Question

A piano tuner stretches a steel piano wire with a tension of 765 N . The steel wire has a length of 0.600 m and a mass of 4.50g .
What is the frequency (f_1) of the string's fundamental mode of vibration?
What is the number n of the highest harmonic that could be heard by a person who is capable of hearing frequencies up to f = 16 kHz?

Answer 1

Answer:

266.14532 Hz

60

Explanation:

m = Mass of string = 4.5 g

T = Tension = 765 N

L = Length of string = 0.6 m

f = Frequency = 16 kHz

Linear density is given by

$\mu=\dfrac{m}{L}\\\Rightarrow \mu=\dfrac{4.5\times 10^{-3}}{0.6}\\\Rightarrow \mu=0.0075\ kg/m$

Fundamental frequency is given by

$f'=\dfrac{1}{2L}\sqrt{\dfrac{T}{\mu}}\\\Rightarrow f'=\dfrac{1}{2\times 0.6}\sqrt{\dfrac{765}{0.0075}}\\\Rightarrow f'=266.14532\ Hz$

The fundamental frequency is 266.14532 Hz

The number of loops is given by

$n=\dfrac{f}{f'}\\\Rightarrow n=\dfrac{16000}{266.14532}\\\Rightarrow n=60.11753$

The number of loops is 60.