density changed with the tension?

I hope I can get some answers. Thanks!

Wow, this brings back memories. I ended up using a rubber band and my EE turned out to pretty good, got an A. My final graph was F^2/T against T, and I ended up with an interesting graph that was linear at a limited interval.

If you are thinking of using a guitar string, then you can reasonably ignore hooke's law/the change in linear density (although you can/and should mention it during evaluation). You probably want to vary only tension because varying length would be very hard (as a string would barely stretch). Also having two independent variables would make your discussion a bit messy. I'm not sure about temperature, but you can always give it a try and see how things turn out. Good luck!

Wow thanks! You are so fast! Yes, I'll probably ignore hooke's law. I was thinking of varying the length by putting something in some point of the string so it stops vibrating there, and shortens/elongates the string without stretching it.

I'll probably only have one variable, but I don't know which one I'll use yet.

Didn't the rubber band get used to the stretching (i don't know how to say it i english)? I mean, when it gets so stretched that can't go back t it's original form.

Ague said:
Wow thanks! You are so fast! Yes, I'll probably ignore hooke's law. I was thinking of varying the length by putting something in some point of the string so it stops vibrating there, and shortens/elongates the string without stretching it.

I'll probably only have one variable, but I don't know which one I'll use yet.

Didn't the rubber band get used to the stretching (i don't know how to say it i english)? I mean, when it gets so stretched that can't go back t it's original form.

For me, the rubber band didn't stretch permanently. You just don't want too much tension on it.

And yes, putting something will create a node and change the wavelength of the standing wave.

Hi!
I'm thinking of doing a similar topic for my Physics IA and I'm worried about its simplicity as I am in HL Physics. Do you have any ideas/tips on how to conduct a more in-depth analysis to the extent which IB is looking for?
Thanks!

winnie13579 said:
Hi!
I'm thinking of doing a similar topic for my Physics IA and I'm worried about its simplicity as I am in HL Physics. Do you have any ideas/tips on how to conduct a more in-depth analysis to the extent which IB is looking for?
Thanks!
Sorry, I have no experience with what Physics IA, IB and HL Physics are all about as far as their levels of difficulty are concerned. Maybe someone else does. If you have a specific physics question to ask, I might be able to answer it.

Dong Min said:
For me, the rubber band didn't stretch permanently. You just don't want too much tension on it.

And yes, putting something will create a node and change the wavelength of the standing wave.

Hi!
I'm thinking of doing a similar topic for my Physics IA and I'm worried about its simplicity as I am in HL Physics. Do you have any ideas/tips on how to conduct a more in-depth analysis to the extent which IB is looking for?
Thanks!

kuruman said:
Sorry, I have no experience with what Physics IA, IB and HL Physics are all about as far as their levels of difficulty are concerned. Maybe someone else does. If you have a specific physics question to ask, I might be able to answer it.
Thanks for the reply! That's ok, just some general ideas on how to add more variety to this simple experiment would be greatly appreciated. Thanks!

kuruman said:
Sorry, I have no experience with what Physics IA, IB and HL Physics are all about as far as their levels of difficulty are concerned. Maybe someone else does. If you have a specific physics question to ask, I might be able to answer it.
In terms of gathering data for tension and frequency values, I was originally thinking of using a sonometer however the one I have is very short(less than half a meter). Will this affect my data collection and do you have any alternative set up ideas? Thanks.

The short sonometer will not affect your data collection; it will affect the range of data that you can collect, specifically the number of resonant harmonics for a given wire and weight load. Using the right combination of weight load and knife edge separation, you should be able to adjust the fundamental frequency of your stretched wire to the frequency of your tuning fork(s). Derive an expression relating the mass density ##\rho## of the wire to the fundamental frequency, weight load, wire diameter and knife edge separation. These you can measure correspondingly with your sonometer, a micrometer and a meter stick. At this point your sonometer has been converted to a "mass densitometer" that allows you to measure the mass densities of wires of various thicknesses and compositions. Compare your values to the known values and see how good this method is. Don't forget to do error propagation analysis if you know how.

