A. 409 Hz
The fundamental frequency of a string is given by:
[tex]f_1=\frac{1}{2L}\sqrt{\frac{T}{m/L}}[/tex]
where
L is the length of the wire
T is the tension in the wire
m is the mass of the wire
For the piano wire in this problem,
L = 0.400 m
T = 1070 N
m = 4.00 g = 0.004 kg
So the fundamental frequency is
[tex]f_1=\frac{1}{2(0.400)}\sqrt{\frac{1070}{(0.004)/(0.400)}}=409 Hz[/tex]
B. 24
For this part, we need to analyze the different harmonics of the piano wire. The nth-harmonic of a string is given by
[tex]f_n = nf_1[/tex]
where [tex]f_1[/tex] is the fundamental frequency.
Here in this case
[tex]f_1 = 409 Hz[/tex]
A person is capable to hear frequencies up to
[tex]f = 1.00 \cdot 10^4 Hz[/tex]
So the highest harmonics that can be heard by a human can be found as follows:
[tex]f=nf_1\\n= \frac{f}{f_1}=\frac{1.00\cdot 10^4}{409}=24.5 \sim 24[/tex]
