Freuuency: Variations of a String

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vertciel
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Hello everyone,

I am struggling with a problem and I would appreciate any help or guidance. Thank you very much!

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1. A guitar string has a frequency of 2048 Hz. If both the tension and length are doubled, what is the new frequency of the string?

I know that the [tex]\frac{f_1}{f_2} = \frac{\sqrt{T_1}}{\sqrt{T_2}}[/tex] and that [tex]\frac{f_1}{f_2} = \frac{L_2}{L_1}[/tex].

My problem is that here, I do not know how I can find the new frequency when both factors are done simultaneously.
 
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vertciel said:
Hello everyone,

I am struggling with a problem and I would appreciate any help or guidance. Thank you very much!

---

1. A guitar string has a frequency of 2048 Hz. If both the tension and length are doubled, what is the new frequency of the string?

I know that the [tex]f1/f2 = \frac{\sqrt{T1}}{sqrt{T2}}[/tex] and that [tex]f1/f2 = L2/L1[/tex].

My problem is that here, I do not know how I can find the new frequency when both factors are done simultaneously.

Don't overcomplicate it! Just remember that the line density [tex]\mu[/tex] is constant.

Hint:

[tex]f_1=\frac{1}{2l}\sqrt{\frac{T}{\mu}}[/tex] find an expression for [tex]f_2[/tex] in terms on an integer times [tex]f_1[/tex]

P.s There is a homework thread you know;)
 
Thanks for your reply, malty.

Could you please expound on your answer a bit further? I am still lost. In particular, I am confused by density and how to "find an expression for [tex]f_2[/tex] in terms on an integer times [tex]f_1[/tex].

With regards to the homework thread, I apologise for posting here as I was not aware of it. If a moderator wishes to move this thread, please do so.
 
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A moderator already moved it:smile:

Ok, take the formula [tex]f=\frac{1}{2l}\sqrt{\frac{T}{\mu}}[/tex]

We know [tex]\mu[/tex] is constant, so that will be same the for both [tex]f_1 \hspace{4} and \hspace{4} f_2[/tex] so we can ignore it completely.

Write an equation for [tex]f_1[/tex] in terms of T and L and [tex]\mu[/tex]
Now look at your tension T and compare them for both [tex]f_1 \hspace{4} and \hspace{4} f_2[/tex] Next do the same for your length. L. You should be able then to take [tex]f_2[/tex] and write an equation for it in terms of [tex]f_1[/tex] e.g Tension is equal to 2T for [tex]f_2[/tex] and length equals 2L sub these into your equation and see what you get . . .

Rearrange it to get [tex]f_2[/tex] in the form of [tex]Cf_1[/tex] where C is some constant.
=> [tex]f_2=C*\frac{1}{2l}\sqrt{\frac{T}{\mu}}[/tex]

Note: line density [tex]\mu[/tex] may also be referred to as mass per unit length