Question

A  string that is 30 cm long and has a mass per unit length of 9.00 3 1 023 kilograms per meter is stretched to a tension  of    20.0 Newton.

(a)  find the fundamental frequency 

(b) the next 3 frequencies that  could cause a  standing wave pattern on the string

Answer 1

(a) We can find the fundamental frequency from the formula:

$$f_n=\dfrac{n}{2L}\sqrt{\dfrac{T}{\mu}},$$ $$f_1=\dfrac{1}{2\cdot0.3\ m}\sqrt{\dfrac{20\ N}{9\cdot10^{-3}\ \dfrac{kg}{m}}}=78.6\ Hz.$$

(b) We can find the next 3 frequencies that could cause a standing wave pattern on the string from the same formula:

$$f_2=\dfrac{2}{2\cdot0.3\ m}\sqrt{\dfrac{20\ N}{9\cdot10^{-3}\ \dfrac{kg}{m}}}=157.1\ Hz,$$ $$f_3=\dfrac{3}{2\cdot0.3\ m}\sqrt{\dfrac{20\ N}{9\cdot10^{-3}\ \dfrac{kg}{m}}}=235.7\ Hz,$$ $$f_4=\dfrac{4}{2\cdot0.3\ m}\sqrt{\dfrac{20\ N}{9\cdot10^{-3}\ \dfrac{kg}{m}}}=314.3\ Hz.$$