Freuuency: Variations of a String

[vertciel]

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 $$\frac{f_1}{f_2} = \frac{\sqrt{T_1}}{\sqrt{T_2}}$$ and that $$\frac{f_1}{f_2} = \frac{L_2}{L_1}$$.

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

 

[malty]

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 $$f1/f2 = \frac{\sqrt{T1}}{sqrt{T2}}$$ and that $$f1/f2 = L2/L1$$.

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 $$\mu$$ is constant.

Hint:

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

P.s There is a homework thread you know;)

 

[vertciel]

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 $$f_2$$ in terms on an integer times $$f_1$$.

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.

 

[malty]

A moderator already moved it

Ok, take the formula $$f=\frac{1}{2l}\sqrt{\frac{T}{\mu}}$$

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

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

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

Note: line density $$\mu$$ may also be referred to as mass per unit length