kuruman said:
The short sonometer will not affect your data collection; it will affect the range of data that you can collect, specifically the number of resonant harmonics for a given wire and weight load. Using the right combination of weight load and knife edge separation, you should be able to adjust the fundamental frequency of your stretched wire to the frequency of your tuning fork(s). Derive an expression relating the mass density ##\rho## of the wire to the fundamental frequency, weight load, wire diameter and knife edge separation. These you can measure correspondingly with your sonometer, a micrometer and a meter stick. At this point your sonometer has been converted to a "mass densitometer" that allows you to measure the mass densities of wires of various thicknesses and compositions. Compare your values to the known values and see how good this method is. Don't forget to do error propagation analysis if you know how.
I was thinking of setting up an apparatus like the one I attached below without the weights since I don't have those resources. The apparatus should allow for a change in the tension of the string thereby producing a different fundamental frequency. Also, as the focus of my paper is varying frequency with tension, would varying the length, tension and diameter be unrelated to the goal of the paper? Thanks.

winnie13579 said:
Thanks for the reply! That's ok, just some general ideas on how to add more variety to this simple experiment would be greatly appreciated. Thanks!
You could maybe find the resonance very exactly if you measure the current into the vibration generator as the frequency is varied. The vibrating string stores energy and creates a dip in current at the resonant frequency. It looks like an electrical resonator, in other words an LC circuit. We see this with loudspeakers very easily. You need to have a series resistor of a few Ohms in the grounded leg, across which you can measure a voltage, maybe with an oscilloscope.
We are doing the string experiment in class at the moment so I will try to do it tomorrow and find out.

tech99 said:
You could maybe find the resonance very exactly if you measure the current into the vibration generator as the frequency is varied. The vibrating string stores energy and creates a dip in current at the resonant frequency. It looks like an electrical resonator, in other words an LC circuit. We see this with loudspeakers very easily. You need to have a series resistor of a few Ohms in the grounded leg, across which you can measure a voltage, maybe with an oscilloscope.
We are doing the string experiment in class at the moment so I will try to do it tomorrow and find out.
Sorry I forgot to specify that I won't be using a vibration generator but rather an apparatus which fixes the violin string to one end and a force sensor on the other. With this set up I can only find tension and frequency values using Vernier and a Phone app however do you have any advice on how I could measure the change in length of the string(as it changes very minimally). Thanks.

winnie13579 said:
Sorry I forgot to specify that I won't be using a vibration generator but rather an apparatus which fixes the violin string to one end and a force sensor on the other. With this set up I can only find tension and frequency values using Vernier and a Phone app however do you have any advice on how I could measure the change in length of the string(as it changes very minimally). Thanks.
I am not sure of your exact equipment layout. Usually the string is held between supports at fixed length, so length does not alter with tension.
We have just done the experiment in school and the pupil measured frequency against length (tension fixed) and tension ( length fixed). Next week we will look at predicting velocity from mass per unit length and tension.

The change in length of the string upon stretching is minimal. You change the length of the string by using a wedge to push up on the string from underneath at some point between the fixed end. Then you end up with two strings under the same tension of lengths ##L_1## and ##L_2## such that their sum is the length of the wire. If you can measure the fundamental of each segment then you can verify that the product ##f_1L_1=f_2L_2=\text{const.}## Next you can move the wedge to a new position and repeat. This result is independent of the tension. If you can measure the tension with a force gauge you can change the tension and repeat. As you may have surmised, the product ##Lf## for the fundamental is half the propagation speed. If you make a plot ##Lf## vs. ##T^2##, you should a straight line the slope of which is related to the linear density (how?).

## 1. What is tension and frequency in relation to a vibrating violin string?

Tension refers to the amount of force applied to the string, while frequency refers to the number of vibrations or cycles the string completes per second.

## 2. How does the tension of a violin string affect its frequency?

The higher the tension of a string, the higher the frequency it produces. This is because the increased tension causes the string to vibrate at a faster rate, resulting in a higher pitch.

## 3. What factors can affect the tension and frequency of a vibrating violin string?

The material and thickness of the string, as well as the length and thickness of the string's vibrating portion, can all affect its tension and frequency. Additionally, the amount of force applied to the string and the tension of the instrument's body can also impact these characteristics.

## 4. How can the tension and frequency of a violin string be measured?

The tension of a string can be measured using a tension gauge, which is a device that measures the amount of force applied to the string. The frequency can be measured using a frequency meter or by comparing the pitch of the string to a known reference pitch.

## 5. Are there any practical applications of understanding the tension and frequency of a vibrating violin string?

Yes, understanding the relationship between tension and frequency is crucial for musicians to produce desired pitches and tones on their instrument. Additionally, this knowledge is important for instrument makers to create strings with specific characteristics for different playing styles and preferences.

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