What is the fundamental frequency of a guitar string that is 1 m long, the mass per unit length is 50 g/m and the Tension in the string is 50 N.
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We have to find the fundamental frequency of a string. The formula for the fundamental frequency of a stretched string is given by
sqrt [ T/ (m/L)]/ 2*L , where T is the tension in the string, m is the mass per length and L is the length of the string.
Using the values given to us:
L = 1m
m= 50g /m = .05 Kg/m
T = 50 N
Therefore sqrt [ T/ (m/L)]/ 2*L
=> sqrt [ 50/ (.05/1)]/ 2*1
=> sqrt [ 50 / .05] / 2
=> sqrt 1000 /2
=> 31.62 / 2
=> 15.81 Hz
Therefore the fundamental frequency is 15.81 Hz
An alternate method is to recognize that for a string fixed at both ends the wavelength is 2 times the length of the string. In that case you have a wavelength of 2 m.
The relationship between the string tension F(T), the m/L ratio and the velocity of the wave is: F(T) = m/L * v^2
Solving for the velocity gives 31.62 m/s.
Since v = frequency * wavelength; 31.62 m/s divided by 2 m = 15.81 Hz as the fundamental frequency.
As you can see, in physics there is often more than one way to end up with the correct answer.
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