# Change guitar string tension given two frequencies

1. Jan 29, 2012

### Sethius

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

f=$\frac{1}{2L}$$\sqrt{\frac{T}{\mu}}$
L=string length
T=Tension
$\mu$=Linear Density

3. 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 $\frac{1}{2L}$ and set them equal to each other with my T$_{2}$being multiplied by a variable "k" being the conversion factor. my equation looked like this:

$\frac{f_{1}}{\sqrt{T_{1}}}$=$\frac{f_{2}}{\sqrt{kT_{2}}}$

I then cancelled the $\sqrt{T}$'s and ended up with $\frac{f_{1}}{f_{2}}$=$\frac{1}{\sqrt{k}}$. 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

2. Jan 29, 2012

### BruceW

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. Jan 29, 2012

### technician

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

4. Jan 29, 2012

### Staff: Mentor

So you are saying that you must increase the tension to reduce the frequency? Check that. (You just have things reversed a bit.)

5. Jan 29, 2012

### Sethius

Ok thank you

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 $\sqrt{k}$=1.0230 shouldn't you square the 1.0230 as opposed to square rooting it like you've done?

6. Jan 29, 2012

### technician

Sorry!! You are correct Seth

7. Jan 29, 2012

### Sethius

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. Jan 29, 2012

### Staff: Mentor

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. Jan 29, 2012

### Sethius

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. Jan 29, 2012

### Staff: Mentor

Compared to 1, that's a percent drop of what?

11. Jan 29, 2012

### Sethius

It would be 4.5%?

12. Jan 29, 2012

### Staff: Mentor

Sounds good. You might what to take that to three sig figs, in case your online system is picky.

13. Jan 29, 2012

### Sethius

Thank you very much for the help!

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