Calculating Steel String Length and Frequency for Optimal Guitar Performance

[redribbbon]

One type of steel has a density of 7000 kg/m^3 and a breaking stress of 7.30×10^8 Pa. A cylindrical guitar string is to be made out of a quantity of steel with a mass of 3.50 g.

What is the length of the longest and thinnest string that can be placed under a tension of 930 N without breaking?

What is the highest fundamental frequency that this string could have?



I know that youngs modulus is tensile stress over strain. Tensile strain is delta l / l , and for the second part, i noe that F = V/(2L), but not sure what to do



im not really sure how to go about this. I know the stress is 7.3 x 10^8, so i treated that as stress, and divided by Youngs modulus for steel (20x10^10) to get strain (.0039), which i noe is equal to delta l / l. But i am not sure how to proceed

 

[OlderDan]

Which aspect of the string is determined by the breaking stress, its length or its area or both? Once you answer that question, you can find the dimensions of the string and the maximum tension it can withstand. The frequency follows from there.

 

[redribbbon]

The stress is determined by area
Well, setting F/A equal to 7.3 x 10^8 Pa gives me a radius of 6.37 x 10^-4
which is right
So knowing area, how am i supposed to find length?
this is probably simple, but i am just not seeing it

nvm, i forgot about the density, the length would be .3922

 

[OlderDan]

The stress is determined by area
Well, setting F/A equal to 7.3 x 10^8 Pa gives me a radius of 6.37 x 10^-4
which is right
So knowing area, how am i supposed to find length?
this is probably simple, but i am just not seeing it

You know the density and the mass of the string, so you can find its volume. You know the area, so you can find the length.

 

[redribbbon]

but now, i am confused about the last part, for V do I use 344 (speed of sound) or 5941 (speed of sound in steel)?
doing V/(2*.392) either way still gives me the wrong answer

 

[redribbbon]

i think the keyword is highest fundamental freq

 

[OlderDan]

but now, i am confused about the last part, for V do I use 344 (speed of sound) or 5941 (speed of sound in steel)?
doing V/(2*.392) either way still gives me the wrong answer

You need to know the relationship between the velocity of a wave on a string and the tension and mass distribution. It is neither of the velocities you identified.

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/string.html

 

[redribbbon]

ahh, i see, thanks for the help