What is the tension of the string needed for a 440 Hz fundamental frequency?

[toothpaste666]

Homework Statement

[/B]
A violin string which is 30 cm long is tuned using a 440 Hz reference tone.

A) What is the wavelength fundamental mode of the string?

B) When the string has a tension of 34 N a violinist hears 5.7 beats per second. What are the frequencies at which the string might be vibrating?

C) The tension is increased and the violinist hears 3.8 beats per second. What is the original frequency of the string?
D) What should the tension of the string be so that the fundamental frequency is 440 Hz

Homework Equations

beatf = f2-f1
f= v/2l
lambda = 2l

The Attempt at a Solution

A) the fundamental wavelength is twice the length of the string. 2 x 30 cm = 60 cm = .6m
B) the difference between the 440 Hz reference tone and the frequency of the violin is 5.7 so the string is either vibrating at 434.3 or 445.7
C) This is where I am stuck. this is my attempt. the beat frequency is 3.8 and the reference tone is 440 so the frequency is now either 443.8 or 436.2 I completely lost on what to do from here though

 

[mfb]

(C) Does the frequency increase or decrease if you increase tension?

 

[toothpaste666]

well the velocity of a wave is v=sqrt(FT/u) where FT is the tension and u is the linear density. also v = lambda * f where lambda is the wavelength and f is the frequencey. therefore
lambda * f = sqrt(FT/u)
f = sqrt(FT/u)/lambda
so if the tension was increased the frequency would increase i think

 

[mfb]

Right. Now you can go back to the frequencies calculated in (B) and see if they are both possible with the additional knowledge from (C).

 

[toothpaste666]

ahh it can't be 445.7 in part B because it is higher than both the new frequencies and the frequency should have increased if the tension was increased. so part C is 434.3?

 

[toothpaste666]

so that would mean for part D)
at 34 N of tension:
f*lambda = sqrt(FT/u)
434.3 * .6 = sqrt(34/u)
260.58 = sqrt(34/u)
67902 = 34/u
u= 34/67902 = 5x10^-4 where u is the linear density

if we want f = 440
FT = u (lambda * f)^2 = (5x10^-4) (.6 * 440)^2 = 34.8

is this correct?

 

[mfb]

Units are missing.
Apart from that, it looks good.

 

[toothpaste666]

oh sorry. 34.8 N thanks for your help!