Change guitar string tension given two frequencies

In summary, the guitar string is supposed to vibrate at 217 Hz, but it is actually vibrating at 222 Hz by a percentage of 4.66%. To get the frequency to the correct value, the tension in the string should be increased by 1.0466%.
  • #1
Sethius
7
0

Homework Statement


A particular guitar string is supposed to vibrate at 217 Hz, but it is measured to actually vibrate at 222 Hz. By what percentage should the tension in the string be changed to get the frequency to the correct value? Do not enter units.



Homework Equations



f=[itex]\frac{1}{2L}[/itex][itex]\sqrt{\frac{T}{\mu}}[/itex]
L=string length
T=Tension
[itex]\mu[/itex]=Linear Density

The Attempt at a Solution


I know that frequency is proportional to the square root of tension given that all other factors remain constant. I solved two frequency equations for [itex]\frac{1}{2L}[/itex] and set them equal to each other with my T[itex]_{2}[/itex]being multiplied by a variable "k" being the conversion factor. my equation looked like this:

[itex]\frac{f_{1}}{\sqrt{T_{1}}}[/itex]=[itex]\frac{f_{2}}{\sqrt{kT_{2}}}[/itex]

I then canceled the [itex]\sqrt{T}[/itex]'s and ended up with [itex]\frac{f_{1}}{f_{2}}[/itex]=[itex]\frac{1}{\sqrt{k}}[/itex]. Solving for k I get 1.0466 which would equal a 4.66% increase in tension however this answer isn't correct. Thank you for any help
 
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  • #2
Your answer looks good to me... I've asked the other homework helpers to see if they can figure out why your answer isn't correct.
 
  • #3
I got f1/f2 = 0.9775 which gives squrt(k) = 1.0230 which gives k = 1.0114
Is this closer to the answer?
 
  • #4
Sethius said:
Solving for k I get 1.0466 which would equal a 4.66% increase in tension however this answer isn't correct.
So you are saying that you must increase the tension to reduce the frequency? Check that. (You just have things reversed a bit.)
 
  • #5
BruceW said:
Your answer looks good to me... I've asked the other homework helpers to see if they can figure out why your answer isn't correct.

Ok thank you

technician said:
I got f1/f2 = 0.9775 which gives squrt(k) = 1.0230 which gives k = 1.0114
Is this closer to the answer?

I don't actually have an answer to look at, I have 10 attempts on a LonCapa website to get the right answer. As for your answer, if [itex]\sqrt{k}[/itex]=1.0230 shouldn't you square the 1.0230 as opposed to square rooting it like you've done?
 
  • #6
Sorry! You are correct Seth
 
  • #7
Doc Al said:
So you are saying that you must increase the tension to reduce the frequency? Check that. (You just have things reversed a bit.)
I'm having a hard time wrapping my brain around this. I would assume that the percent change would remain the same regardless of increase or decrease in tension. Should I add a negative sign?
 
  • #8
Sethius said:
I would assume that the percent change would remain the same regardless of increase or decrease in tension.
Rethink that. To go from 100 to 110 is a ten percent increase. But to go from 110 to 100 is what percent decrease?
 
  • #9
Ok I understand now. So if I'm going from 222Hz to 217Hz I would get .977 as my ratio, then squaring that I would get .955. It's here I'm not entirely sure what to do. Do I subtract that from 1? Or divide one by it?
 
  • #10
Sethius said:
Ok I understand now. So if I'm going from 222Hz to 217Hz I would get .977 as my ratio, then squaring that I would get .955. It's here I'm not entirely sure what to do. Do I subtract that from 1? Or divide one by it?
Compared to 1, that's a percent drop of what?
 
  • #11
Doc Al said:
Compared to 1, that's a percent drop of what?

It would be 4.5%?
 
  • #12
Sethius said:
It would be 4.5%?
Sounds good. You might what to take that to three sig figs, in case your online system is picky.
 
  • #13
Thank you very much for the help!
 

Related to Change guitar string tension given two frequencies

What is guitar string tension?

Guitar string tension refers to the amount of force or pull that is exerted on a string in order to produce a certain pitch or frequency. This tension is created by tightening or loosening the string using the tuning pegs on the guitar.

How do you change guitar string tension?

To change guitar string tension, you can use the tuning pegs on the headstock of the guitar. Turning the peg clockwise will tighten the string and increase the tension, while turning it counterclockwise will loosen the string and decrease the tension. It's important to make small adjustments and tune the string in between to avoid over-tightening or loosening.

How do you know the tension of a guitar string?

The tension of a guitar string can be measured using a tension gauge or calculated using the frequency and length of the string. The higher the frequency and shorter the length, the higher the tension. Some string manufacturers also provide tension ratings for their strings.

What is the relationship between guitar string tension and frequency?

The relationship between guitar string tension and frequency is indirect. As the tension on a string increases, the frequency or pitch of the string will also increase. Similarly, decreasing the tension will result in a decrease in frequency. This is because the tension affects the speed at which the string vibrates, which in turn affects the frequency produced.

How do you adjust string tension to change the frequency of a guitar string?

To change the frequency of a guitar string, you can adjust the tension by tightening or loosening the string with the tuning pegs. If you want a higher frequency, tighten the string, and if you want a lower frequency, loosen the string. It's important to tune the string after each adjustment to ensure the desired frequency is achieved.

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