(a) 700 Hz
For standing waves on a string, the number of antinodes (n) corresponds to the order of the harmonic. So, three antinodes corresponds to the third harmonic. Also, the frequency of the nth-harmonic is the nth-integer multiple of the fundamental frequency, so we have:
[tex]f_3 = 3 f_1 = 420 Hz[/tex]
where [tex]f_1[/tex] is the fundamental frequency. Solving for f1, we find
[tex]f_1 = \frac{420 Hz}{3}=140 Hz[/tex]
And so now we can find the frequency of the 5th-harmonic:
[tex]f_5 = 5 f_1 = 5 (140 Hz)=700 Hz[/tex]
(b) 56.4 N
The fundamental frequency of a string is given by:
[tex]f_1 = \frac{1}{2L} \sqrt{\frac{T}{\mu}}[/tex]
where we have:
L = 60 cm = 0.60 m is the length of the string
[tex]\mu = 2.0 g/m = 0.002 kg/m[/tex] is the linear density
T = ? is the tension in the string
Solving the formula for T and using the fundamental frequency, f1=140 Hz, we find
[tex]T=\mu (2Lf_1)^2=(0.002 kg/m)(2(0.60 m)(140 Hz))^2=56.4 N[/tex]
